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10. Shape and Measurement
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VCE 11 General 2023

10.08 Similarity

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Lesson

Similar shapes

Shapes or objects are described as similar if they are exactly the same shape but different size. Objects or shapes are congruent if they are exactly the same.

In similar shapes:

  • All corresponding sides are proportional
  • All corresponding angles are equal

Some of our standard shapes have similarity because of the nature of their properties:

  • All circles are similar because every circle is an enlargement of another. That is, one circle can create any other circle by enlarging or reducing its radius.
  • All squares are similar because every square is an enlargement of another.
Remember!

Sometimes a shape can be rotated, translated or reflected but still be similar.

These two following triangles are similar as one is an enlargement of the other. The methods of proving similar triangles are detailed in the section below.

All the corresponding sides between the two shapes are in the same ratio and all corresponding angles between the two shapes are equal.

These triangles are not similar as each side has not been increased by the same factor.

To make the larger triangle, two of the sides of the smaller triangle are enlarged by a factor of $3$3 and one of the sides is enlarged by a factor of $2$2. The corresponding sides between the two shapes are not in the same ratio and all corresponding angles between the two shapes are not equal.

These rectangles are not similar for the same reason as the triangles.

The corresponding sides between the two shapes are not in the same ratio. That is, the same enlargement factor has not been applied to each side of the smaller rectangle.

These quadrilaterals are similar.

To make the larger quadrilateral, each side length of the smaller quadrilateral has been doubled. There are four pairs of equal corresponding angles, and all corresponding sides are in the same ratio.

Explore this interactive to create similar polygons.

Created with Geogebra

Practice questions

Question 1

Consider the two similar triangles.

  1. By filling in the gaps, match the corresponding angles.

    $\angle$∠$D$D corresponds to $\angle$∠$\editable{}$

    $\angle$∠$E$E corresponds to $\angle$∠$\editable{}$

    $\angle$∠$C$C corresponds to $\angle$∠$\editable{}$

  2. $CD$CD corresponds to which side in $\triangle LMN$△LMN?

    $LN$LN

    A

    $ML$ML

    B

    $MN$MN

    C

  3. $CE$CE corresponds to which side in $\triangle LMN$△LMN?

    $LN$LN

    A

    $MN$MN

    B

    $ML$ML

    C

Question 2

Consider the two shapes:

Two polygon is depicted on the image. The small polygon on the left is called hexagon with equal sides indicated by hash marks and an equal angles. The large polygon is called hexagon with equal sides indicated with two hash marks and an equal angle on each of its vertices. Hexagons has six sides. 

  1. Are the two shapes similar?

    Yes

    A

    No

    B

  2. Which of the following statements are true?

    One of the shapes is a rotation of the other.

    A

    One of the shapes is a translation of the other.

    B

    One of the shapes is an enlargement of the other.

    C

Question 3

The smaller quadrilateral has been reflected, then enlarged and finally rotated.

Two labeled quadrilaterals with their vertices labeled. The smaller quadrilateral is labeled ABCD, while the larger quadrilateral is labeled EFGH. The process transforming the smaller quadrilateral ABCD into the larger quadrilateral EFGH consists of a reflection, followed by an enlargement, and concluded with a rotation. Vertex A corresponds to vertex E, B corresponds to F, C corresponds to G, and D also corresponds to H. Side AD corresponds to side EH, side AB corresponds to side EF, side BC corresponds to side FG, and side CD also corresponds to GH.

  1. The two shapes are:

    Congruent

    A

    Similar

    B

    Neither

    C

  2. Which side in the larger shape is corresponding to side $AB$AB?

    $GH$GH

    A

    $EF$EF

    B

    $FG$FG

    C

    $HE$HE

    D

  3. Which angle in the larger shape is corresponding to $\angle ADC$∠ADC?

    $\angle EHG$∠EHG

    A

    $\angle FEH$∠FEH

    B

    $\angle HGF$∠HGF

    C

    $\angle GFE$∠GFE

    D

 

Similar triangles

In similar triangles, just as any similar object:

  • all corresponding angles are equal
  • all corresponding sides are in the same ratio

Proofs for similar triangles

There are three methods for proving two triangles are similar. You only need to use one of these proofs to show two triangles are similar.

