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Home > Maths II > Measure and Shape > Similarity

Maths II

Similarity

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Similar shapes

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

For example, two circles are always similar -

Two circles. A big orange one followed by a smaller yellow one.

Two squares are always similar -

Two squares. A big orange one followed by a smaller yellow one.

And two rectangles could be similar -

Two rectangles of the same proportions. A big orange one followed by a smaller yellow one.

But often will not be -

Two rectangles of different proportions and roughly equals area. The first is orange and the second is yellow.

Similar figures

Look at these similar figures -

Two shapes of the same shape but different sizes. Shape A is 2 centimetres at the top, 7 centimetres down the left, 4 centimetres down the right, 5 centimetres along the bottom, then has a diagonal edge joining the right of the bottom edge to the bottom of the right, 4 centimetre edge. The angle of this line is at 45 degrees, and the other inner angle created is 225 degrees at this join. Shape B has all the lengths doubled and the angles are the same.

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.

Two quadrilateral shapes, of the same shape but different sizes. They both have two right angles on the right side. The first shape is 4 centimetres along the top, between corners b and c, and 8 centimetres along the bottom, between corners a and d. The second shape has the equivalne measurements of 6 centimetres on top and 12 centimetres along the bottom.

{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}, i.e. 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},

therefore, {AD \over ad} = {BC \over bc}.

These facts can be used when solving problems.

Question
Two rectangles of the same proportions. The first rectangle has a width of 4 centimetres and a length of 9 centimetres. The larger rectangle has a width of 7 centimetres and the length of the length is indicated by a question mark.

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

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 \over ps} = {PQ \over pq}

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

{PS \over 9} = {7 \over 4}

PS = (7 \times 9) \div 4 = 63 \div 4 = 15.75cm

Remember : Try to use the formula which has the 'unknown' at the top of the fraction.

Question
Two quadrilaterals of the same shape and proportions. The angles are labelled w, z, x and y. Angles w and z are right angles. Angle x is 57 degrees. The top edge between w and x on the first shape is 8 centimetres and between x and y is 4 centimetres. The top edge os the second shape is 9 centimetres.

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

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 \times 9) \div 8 = 4.5cm

Question

What is the size of angle The lettwe w, x and y in sequence. The x has a carrot mark over it.?

Answer

Remember that the angles in similar figures stay the same. So The lettwe w, x and y in sequence. The x has a carrot mark over it. is 57°.

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