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\title{Enlargements and similarity }

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\maketitle
This chapter introduces a transformation called an enlargement. This allows us to develop the idea of similar figures, which is a more general idea than congruence.

For example, the two right-angled triangles shown here are similar because they have exactly the same shape, but they are not congruent because they have different side lengths.

Similarity and proportion are closely related, as we shall see. This idea has been used in art and architecture for centuries. It is now used extensively in photography and in the zoom function of a computer application.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-01}
\end{center}

\section*{Enlargements}
You have already met three types of transformations: translations, rotations and reflections.

This section introduces a fourth type of transformation called an enlargement, in which the lengths of the sides are increased or decreased by the same factor. In ICE-EM Mathematics Year 8, we looked at these ideas under the topic heading of scale drawings. Now we are going to examine these ideas a little more closely.

The figure to the right below is a right-angled triangle \(A B C\), whose sides have length \(1.5 \mathrm{~cm}, 2 \mathrm{~cm}\) and \(2.5 \mathrm{~cm}\).

To define an enlargement of this figure, two things need to be specified.

First, we have to specify a centre of enlargement - in our example, it is the point \(O\) to the left of the triangle. The centre of enlargement stays fixed in the one place while the enlargement expands or shrinks everything else around it.

Secondly, we have to specify an enlargement ratio or enlargement factor - we will use an enlargement ratio of 2 here. This means that every point will double its distance

\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-02}
from \(O\). As a consequence, the side lengths of the triangle will double, as we saw in scale drawings.

The second diagram shows how this is done.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-02(1)}
\end{center}

\begin{itemize}
  \item The interval \(O A\) is joined, then produced to \(A^{\prime}\) so that \(O A^{\prime}=2 \times O A\).

  \item The interval \(O B\) is joined, then produced to \(B^{\prime}\) so that \(O B^{\prime}=2 \times O B\).

  \item The interval \(O C\) is joined, then produced to \(C^{\prime}\) so that \(O C^{\prime}=2 \times O C\).

\end{itemize}

The triangle \(A^{\prime} B^{\prime} C^{\prime}\) is then joined up, and is called the image of triangle \(A B C\) with enlargement factor 2 and centre of enlargement \(O\).

The image triangle \(A^{\prime} B^{\prime} C^{\prime}\) is related to the original triangle in three important ways.

First, each side of the image triangle \(A^{\prime} B^{\prime} C^{\prime}\) is twice the length of the matching sides of the original triangle. We will usually write this using ratios:

\[
\frac{A^{\prime} B^{\prime}}{A B}=2 \quad \frac{B^{\prime} C^{\prime}}{B C}=2 \quad \frac{A^{\prime} C^{\prime}}{A C}=2
\]

Secondly, each angle of the image is equal to the matching angle of the original:

\[
\angle B^{\prime} A^{\prime} C^{\prime}=\angle B A C \quad \angle A^{\prime} C^{\prime} B^{\prime}=\angle A C B \quad \angle C^{\prime} B^{\prime} A^{\prime}=\angle C B A
\]

Thus the image \(A^{\prime} B^{\prime} C^{\prime}\) is a scale drawing of the original figure \(A B C\), with ratio 2 .

Thirdly, since matching angles are equal we can see that \(A^{\prime} B^{\prime}\left\|A B, B^{\prime} C^{\prime}\right\| B C\) and \(C^{\prime} A^{\prime} \| C A\).

The same kinds of results hold if we perform an enlargement by any positive factor.

\section*{Enlargement transformations}
\begin{itemize}
  \item Each enlargement has a centre of enlargement \(O\) and an enlargement factor \(k>0\).
  \item The enlargement moves each point \(P\) to a new point \(P^{\prime}\) on the ray with vertex \(O\) passing through \(P\).
  \item The distance of \(P^{\prime}\) from \(O\) is \(k\) times the distance of \(P\) from \(O\). That is, \(O P^{\prime}=k \times O P\).
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-03(1)}
\end{center}

When a figure is enlarged by a factor \(k\) :

\begin{itemize}
  \item The image of each interval has \(k\) times the length of the original interval.
  \item The image of an angle has the same size as the original angle.
  \item The image of an interval is parallel to the original interval.
\end{itemize}

The first three dot points follow from the definition of an enlargement. If \(k>1\), then the image is larger than the original figure. If \(k<1\), then the image is smaller. If \(k=1\), then the original figure is unchanged by the transformation.

\section*{Exercise 9A}
1 The following diagram shows an isosceles triangle \(A B C\) and a point \(O\). Copy or trace the diagram into your exercise book, leaving plenty of room to the right and below your diagram.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-03}
\end{center}

a Use ruler and compasses to apply an enlargement transformation with centre \(O\) and enlargement factor 3. Label your image \(A^{\prime} B^{\prime} C^{\prime}\).

b Verify that each side length of \(\triangle A^{\prime} B^{\prime} C^{\prime}\) is three times the matching side length of \(\triangle A B C\).

c Use a protractor to verify that each angle of \(\Delta A^{\prime} B^{\prime} C^{\prime}\) is equal to the matching angle of \(\triangle A B C\).

