Similar Figures and Scale Factor

Two figures are similar when one is an enlargement of the other. Their angles are equal, and every pair of matching sides is in the same ratio. The second triangle here has been flipped over and turned, so you cannot match sides by where they are on the page. Match the angles instead. The single arc at A matches the single arc at R, the double arc at B matches P, and the triple arc at C matches Q. So side A B goes with side R P, B C goes with P Q, and C A goes with Q R. Six over four, nine over six and twelve over eight all equal one point five, so the triangles are similar.

The scale factor tells you how many times bigger the image is than the original. Divide an image length by the matching original length. Twelve divided by four is three, and nine divided by three is also three, so the scale factor is three. If you go the other way, from the big rectangle back to the small one, the scale factor is four divided by twelve, which is one third. A scale factor bigger than one makes an enlargement, and a scale factor between zero and one makes a reduction.

To find a missing side, first find the scale factor from a pair of sides you know completely. D E is four and its match K L is seven, so k is seven divided by four, which is one point seven five. The side x matches D F, which is six. Multiply: six times one point seven five is ten point five. You can also write it as a ratio: x over six equals seven over four. Either way, x is ten point five.

At the same moment, the sun's rays hit the ground at the same angle everywhere nearby. Each upright object and its shadow make a right-angled triangle, and those triangles have equal angles, so they are similar. The person is one point six metres tall with a two metre shadow, so height divided by shadow is zero point eight. The tree's shadow is fifteen metres, so its height is zero point eight times fifteen, which is twelve metres.

When a line inside a triangle is parallel to one side, it cuts off a small triangle that is similar to the big one. Triangle A D E and triangle A B C share the angle at A, and because D E is parallel to B C, the angles at D and B are equal corresponding angles. The trap is the matching. A D in the small triangle matches the whole side A B in the big triangle, which is four plus six, or ten, not just the six. So k is ten divided by four, which is two point five, and B C is five times two point five, which is twelve point five. Quiz and exercises now.