Consider the following pairs of shapes:
Pair A: Similar Triangles
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
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
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
There are four pairs of equal corresponding angles, and all corresponding sides are in the same ratio.
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
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.
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.
Let's have a look at these worked examples.
Consider the following shapes:
Are the two shapes similar?
Give a reason for your answer.
All sides are in the same ratio but not all matching angles are equal.
All matching angles are not equal and all sides are not in the same ratio
All matching angles are equal but all sides are not in the same ratio
Consider the shapes attached:
Are the 2 shapes similar?
Find the enlargement factor.
Consider the following triangles:
Which of these triangles are similar?
Give a suitable reason for their similarity.
All corresponding angles are equal.
Two angles are equal and one side is a multiple of the corresponding side of the other.
All corresponding sides are in the same ratio.