2 The diagram below shows a rectangle \(A B C D\) and a point \(O\). Copy or trace the diagram into your exercise book.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-04}
\end{center}

a Now apply an enlargement with centre \(O\) and factor \(\frac{1}{2}\) using these steps:

i Join \(O A\) and mark the midpoint \(A^{\prime}\) of \(O A\).

ii Join \(O B\) and mark the midpoint \(B^{\prime}\) of \(O B\).

iii Join \(O C\) and mark the midpoint \(C^{\prime}\) of \(O C\).

iv Join \(O D\) and mark the midpoint \(D^{\prime}\) of \(O D\).

b Use compasses to verify that each side length of \(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) length of \(A B C D\).

c Use a protractor to verify that each angle of \(A^{\prime} B^{\prime} C^{\prime} D^{\prime}\) is a right angle, matching the angles of \(A B C D\).

3 I have built a primitive slide projector by shining light from a point source through a transparent slide held up \(10 \mathrm{~cm}\) away. The light then falls onto a wall that is parallel to the slide and distant 1 metre from it.

a Draw a labelled diagram of a triangle on the slide and its image on the wall.

b What are the centre of enlargement and the enlargement factor of the transformation?

4 A ginger cat is sitting on a thin railing 2 metres horizontally from a desk lamp, producing a shadow of the cat on a wall 3 metres behind the cat. What is the enlargement factor of the shadow with respect to the profile of the cat?

5 Ernest is making scale drawings of atoms, in which his drawing of a hydrogen atom is a circle of radius \(1 \mathrm{~cm}\). Here are the actual approximate radii of some of the atoms in his drawings, given in picometres ( \(1 \mathrm{pm}=10^{-12}\) metres).

Hydrogen: 25 pm Carbon: 70 pm Gold: 135 pm Radium: 215 pm

Calculate the radii of Ernest's drawings of carbon, gold and radium atoms.

6 A map of a country is drawn to a scale of \(1: 12500000\). Find the actual distance in kilometres between two points whose separation on the map is:
a \(1 \mathrm{~cm}\)
b \(4 \mathrm{~cm}\)
c \(0.6 \mathrm{~cm}\)
d \(1 \mathrm{~mm}\)

\section*{Using ratios in similar figures}
Two figures are called similar if we can enlarge one figure so that its enlargement is congruent to the other figure.

In simple terms this means that by enlarging or shrinking one figure, we get the other figure perhaps translated, rotated or reflected.

Thus similar figures have the same shape, but not necessarily the same size, just as a scale drawing has the same shape as the original, but has a different size.

\begin{itemize}
  \item Matching lengths of similar figures are therefore always in the same ratio, called the similarity ratio.
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-05(2)}
\end{center}

\begin{itemize}
  \item Matching angles of similar figures are equal.
\end{itemize}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-05(1)}
\end{center}

We will use the same convention for matching angles as we did for congruence.

If \(\triangle A B C\) is similar to \(\triangle P Q R\) then, \(\angle A=\angle P, \angle B=\angle Q\) and \(\angle C=\angle R\)

The two triangles in the diagram below are similar.
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-05}

\[
\frac{P Q}{A B}=\frac{Q R}{B C}=\frac{P R}{A C}=2
\]

Hence \(\triangle A B C\) is similar to \(\triangle P Q R\) and the similarity ratio is 2 .

\section*{Example 1}
These two parallelograms are known to be similar.
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-06}

a What is the size of \(\angle P\) ?

b Find the base length of the large parallelogram.

c What is the similarity ratio of the transformation taking \(P Q R S\) to \(A B C D\) ?

\section*{Solution}
a \(\angle P=110^{\circ} \quad\) (matching angles of similar figures)

b \(\frac{D C}{S R}=\frac{B C}{Q R} \quad\) (ratio of matching sides)

\(\frac{D C}{2}=\frac{2.5}{1}\)

\(D C=5 \mathrm{~cm}\)

c The similarity ratio is 2.5 .

Note 1: In part \(\mathbf{b}\), we could have written the alternative ratio statement

\[
\frac{D C}{B C}=\frac{S R}{Q R} \text { (ratios within figures, 'larger' figure on the left, 'smaller' figure on the right) }
\]

It does not matter at all which ratio statement you choose to use. The subsequent algebra will be easier, however, if you always start by placing the unknown length on the top of the left-hand side of the equation.

Note 2: An alternative solution to part \(\mathbf{b}\) would be to calculate \(\frac{B C}{Q R}=2.5\) as the similarity ratio.

Thus \(D C=2.5 \times S R\)

\section*{Similar figures}
Two figures are called similar if an enlargement of one figure is congruent to the other figure.

\section*{Matching lengths and angles of similar figures}
\begin{itemize}
  \item Matching lengths in similar figures are in the same ratio, called the similarity ratio.
  \item Matching angles in similar figures are equal.
\end{itemize}

\section*{Example 2}
The car shown in the photograph below is 3.8 metres long.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-07}
\end{center}

a Use your ruler and your calculator to find approximate ratios for the following, each correct to 1 decimal place. Your measurements are for the diagram, but the ratio is for both the diagram and the real car.
i \(\frac{\text { height of the car }}{\text { diameter of wheel }}\)
ii \(\frac{\text { length of the car }}{\text { height of the car }}\)

b Find the height of the real car.

