Congruent and similar shapes
We are going to look at:
Similar figures are identical in shape, but not in size.
For example, two circles are always similar.
Two squares are always similar:
And two rectangles could be similar:
But will probably not be.
Look at these similar figures:
B is an enlargement of A. The lengths have doubled, but the angles have stayed the same.
Remember: For any pair of similar figures corresponding sides are in the same ratio and corresponding angles are equal.
Look at the diagram below. Figures abcd and ABCD are similar.
Therefore, ie, the sides are in the same ratio.
We can also say that and
These facts can be used when solving problems.
The rectangles pqrs and PQRS are similar. What is the length of PS?
The answer is PS = 15.75cm
Here's how to work it out. We know that the sides are in the same ratio, so:
Include the numbers you already know from the diagram like this:
PS = (7 × 9) ÷ 4 = 63 ÷ 4 = 15.75 cm
Remember : Try to use the formula which has the 'unknown' at the top of the fraction.
wxyz and WXYZ are similar figures. What is the length of XY?
The answer is 4.5cm. Here's how to work it out. Because the shapes are similar we can write
xy = (4 × 9) ÷ 8 = 4.5
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