Similar Shapes

Two shapes are mathematically similar when one is an enlargement of the other. They are exactly the same shape, but they can be different sizes.

In the exam you are usually told that two shapes are similar and asked to work out the length of a side, for 2 marks. These questions often have two parts and come near the end of the paper.

You may also be asked to show that two shapes are similar, or to say whether someone is right about two shapes, for 1 or 2 marks.

The diagrams are not drawn accurately, so never measure them with a ruler. Work every length out from the numbers you are given.

In similar shapes, matching angles are equal. Enlarging a shape does not change its angles, so a 40^\circ angle on the small shape is still 40^\circ on the big shape.

Every length is multiplied by the same number. This number is the scale factor. If one side is doubled, every side is doubled.

Sides in the same position on the two shapes are called corresponding sides. Here the 4 cm side matches the 8 cm side, the 6 cm side matches the 12 cm side, and the 5 cm side matches the 10 cm side.

To find the scale factor, you need one pair of corresponding sides where you know both lengths. Divide the length on the new shape by the matching length on the original shape.

\text{scale factor}=\frac{\text{length on the new shape}}{\text{matching length on the original shape}}

For example, if a 6 cm side matches a 15 cm side, the scale factor from the small shape to the big shape is 15\div6=2.5. A scale factor does not have to be a whole number.

Sometimes you are given the ratio of matching lengths instead, such as 2:3. This means every length on the second shape is 3\div2=1.5 times the matching length on the first shape.

Once you know the scale factor from the small shape to the big shape, you can find any missing length.

• Small shape to big shape: multiply by the scale factor, so the answer is longer.
• Big shape to small shape: divide by the scale factor, so the answer is shorter.

In the exam, one shape is often turned round or flipped over, so corresponding sides are not in the same place on the page. Match the sides using the equal angles, not their position.

Letters help. If angle A equals angle P and angle B equals angle Q, then side AB matches side PQ, BC matches QR, and AC matches PR. Keep the letters in the same order.

To check whether two shapes are similar, divide each length on one shape by the matching length on the other. If every answer is the same, the shapes are similar. If the answers are different, they are not.

Adding the same amount to every side does not make a similar shape. A 2 cm by 4 cm rectangle with 3 cm added to each side becomes 5 cm by 7 cm. The long side was twice the short side, but 7 is not twice 5, so the shape has changed.

For rectangles, match the short side with the short side, and the long side with the long side.

Many exam questions ask for two missing lengths, one on each shape. Find the scale factor once. Then multiply for the length on the big shape, and divide for the length on the small shape.

Write the scale factor down before you use it. It can earn a mark on its own, but it is not the answer: carry on and work out the length you were asked for.

A perimeter is a length too. If every side is multiplied by 3, the perimeter is multiplied by 3 as well.

Sometimes a small triangle sits inside a bigger one and shares a corner with it. In triangle ABC below, D is on AB, E is on AC, and DE is parallel to BC. The arrows show the parallel lines.

Triangles ADE and ABC are similar. Angle A is in both triangles, and angle ADE equals angle ABC because they are corresponding angles on parallel lines.

Draw the two triangles separately and label the sides you know. Always use whole sides: AD matches AB, not DB.

• Adding or subtracting instead of multiplying. Similar shapes are linked by multiplying, so find the scale factor by dividing two matching sides.
• Giving the scale factor as the answer. Use the scale factor to work out the length the question asks for.
• Multiplying when you should divide. Multiply for a length on the bigger shape and divide for a length on the smaller shape.
• Matching the wrong sides. On a turned or flipped shape, match sides using the equal angles or the letters, not their position.
• Using part of a side. In a triangle inside a triangle, compare whole sides, such as AD with AB, not AD with DB.
• Measuring the diagram. The diagrams are not drawn accurately, so work lengths out from the numbers given.