Similarity, Right Triangles, and Trigonometry
Similar triangles
Use equal angles and side ratios in similar triangles, including the ratio of their areas.
Two triangles are similar when they are the same shape but not necessarily the same size — one is a scaled photograph of the other. Every pair of matching angles is equal, and every pair of matching sides is in the same ratio. Congruent triangles are the special case where that ratio happens to be 1 : 1.
One test is enough
If two angles of one triangle match two angles of the other, the triangles are similar — the third pair has to match, since all three add to 180°. That is the AA test, and it is why a tree and a ruler casting shadows at the same moment give you similar triangles for free.
Worked example
Triangle ABC is similar to triangle PQR. AB is 4 and PQ is 12. BC is 5. How long is QR?
AB and PQ are matching sides, so the scale factor is 12 ÷ 4 = 3.
Find the scale factor from the one pair you know completely, before touching any other side.
Areas square the ratio
If the sides are in the ratio 2 : 3, the areas are in the ratio 4 : 9. Doubling the sides of a shape gives four times the area, not twice — an area is two lengths multiplied together, so the scale factor gets used twice. Perimeter, being a length, scales by the plain ratio.
Try it together
Two similar triangles have sides in the ratio 2 : 5, and the smaller one has an area of 12 square units. Let us find the area of the larger.
Sides first, then square the ratio, then scale the area.
1.The ratio of the areas is the ratio of the sides squared. What is 2 × 2?
Have a go
Have a go on your own: two similar triangles have sides in the ratio 1 : 3, and the smaller has an area of 7 square inches. What is the area of the larger, in square inches?
Hint: Three times the sides means nine times the area.
Ready to practice?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.