\section*{Solution}
a In the picture, height of the \(\mathrm{car} \approx 2.8 \mathrm{~cm}\) diameter of a wheel \(\approx 1.2 \mathrm{~cm}\) length of the car \(\approx 8.2 \mathrm{~cm}\)

i \(\frac{\text { height of the car }}{\text { diameter of wheel }} \approx \frac{2.8}{1.2}\)

ii \(\frac{\text { length of the car }}{\text { height of the car }} \approx \frac{8.2}{2.8}\)

\[
\approx 2.3
\]

b Enlargement factor \(\approx \frac{380}{8.2} \approx 46\)

The height of the \(\mathrm{car} \approx 2.8 \times 46 \approx 128 \mathrm{~cm}=1.28 \mathrm{~m}\)

\section*{Exercise 9B}
Example 1 1 The diagram in each part shows two figures that are known to be similar. Copy and complete the statements below each diagram.

a \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-08(5)}
\end{center}

i \(\quad \frac{P R}{\cdots}=\frac{P Q}{\cdots}\)

c \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-08(6)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-08(3)}
\end{center}

ii \(\frac{Q R}{\cdots}=\frac{R P}{\cdots}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-08(2)}
\end{center}

b
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-08(9)}

ii \(\frac{F E}{\cdots}=\frac{F D}{\cdots}\)

i \(\frac{D E}{\cdots}=\frac{D F}{\cdots}\)
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-08}

i \(\quad \frac{A B}{\cdots}=\frac{A D}{\cdots} \quad\) ii \(\quad \frac{C B}{\cdots}=\frac{C D}{\cdots}\)

i \(\frac{J K}{\cdots}=\frac{M N}{\cdots}\)

ii \(\frac{J L}{\cdots}=\frac{M N}{\cdots}\)

2 The diagram in each part shows two figures that are known to be similar. Write a ratio statement. Hence find the value of the pronumeral.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-08(8)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-08(4)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-08(7)}
\end{center}

\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-08(1)}
c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-09(3)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-09}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-09(1)}
\end{center}

3 The diagram below shows some of the keys of a piano.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-09(5)}
\end{center}

By measuring lengths with your ruler, find approximately:

a the ratio of the lengths of the black and the white keys

b the ratio of the widths of the black and white keys at their front edges

c the ratio of the length of a white key and the width of its front edge

d the width of its front edge if the length of a white key is \(13.3 \mathrm{~cm}\)

In questions 4 to 6 , assume that the triangles are named in matching order.

\(4 \triangle L M N\) is similar to \(\triangle T U V\), where \(\triangle L M N\) is shown below, and \(T U=24 \mathrm{~cm}\), find:

a \(\angle T U V\)

b \(U V\)

c \(\angle V T U\)

d \(V T\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-09(4)}
\end{center}

\(5 \quad \triangle X Y Z\) is similar to \(\triangle G H K\), where \(\triangle X Y Z\) is shown below, and \(H K=18 \mathrm{~cm}\), find:

a \(\angle G H K\)

b \(\angle G K H\)

c \(G H\)

d \(G K\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-09(2)}
\end{center}

6 In two triangles \(A B C\) and \(L M N, A B=6 \mathrm{~cm}, L M=12 \mathrm{~cm}, L N=8 \mathrm{~cm}\) and \(M N=16 \mathrm{~cm}\). What is the similarity ratio between the two triangles if:
a \(\triangle A B C\) is similar to \(\triangle L M N\) ?
b \(\triangle A B C\) is similar to \(\triangle L N M\) ?
c \(\triangle A B C\) is similar to \(\triangle M N L\) ?
d \(\triangle A B C\) is similar to \(\triangle M L N\) ?

\section*{The AAA similarity test for triangles}
As with congruence, most of the problems involving similarity come down to establishing that two triangles are similar. In this section and the next two sections, we shall establish four similarity tests for triangles and use them in problems.

The two triangles below have the same interior angles, but have different sizes.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-10(1)}
\end{center}

Let us now enlarge the left-hand triangle by an enlargement factor of \(\frac{3}{2}\) so that \(B^{\prime} C^{\prime}=Q R\), as in the diagram following. We have seen that the angles do not change, and that all three side lengths increase by the same factor.
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-10}

By the AAS congruence test, the new triangle \(\triangle A^{\prime} B^{\prime} C\) is congruent to \(\triangle P Q R\).

This is because the two triangles have two pairs of angles equal, and the matching sides \(B^{\prime} C^{\prime}\) and \(Q R\) are also equal.

Hence the original triangle \(\triangle A B C\) is similar to \(\triangle P Q R\), because its enlargement \(\triangle A^{\prime} B^{\prime} C\) is congruent to \(\triangle P Q R\).

\section*{The AAA similarity test}
If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.

Note: When using this test, it is sufficient to prove the equality of just two pairs of angles the third pair must then also be equal since the angle sum of any triangle is \(180^{\circ}\). Thus the test is sometimes called 'the AA similarity test'.

\section*{Similarity statements}
Arguments using similar triangles depend on a similarity statement that two triangles are similar. As with congruence, always give the similarity test in brackets afterwards, and be particularly careful to name the vertices in matching order.

Thus the conclusion of our argument above is written as:

\(\triangle A B C\) is similar to \(\triangle P Q R(\mathrm{AAA})\)

There are two common notations for similarity. These are:

\[
\triangle A B C \sim \triangle P Q R \text { and } \triangle A B C\|\| \triangle P Q R
\]

These are both read as 'triangle \(A B C\) is similar to triangle \(P Q R\) '.

\section*{Example 3}
In each diagram below, write a similarity statement beginning with ' \(\triangle A B C\) is similar to ...'

Be careful to name the vertices in matching order.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-11(1)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-11}
\end{center}

\section*{Solution}
a \(\triangle A B C\) is similar \(\triangle C P B\).