AAA (angle, angle, angle)

If all corresponding angles in two different triangles are equal the triangles are similar.

The triangles show that all three corresponding pairs of angles are equal.

$\angle A=\angle P$∠A=∠P, $\angle B=\angle Q$∠B=∠Q, $\angle C=\angle R$∠C=∠R

Therefore, the $\triangle ABC$△ABC is similar to $\triangle PQR$△PQR (AAA). Note that we only need to show two angles are congruent to prove (AAA), as if we have two congruent angles then the third angle must also be congruent as angles in a triangle always add to $180$180 degrees.

 

SSS (side, side, side)

If two triangles have all three pairs of corresponding sides in the same ratio, then these triangles are similar.

For example, all corresponding sides in the triangle below are in the same ratio:

$\frac{12}{4}=\frac{12}{4}$124​=124​$=$=$\frac{21}{7}$217​$=$=$3$3

 

SAS (side, angle, side)

If two triangles have two pairs of sides in the same ratio and equal included angles, then these triangles are similar.

There are two pairs of sides in the same ratio and the included angle (marked by the dot) is equal.

$\frac{PQ}{AB}=\frac{6}{3}$PQAB​=63​$=$=$2$2, $\angle Q=\angle B$∠Q=∠B , $\frac{QR}{BC}=\frac{10}{5}$QRBC​=105​$=$=$2$2

Practice questions

Question 4

Consider the shapes attached:

Two triangles. On the left, the triangle has sides measuring $18$18 units, $42$42 units, and $48$48 units in clockwise order. The triangle on the right has sides measuring $3$3 units, $7$7 units, and $8$8 units, also in clockwise order.

  1. Are the 2 shapes similar?

    Yes

    A

    No

    B

  2. Find the enlargement factor.

    $6$6

    A

    $5$5

    B

    $7$7

    C

Question 5

Consider the following triangles:

  1. Which of these triangles are similar?

    A triangle is depicted with its interior angles labeled at each vertex: 55 degrees at the top vertex with a single arc, 53 degrees at the bottom left vertex with a double arc, and 72 degrees at the bottom right vertex with a triple arc. The side between the 55-degree and 53-degree angles is labeled with a length of 4 units.

    A

    A triangle is depicted with its interior angles labeled at each vertex: 55 degrees at the top vertex with a single arc, 70 degrees at the bottom left vertex with a double arc, and 55 degrees at the bottom right vertex with a triple arc. The side between the 55-degree and 55-degree angles is labeled with a length of 8 units.

    B

    A triangle is depicted with its interior angles labeled at each vertex: 72 degrees at the top vertex with a single arc, 38 degrees at the bottom left vertex with a double arc, and 70 degrees at the bottom right vertex with a triple arc. The side between the 72-degree and 70-degree angles is labeled with a length of 4 units.

    C

    A triangle is depicted with its interior angles labeled at each vertex: 55 degrees at the top vertex with a single arc, 70 degrees at the bottom left vertex with a double arc, and 55 degrees at the bottom right vertex with a triple arc. The side between the 55-degree and 55-degree angles is labeled with a length of 32 units.

    D

  2. Give a suitable reason for their similarity.

    All corresponding angles are equal.

    A

    Two angles are equal and one side is a multiple of the corresponding side of the other.

    B

    All corresponding sides are in the same ratio.

    C

Outcomes

U2.AoS4.7

similarity and scaling, and the linear scale factor 𝑘 and its extension to areas and volumes

U2.AoS4.14

use a linear scale factor to scale lengths, areas and volumes of similar figures and shapes in practical situations

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