Congruence and similarity

2) Similar shapes

When a shape is enlarged, the image is similar to the original shape. It is the same shape but a different size.

These two shapes are similar as they are both rectangles but one is an enlargement of the other.

Square and enlarged square

Similar triangles

Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio. Either of these conditions will prove two triangles are similar.

Original triangle and triangle enlarged by a scale factor of 2

Triangle B is an enlargement of triangle A by a scale factor of 2. Each length in triangle B is twice as long as in triangle A.

The two triangles are similar.

Example 1

State whether the two triangles are similar. Give a reason to support your answer.

Two similar triangles showing sides in the same proportion and included angle is equal

Yes, they are similar. The two lengths have been increased by a scale factor of 2. The corresponding angle is the same.

Example 2

State whether the two triangles are similar. Give a reason to support your answer.

Two similar triangles showing all three angles are equal in both triangles

To decide whether the two triangles are similar, calculate the missing angles.

Remember angles in a triangle add up to 180°.

Angle yzx = 180-85-40=55°
Angle YZX = 180-85-55=40°

Yes, they are similar. The three angles are the same.

Example 3

State whether the two triangles are similar. Give a reason to support your answer.

Triangle 1: 1.5x5x4cm Triangle 2: 3x7.5x6cm

No. The two sides of the triangle are increased by a scale factor of 1.5. The other side has been increased by a scale factor of 2.


Calculating lengths and angles in similar shapes

In similar shapes, the corresponding lengths are in the same ratio. This fact can be used to calculate lengths.

Example

Calculate the length PS.

Rectangle 1: 4x9cm Rectangle 2: 7x?cm

The scale factor of enlargement is 2.
Length PS is twice as long as length ps.
PS = 9 x 2 = 18 cm

Similar shapes may be inside one another.

Question 1

Show that triangles ABC and DBE are similar and calculate the length DE.

Triangle ABC and DBE overlapped

Answer: Angle BCA = angle BED because of corresponding angles in parallel lines.
Angle BAC = angle BDE because of corresponding angles in parallel lines.
Angle DBE = angle ABC because both triangles share the same angle.
All three angles are the same in both triangles so they are similar.

To calculate a missing length, draw the two triangles separately and label the lengths.

Triangle ABC and DBE separated

To calculate the scale factor, divide the two corresponding lengths.
The scale factor of enlargement is \frac{6}{4} = 1.5 .
Therefore, DE = 7.5 : 1.5 = 5 cm

Question 2

Calculate the length TR.

Triangle QTS and PQR overlapped

Answer: To calculate a missing length, draw the two triangles separately and label the lengths.

The scale factor is \frac{6}{3} = 2 .
To calculate TR, first find QR.
QR = 6 x 2 = 12 cm
QR = QT + TR
TR = QR – QT
TR = 12 – 6 = 6 cm

Triangle QTS and PQR separated

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