\((\mathrm{AAA})\)

b \(\triangle A B C\) is similar \(\triangle B M C\).

(AAA)

Notice that in part \(\mathbf{b}\) the angle \(\angle C\) is common to both triangles.

\section*{Proving similarity}
In many geometric problems, we need to prove that two triangles are similar. The proofs are set out just like congruence proofs.

\section*{Example 4}
Prove that the two triangles in the diagram to the right are similar.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-12(2)}
\end{center}

\section*{Solution}
In the triangles \(A B C\) and \(P Q C\) :

\begin{center}
\begin{tabular}{ll}
\(\angle B A C=\angle Q P C\) & (corresponding angles, \(A B \| P Q\) ) \\
\(\angle C=\angle C\) & (common) \\
so \(\triangle A B C\) is similar to \(\triangle P Q C\). & (AAA) \\
\end{tabular}
\end{center}

\section*{Using similar triangles to find lengths}
Once we have established that two triangles are similar, we know that matching sides have the same ratio.

If \(\triangle A B C\) is similar to \(\triangle X Y Z\), then:

This ratio is the similarity ratio.

\[
\frac{A B}{X Y}=\frac{B C}{Y Z}=\frac{C A}{Z X}
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-12(1)}
\end{center}

\section*{Example 5}
In the diagram to the right:

a Prove that \(\triangle A B O\) is similar to \(\triangle G F O\).

b Hence find the length \(A B\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-12}
\end{center}

\section*{Solution}
a In the triangles \(A B O\) and \(G F O\) :

\[
\begin{array}{cl}
\angle A O B=\angle G O F & \text { (vertically opposite at } O \text { ) } \\
\angle A=\angle G & \text { (alternate angles, } A B \| F G \text { ) } \\
\text { so } \triangle A B O \text { is similar to } \triangle G F O . & \text { (AAA) }
\end{array}
\]

b Hence \(\frac{A B}{3}=\frac{5}{2}\)

(matching sides of similar triangles)

\[
\begin{aligned}
A B & =3 \times \frac{5}{2} \\
& =7 \frac{1}{2}
\end{aligned}
\]

\section*{Example 6}
Find the values of \(a\) and \(b\) in the following diagrams.
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-13}

\section*{Solution}
\(\triangle A B C\) is similar to \(\triangle U S T\). (AAA)

Hence \(\frac{S T}{B C}=\frac{S U}{B A} \quad\) (matching sides of similar triangles)

\[
\begin{aligned}
\frac{b}{10} & =\frac{40}{16} \\
b & =25
\end{aligned}
\]

Also \(\quad \frac{A C}{U T}=\frac{B A}{S U} \quad\) (matching sides of similar triangles)

\[
\frac{a}{22.5}=\frac{16}{40}
\]

\[
\text { So } \quad a=9
\]

Alternatively, since \(\frac{S U}{B A}=\frac{40}{16}=2.5\), the similarity ratio from \(\triangle A B C\) to \(\triangle U S T\) is 2.5 ; that is, every length in \(\triangle U S T\) will be 2.5 times the matching length in \(\triangle A B C\).

\[
\begin{aligned}
& \text { Thus } S T=2.5 \times B C \quad \text { and } \quad U T=2.5 \times A C \\
& b=2.5 \times 10 \quad 22.5=2.5 \times a \\
& =25 \quad a=9
\end{aligned}
\]

\section*{Example 7}
Find the value of \(x\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-14}
\end{center}

\section*{Solution}
In the triangles \(A B C\) and \(A X Y\) :

\(\angle A=\angle A\) (common)

\(\angle A B C=\angle A X Y\) (corresponding angles, \(X Y \| B C\) )

so \(\triangle A B C\) is similar to \(\triangle A X Y\) (AAA).

Hence \(\frac{A B}{A X}=\frac{B C}{X Y} \quad\) (matching sides of similar triangles)

\[
\begin{aligned}
\frac{x+3}{3} & =\frac{8}{6}=\frac{4}{3} \\
x+3 & =4 \\
x & =1
\end{aligned}
\]

\section*{Example 8}
A tall vertical flagpole throws a 5 metre shadow. At the same time of day, a fencepost of height 2 metres throws a \(30 \mathrm{~cm}\) shadow.

a Draw a diagram, identifying two similar triangles and proving that they are similar.

b Hence find the height of the flagpole.

\section*{Solution}
a In the diagram, \(F C\) is the flagpole, \(P B\) is the fencepost, and \(T\) and \(S\) mark the end of the shadows.

In the triangles \(F C T\) and \(P B S\) :

\(\angle C=\angle B=90^{\circ}\) (pole and post are vertical)

\(\angle T=\angle \mathrm{S}\) (both are the angle of elevation of the Sun)

so \(\triangle F C T\) is similar to \(\triangle P B S\)

(AAA)

b Hence \(\frac{F C}{2}=\frac{5}{0.3} \quad\) (matching sides of similar triangles)

\[
F C=33 \frac{1}{3} \text { metres }
\]

Thus the height of the flagpole is \(33 \frac{1}{3}\) metres.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-14(1)}
\end{center}

\section*{Exercise \(9 \mathrm{C}\)}
Example 3 1 Copy and complete each similarity statement, giving the similarity test and naming the vertices in matching order. Then copy and complete the ratio statement.

