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Maths

Congruent and similar shapes

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We are going to look at:

Similar shapes

Similar figures are identical in shape, but not in size.

image: circle

For example, two circles are always similar.

image: 2 squares

Two squares are always similar:

image: 2 rectangle

And two rectangles could be similar:

image: 2 rectangle

But will probably not be.

Similar figures

Look at these similar figures:

Shape A: Five sided shape, from clockwise directions mesurements from top: 2cm, left-angle: 225 degrees, right-side: 4cm, bottom-left corner: 45 degrees, bottom: 5cm, left-side: 7cm. Shape B: Five sided shape, from clockwise directions lengths from top: 4cm, left-angle: 225 degrees, right-side: 8cm, bottom-left corner: 45 degrees, bottom: 10cm, left-side: 14cm.

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.

images: 4 sided shape: left shape: top length: corners BC:4cm, right-side: CD: corners CD: both 90 degree angles, bottom length: DA: 8cm, left-side: at an angle: AB. Right shape: top length: corners BC:6cm, right-side: CD: corners BC:4cm, right-side: CD, corners CD: both 90 degree angles, bottom length: DA: 12cm, left-side: at an angle: AB

ad over AD = 8 over 12 = 2 over 3 and bc over BC = 4 over 6 = 2 over 3

Therefore, ad over AD = bc over BC   ie, the sides are in the same ratio.

We can also say that ad over AD = 12 over 8 = 3 over 2 and bc over BC = 6 over 4 = 3 over 2

and therefore AD over ad = BC over bc

These facts can be used when solving problems.

Question
image: two rectangles, left rectangle: top-left corner: Q, top-right: R, bottom-right: S, bottom-left: P, length QP: 4cm, PS: 9cm. Right rectangle: same labels, QP: 7cm, PS: ?.

The rectangles pqrs and PQRS are similar. What is the length of PS?

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Answer

The answer is PS = 15.75cm

Here's how to work it out. We know that the sides are in the same ratio, so:

PS pver ps = PQ over pq

Include the numbers you already know from the diagram like this:

PS over 9 = 7 over 4

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.

Question
image: two four-sided shape: top-right: X, right: Y, bottom-left: Z, left: W, length WX: 8cm, XY: 4cms, corner X: 57 degrees. Right shape: same lettered labels: length WX: 9cm.

wxyz and WXYZ are similar figures. What is the length of XY?

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Answer

The answer is 4.5cm. Here's how to work it out. Because the shapes are similar we can write

9 over 8 = xy over 4

xy = (4 × 9) ÷ 8 = 4.5

Question

What is the size of angle wxy ?

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Answer

Remember that the angles in similar figures stay the same. Sowxy is 57°.

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