Topics
Congruence and Similarity
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India
Class IX

Introduction to Similarity

Lesson
Practice
Lesson

Consider the following pairs of shapes:

Pair A: Similar Triangles

Note:

  • Since all sides in the smaller triangle are equal, all angles are equal. This is an equilateral triangle.
  • Each side in the smaller triangle is multiplied by 3 to give the side length of the larger triangle.
  • This makes the sides of the larger triangle all equal. So the larger triangle is also equilateral.

 

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

 

Pair B: Non-Similar Triangles

Note:

  • Since all sides in the smaller triangle are equal, all angles are equal. This is an equilateral triangle.
  • 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. 
  • This makes the sides of the larger triangle unequal. So the larger triangle is not equilateral, and all angles are not $60^\circ$60°.

 

The corresponding sides between the two shapes are NOT in the same ratio and all corresponding angles between the two shapes are NOT equal.

 

Pair C: Non-Similar Rectangles

Note:

  • Both shapes are rectangles
  • To make the larger rectangle, two of the sides of the smaller rectangle are enlarged by a factor of 2 and the other two sides are not changed.

Even though all corresponding angles between the two shapes are equal, 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.

 

Pair D: Similar Quadrilaterals

Note:

  • 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.

 

Similar Shapes

If the only difference between two shapes is the size, (one is an enlargement of the other) then the two shapes are similar.

In similar shapes

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

 

Did you know?

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

 

Some of our standard shapes create 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 a similar because every square is an enlargement of another.

 

Example:

How can a square of side lengths $4$4 cm create a similar square of side length $10$10 cm?

Solution: Multiply each side of the smaller square by $2.5$2.5.

 

In fact, all REGULAR shapes create similarity (shapes that have all side lengths of equal value).

 

Explore this interactive to create similar polygons.  

Created with Geogebra

 

 

 

Let's have a look at these worked examples.

Practice questions

Question 1

Consider the following shapes:

Two parallelograms are depicted side by side. The one on the left has an angle at its bottom left corner labeled as 78 degrees and the length of its top side is 15 units. All sides are marked with single hashmarks. The one on the right has an angle at its bottom left corner labeled as 71 degrees and the length of its top side is 31 units. All sides are marked with double hashmarks.

  1. Are the two shapes similar?

    Yes

    A

    No

    B

  2. Give a reason for your answer.

    All sides are in the same ratio but not all matching angles are equal.

    A

    All matching angles are not equal and all sides are not in the same ratio

    B

    All matching angles are equal but all sides are not in the same ratio

    C

Question 2

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 3

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

9.G.T.1

Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).

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