\(\mathbf{a}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-15(4)}
\end{center}

\(\triangle A B C\) is similar to \(\ldots(\ldots)\)

\[
\frac{A B}{\ldots}=\frac{B C}{\ldots}
\]

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-15(5)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-15(1)}
\end{center}

\(\triangle A B C\) is similar to \(\ldots(\ldots)\)

\(\frac{A B}{\cdots}=\frac{B C}{\cdots}\)

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-15}
\end{center}

\(\triangle A B C\) is similar to \(\ldots(\ldots)\)

\(\frac{A B}{\cdots}=\frac{B C}{\cdots}\) \(\triangle A B C\) is similar to \(\ldots(\ldots)\)

\(\frac{A B}{\cdots}=\frac{B C}{\cdots}\)

2 Prove that the two triangles in each diagram are similar, naming the vertices in matching order. Then copy and complete the ratio statement.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-15(2)}
\end{center}

\(\frac{A B}{\cdots}=\frac{A P}{\cdots}\) b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-15(3)}
\end{center}

\(\frac{G L}{\cdots}=\frac{G M}{\cdots}\)

3 Prove that the two triangles in each diagram are similar. Then write a ratio statement and find the value of the pronumeral.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-16(3)}
\end{center}

c \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-16(1)}
\end{center}

\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-16}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-16(2)}
\end{center}

d
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-16(4)}

4 In each part, you will need to draw a diagram and prove that two triangles are similar.

a A stick 1.5 metres long leans against a wall and reaches 1 metre up the wall. How far up the wall will a ladder 10 metres long reach if it is parallel to the stick?

b A 3 metre high road-sign casts a shadow 1.4 metres long. How high is a telegraph pole that casts a shadow 3.5 metres long at the same time of day?

c A steep road rises 1.2 metres for every 6 metres travelled along the road surface. How far will a car have to travel to rise 600 metres?

d The front of an A-frame hut is an isosceles triangle of height 5 metres and base 8 metres. How wide will a model of the hut be if its height is 0.8 metres?

5 a Find the sizes of \(\alpha, \beta\) and \(\gamma\) in the diagram on the right.

b The three triangles in the diagram are similar.

Copy and complete the statement:

\(\triangle A B C\) is similar to ... which is similar to ... (AAA).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-17(3)}
\end{center}

c Hence copy and complete:
i \(\frac{a}{b}=\frac{\cdots}{\cdots}=\frac{\cdots}{\cdots}\)
ii \(\frac{h}{a}=\frac{\cdots}{\cdots}=\frac{\cdots}{\cdots}\)

6 Prove that: The diagonals of a trapezium dissect the trapezium into four triangles, two of which are similar.

Let the diagonals of the trapezium \(A B C D\) meet at \(M\).

Identify the two similar triangles and prove that they are similar.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-17}
\end{center}

7 Prove that: The altitude to the hypotenuse of a right-angled triangle divides the hypotenuse into two intervals whose product equals the square of the altitude.

Let \(A N\) be the altitude to the hypotenuse \(B C\) of the rightangled triangle \(\triangle A B C\).

a Prove that \(\triangle A B N\) is similar to \(\triangle C A N\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-17(2)}
\end{center}

b Hence prove that \(A N^{2}=B N \times C N\).

8 Let \(A B C\) be a triangle, and let \(B P\) and \(C Q\) be altitudes.

a Prove that \(\triangle A B P\) is similar to \(\triangle A C Q\).

b Hence prove that \(\frac{B P}{C Q}=\frac{A B}{A C}\).

c As an alternative approach, explain why \(\frac{1}{2} \times A C \times B P\) and \(\frac{1}{2} \times A B \times C Q\) are both equal to the area of \(\triangle A B C\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-17(1)}
\end{center}

d Hence prove that \(\frac{B P}{C Q}=\frac{A B}{A C}\).

\section*{The SAS similarity test for triangles}
The two triangles below have a pair of equal angles, and the sides including this angle are in the same ratio \(1: 2\).

\[
\angle B=\angle Q, A B=\frac{1}{2} P Q \text { and } B C=\frac{1}{2} Q R
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-18}
\end{center}

Using the same procedure as before, we can enlarge the first triangle by a factor of 2 to produce a new triangle \(\triangle A^{\prime} B^{\prime} C^{\prime}\), and \(\triangle A^{\prime} B^{\prime} C^{\prime}\) is congruent to \(\triangle P Q R\) by the SAS congruence test.
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-18(2)}

Hence the original triangle \(\triangle A B C\) is similar to \(\triangle P Q R\).

This gives us our second similarity test.

\section*{The SAS similarity test}
If the ratio of the lengths of two pairs of matching sides are equal and the included angles are equal then the two triangles are similar.

The statement that the two triangles above are similar is thus written as

\(\triangle A B C\) is similar to \(\triangle P Q R\) (SAS).

\section*{Example 9}
In triangle \(A B C\),

\(A B=A C=10\) and \(A P=A Q=7\).

a Prove that \(\triangle A B C\) is similar to \(\triangle A P Q\).

b Prove that \(P Q\) is parallel to \(B C\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-18(1)}
\end{center}

\section*{Solution}
a In the triangles \(A B C\) and \(A P Q\) :

\[
\begin{aligned}
& \frac{A B}{A P}=\frac{10}{7} \\
& \text { (given) } \\
& \frac{A C}{A Q}=\frac{10}{7} \\
& \text { (given) } \\
& \angle A=\angle A \\
& \text { (common) }
\end{aligned}
\]

so \(\triangle A B C\) is similar to \(\triangle A P Q\) (SAS).

