Similar shapes
When a shape is enlarged, the image is Having the same shape but not necessarily the same size. The corresponding angles within the shapes are equal. to the original shape. It is the same shape but a different size.
These 2 shapes are similar as they are both rectangles but 1 is an enlargement of the other.
Similar triangles
2 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 2 triangles are similar.
Triangle B is an enlargement of triangle A by a The ratio of corresponding lengths in similar shapes, ie how much larger or smaller the shapes are. of 2. Each length in triangle B is twice as long as in triangle A.
The 2 triangles are similar.
Example one
State whether the 2 triangles are similar. Give a reason to support your answer.
Yes, they are similar. The 2 lengths have been increased by a scale factor of 2. The corresponding angle is the same.
Example two
State whether the 2 triangles are similar. Give a reason to support your answer.
To decide whether the 2 triangles are similar, calculate the missing angles.
Remember angles in a triangle add up to 180°.
Angle yxz = \(180 - 85 - 40 = 55^\circ\)
Angle YZX = \(180 - 85 - 55 = 40^\circ\)
Yes, they are similar. The 3 angles are the same.
Example three
State whether the 2 triangles are similar. Give a reason to support your answer.
No. 2 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.
The scale factor of enlargement is 2.
Length PS is twice as long as length ps.
PS = \(9 \times 2 = 18~\text{cm}\)
Similar shapes may be inside one another.
Question
Show that triangles ABC and DBE are similar and calculate the length DE.
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 3 angles are the same in both triangles so they are similar.
To calculate a missing length, draw the 2 triangles separately and label the lengths.
To calculate the scale factor, divide the 2 corresponding lengths.
\(\frac{6}{4} = 1.5\)
The scale factor of enlargement is 1.5.
DE = \(7.5 \div 1.5 = 5~\text{cm}\)
Question
Calculate the length TR.
To calculate a missing length, draw the 2 triangles separately and label the lengths.
The scale factor is \(\frac{6}{3} = 2\)
To calculate TR, first find QR.
QR = \(6 \times 2 = 12~\text{cm}\)
QR = QT + TR
TR = QR - QT
TR = \(12 - 6 = 6~\text{cm}\)