\(\begin{array}{ll}\text { b } \angle A B C=\angle A P Q & \text { (matching angles of similar triangles) } \\ \text { So } B C \| P Q . & \text { (corresponding angles are equal) }\end{array}\)

\section*{Exercise 9D}
1 Prove that the two triangles in each diagram are similar, and write the similarity statement and the similarity factor (left to right).

\(\mathbf{a}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-19(3)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-19}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-19(2)}
\end{center}

d
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-19(5)}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-19(6)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-19(4)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-19(1)}
\end{center}

3 Prove that \(\triangle A B C\) is similar to \(\triangle A D E\) and copy and complete the ratio statement.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-20(2)}
\end{center}

\[
\frac{\mathrm{AB}}{\ldots}=\frac{\mathrm{AC}}{\ldots}=\ldots
\]

4 Prove that triangles \(A B C, A D E, A F G\), and \(A H I\) are similar and find \(x, y\) and \(z\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-20(1)}
\end{center}

5 In triangles \(P Q R\) and \(X Y Z, \angle P=\angle Z\) and \(P Q \times Y Z=P R \times X Z\). Name the other pairs of equal angles in the triangles.

6 Prove that if \(A B C D\) is a quadrilateral such that \(\angle A=\angle C\) and \(\frac{D A}{A B}=\frac{B C}{C D}\), then \(A B C D\) is a parallelogram.

\section*{The SSS and RHS similarity tests}
We now introduce two further similarity tests, giving four tests altogether.

\section*{The SSS similarity test}
The two triangles below have different sizes, but their sides can be paired up so that matching sides have the same ratio \(1: 2\).

\[
A B=\frac{1}{2} P Q, B C=\frac{1}{2} Q R \text { and } C A=\frac{1}{2} R P
\]

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-20}
\end{center}

Let us now enlarge the first triangle by a factor 2 , so that the side lengths all double.
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-21(1)}

By the SSS congruence test, the new triangle \(\triangle A^{\prime} B^{\prime} C^{\prime}\) is congruent to \(\triangle P Q R\).

Hence the original triangle \(\triangle A B C\) is similar to \(\triangle P Q R\).

The statement that the two triangles above are similar is thus written as

\(\triangle A B C\) is similar to \(\triangle P Q R\) (SSS).

As an interesting example, the Pythagorean triples \(3,4,5 ; 6,8,10 ; 9,12,15 ; \ldots\) give an infinite family of similar right-angled triangles.

\section*{The SSS similarity test}
If we can match up the sides of one triangle with the sides of the other so that the ratio of matching lengths is constant, then the triangles are similar.

\section*{The RHS similarity test}
The two triangles below are both right-angled, and the hypotenuse and one side are in the same ratio \(3: 4\).
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-21}

Once again, we can enlarge the first triangle by a factor of \(\frac{4}{3}\) to produce a triangle \(\Delta A^{\prime} B^{\prime} C^{\prime}\), and \(\triangle A^{\prime} B^{\prime} C^{\prime}\) is congruent to \(\triangle P Q R\) by the RHS congruence test.

Hence the original triangle \(\triangle A B C\) is similar to \(\triangle P Q R\).

This gives us our fourth and last similarity test.

\section*{The RHS similarity test}
If the ratio of the hypotenuses of two right-angled triangles equals the ratio of another pair of sides, then the two triangles are similar.

The statement that the two triangles above are similar is thus written as

\(\triangle A B C\) is similar to \(\triangle P Q R\) (RHS).

Recall that given the hypotenuse and one side of a right-angled triangle, the third side can be found by Pythagoras' theorem.

\section*{Example 10}
a Prove that the two triangles in the diagram to the right are similar.

b Identify the equal angles in two triangles.

c Prove that \(A B \| P Q\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-22}
\end{center}

\section*{Solution}
a In the triangles \(A B M\) and \(Q P M\)

\(\angle A M B=\angle Q M P=90^{\circ} \quad\) (vertically opposite)

\(\frac{A B}{Q P}=\frac{2}{3}=\frac{B M}{P M}\)

(given)

so \(\triangle A B M\) is similar to \(\triangle Q P M\) (RHS similarity).

b Hence \(\alpha=\phi\) and \(\beta=\theta\) (matching angles of similar triangles).

c Hence \(A B \| P Q\) (alternate angles are equal).

\section*{Using the four similarity tests in problems}
Here are examples using the AAA, SSS, SAS and RHS similarity tests in problems.

\section*{Example 11}
a Prove that the two triangles in the diagram are similar.

b Which of the marked angles are equal?

\section*{Solution}
a In the triangles \(\triangle A B C\) and \(\triangle C B D\) :

\(A B=2 \times C B \quad\) (given)

\(B C=2 \times B D \quad\) (given)

\(C A=2 \times D C \quad\) (given)

so \(\triangle A B C\) is similar to \(\triangle C B D\) (SSS).

b Hence \(\gamma=\theta\) (matching angles of similar triangles).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-22(1)}
\end{center}

\section*{Example 12}
a Prove that the two triangles in the diagram are similar.

b Hence prove that \(L M \| B C\).

\section*{Solution}
a In the triangles \(\triangle A B C\) and \(\triangle A L M\) :

\(A B=4 \times A L\)

\(A C=4 \times A M\)

\(\angle B A C=\angle L A M \quad\) (common)

so \(\triangle B A C\) is similar to \(\triangle L A M\) (SAS).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(8)}
\end{center}

b Hence \(\angle A B C=\angle A L M\) (matching angles of similar triangles) and so \(L M \| B C\)

(corresponding angles are equal).

\section*{Exercise 9E}
1 Name the pair of similar triangles in each part, naming the vertices in matching order and stating the similarity test.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(5)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(9)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(1)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(3)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(4)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23}
\end{center}

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(6)}
\end{center}

d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-23(2)}
\end{center}

\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-23(7)}
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-23(10)}
\(\mathbf{e}\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24(1)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24(6)}
\end{center}

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24(5)}
\end{center}

2 Copy and complete each similarity statement, stating the similarity test, and naming the vertices in matching order. They copy and complete the ratio statement.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24(2)}
\end{center}

\(\triangle A B C\) is similar to \(\ldots(\ldots)\)

\(\frac{A B}{\cdots}=\frac{B C}{\cdots}\)

c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24}
\end{center}

\(\triangle A B C\) is similar to \(\ldots(\ldots)\)

\(\frac{A B}{\cdots}=\frac{B C}{\cdots}\) b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24(4)}
\end{center}

\(\triangle A B C\) is similar to \(\ldots(\ldots)\)

\(\frac{A B}{\cdots}=\frac{B C}{\cdots}\)

d \(A\)

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24(3)}
\end{center}

\(\triangle A B C\) is similar to ... (...)

\(\frac{A B}{\cdots}=\frac{B C}{\cdots}\)

3 Find the values of the pronumerals in each part.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-24(7)}
\end{center}

b

\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-24(8)}
c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-25(1)}
\end{center}

e

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-25(2)}
\end{center}

4 a Find the size of:

i \(\angle A B D\)

b Justify the statement that \(\triangle A B C\) is similar to \(\angle A D B\).

c Find \(A C\).

d Show that \(B D=3 \sqrt{3} \mathrm{~cm}\).

e Find \(B C\).

5 a Find \(L M\).

b Find \(L N\).

c Find \(M N\). d

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-25(4)}
\end{center}

f

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-25}
\end{center}

ii \(\angle A C B\)
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-25(5)}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-25(3)}
\end{center}

\section*{\(\angle C D M\).}
ii Hence prove that \(\angle B=\angle D\).

iii Is it true that \(A B \| C D\) ?
c

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-26(2)}
\end{center}

i Prove that \(\triangle A B C\) is similar to \(\triangle C B F\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-26(1)}
\end{center}

i Prove that \(\triangle A B C\) is similar to \(\triangle \ldots\)

ii Is \(\alpha\), or \(\beta\) equal to \(\theta\) ?

ii Hence find \(x\) and \(y\).

7 a Prove that: The line interval joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

b Prove that: The line through the midpoint of one side of a triangle parallel to another side of the triangle meets the third side at its midpoint.

c Prove that: The midpoints of the sides of a quadrilateral form the vertices of a parallelogram. Let \(A B C D\) be a quadrilateral, and let \(P, Q, R\) and \(S\) be the midpoints of the sides \(A B, B C, C D\) and \(D A\) respectively. Construct diagonals \(A C\) and \(B D\).

8 Prove that: The intervals joining the midpoints of the sides of a triangle dissect the triangle into four congruent triangles each similar to the original triangle.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-26}
\end{center}

9 If you think the statement is true, justify your answer using a similarity test. If you think it is false, draw two triangles that provide a counter-example.

a Any two equilateral triangles are similar.

b Any two isosceles triangles are similar.

c Any two isosceles triangles with equal apex angles are similar.

d Any two isosceles triangles with equal length bases are similar.

e Any two right-angled triangles are similar.

f Any two right-angled triangles with equal length hypotenuses are similar.

g Any two isosceles right-angled triangles are similar.

h Any two right-angled triangles in which one other angle is \(40^{\circ}\) are similar.

10 If you think the statement is true, justify your answer. If you think it is false, draw two figures that provide a counterexample.

a Any two squares are similar.

b Any two rectangles are similar.

c Any two rectangles in which one side is three times another are similar.

d Any two rectangles whose diagonals meet at \(30^{\circ}\) are similar.

e Any two rectangles whose diagonals have length \(6 \mathrm{~cm}\) are smilar.

f Any two rhombuses are similar.

g Any two rhombuses with vertex angles \(50^{\circ}\) and \(130^{\circ}\) are similar.

11 a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-27(3)}
\end{center}

Find \(A D, D C\) and \(B C\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-27(2)}
\end{center}

Find \(A M, B M\) and \(D M\).

\section*{Review exercise}
1 a Prove that triangle \(A B C\) is similar to triangle \(D E C\) and find the value of \(e\) and \(c\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-27(1)}
\end{center}

b Prove that triangle \(J N K\) is similar to triangle \(J M L\) and find the values of \(x\) and \(y\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-27}
\end{center}

2 Write down a pair of similar triangles and find the value of each pronumeral.

a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-28(2)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-28(5)}
\end{center}

3 Write down a pair of similar triangles and find the value of each pronumeral.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-28(4)}
\end{center}

b

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-28(1)}
\end{center}

4 A vertical stick of length \(30 \mathrm{~cm}\) casts a shadow of length \(5 \mathrm{~cm}\) at 12 p.m. Find the length, in centimetres, of the shadow cast by a 1 metre ruler placed in the same position at the same time of the day.

5 a

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-28}
\end{center}

i Prove that \(\triangle A L M\) is similar to \(\triangle A B C\).

ii Hence prove that \(L M \| B C\).

iii Find \(x\). b G

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-28(3)}
\end{center}

i Prove that \(\triangle O P Q\) is similar to \(\triangle O F G\).

ii Hence prove that \(P Q \| F G\).

iii Find \(x\).

6 Prove that \(\triangle A B C\) is similar to \(\triangle D E F\).

Find \(x\).
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-29(1)}

7 A tower standing on level ground casts a shadow \(40 \mathrm{~m}\) long. A vertical stick \(3 \mathrm{~m}\) high is placed at the tip of the shadow. The stick is found to cast a shadow \(6 \mathrm{~m}\) long. Find the height of the tower.

8 A line from the top of a church steeple to the ground just passes over the top of a pole \(2.5 \mathrm{~m}\) high and meets the ground at a point \(A, 1.5 \mathrm{~m}\) from the base of the pole. If the distance from \(A\) to a point directly below the church steeple is \(30 \mathrm{~m}\), find the height of the steeple.

\(9 A B C D\) is a parallelogram with \(\angle B A D\) acute. The point \(E\) lies on the ray \(C B\) such that \(\triangle A B E\) is isosceles with \(A B=A E\). The point \(F\) lies on the ray \(C D\) such that \(\triangle A D F\) is isosceles with \(A D=A F\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-29}
\end{center}

a Prove that \(\triangle A B E\) is similar to \(\triangle A D F\).

b Prove that \(D E=B F\).

\(10 A B C\) is a triangle, \(M\) is a point in the interval \(A B\) such that \(A B=3 A M\) and \(N\) is a point in the line segment \(A C\) such that \(A C=3 A N\).

a Prove that \(B C \| M N\).

b If \(B N\) and \(C M\) intersect at \(P\), prove that \(B P=3 P N\).

\section*{Challenge exercise}
1 You proved in Exercise 9E, Question 8 that the midpoints of the sides of a quadrilateral form the vertices of a parallelogram. Now prove that this parallelogram has area half the area of the original quadrilateral.

2 Let \(A B C D\) be a trapezium with \(A B \| C D\). Let the diagonals \(A C\) and \(B D\) meet at \(O\).

Let the line through \(O\) parallel to \(A B\) meet \(A D\) at \(P\) and \(B C\) at \(Q\).

By considering the pairs of triangles \(A D B\) and \(P O D, B D C\) and \(Q O B\), and \(A O B\) and \(C O D\), prove that \(O\) is the midpoint of \(P Q\).

3 Let \(A B C D\) be a rectangle, and let \(K\) and \(L\) be the midpoints of \(A B\) and \(C D\) respectively. Let \(A C\) meet \(K L\) at \(M\).

a Show that \(M\) is the midpoint of \(K L\).

b Let \(D K\) meet \(A C\) at \(X\). Find the ratio \(A X: X C\).

c Find the ratio of the area of \(\triangle K X A\) to the area of the rectangle \(A B C D\).

4 Take any two points \(A\) and \(B\) in the plane and a piece of string longer than the distance \(A B\). Imagine fixing one end of the string at \(A\) and the other at \(B\), and pulling the string taut by using a pencil.

Now move the pencil around keeping the string taut

\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-30}
and letting the string slide around the point of the pencil. In this way we trace out a curve called an ellipse.

Prove that an enlargement transformation takes an ellipse to another ellipse.

5 Take three parallel lines and any two transversals that are not parallel.

a Prove that the ratio of the lengths cut off

by the parallel lines is the same for both transversals.

That is, prove that \(\frac{a}{b}=\frac{c}{d}\) in the diagram opposite.

b Is the result still true if the transversals are parallel?

c In the diagram below, find \(a\).
\includegraphics[max width=\textwidth, center]{2023_12_09_8eb2d5cdd18b2141f38dg-30(1)}

6 An altitude of a triangle is an interval from a vertex meeting the opposite side at right angles. Prove that if two altitudes of a triangle are equal, then the triangle is isosceles.

7 In this question we will prove that: The medians of a triangle are concurrent and trisect each other.

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-31(1)}
\end{center}

A median of a triangle is the line which joins a vertex to the middle of the opposite side.

a Let the medians \(B E\) and \(C F\) meet at \(G\). Show that \(B G=2 G E\) and \(C G=2 G F\).

\begin{center}
\includegraphics[max width=\textwidth]{2023_12_09_8eb2d5cdd18b2141f38dg-31}
\end{center}

b Deduce that the third median \(A D\) also passes through \(G\). The point \(G\) is called the centroid of the triangle \(A B C\).

c Prove that if two of the medians of a triangle have equal length, then the triangle is isosceles.

\section*{8 Theorem}
Let \(A B C\) and \(P Q R\) be equiangular triangles, with \(\angle A=\angle P=\alpha, \angle B=\angle Q=\beta\), \(\angle C=\angle R=\gamma\). Let \(B C=a, C A=b\) and \(A B=c\), and let \(Q R=k a\), where \(k\) is a positive rational number. Then \(R P=k b\) and \(P Q=k c\).

Use congruence (not similarity) to prove the result for \(k=2\) and then \(k=3\).

This is a beginning of the proof that the four similarity tests can be proven from the congruence tests without enlargements when the similarity factor is a rational number.


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