Similar Triangles

Similarity is found in scale models, blueprints, maps, microscopes, and when enlarging or reducing a photocopy. All of the angles are exactly the same size, so the object looks exactly like the original, only larger or smaller. If we know the scale factor and the side measures of one triangle, we can easily find the side measures of the other triangle. These blue triangles have a scale factor of 3/4.

EXAMPLE

The triangles below share an corner angle and both have right angles. Therefore the third angles are also congruent by the sum of interior angles. We find the scale factor by setting a ratio of corresponding sides 8/12 = 3/4. This means that the smaller triangle is 3/4ths the size of the larger triangle. The other sides of the triangles are also in proportion. The base of the large triangle is 28 inches, and we find the base side of the small triangle by taking (3/4)(28) = 21 inches. If needed, we can find the third side measurement using the Pythagorean Theorem.

EXAMPLE

Julie wants to know how tall her grandfather's maple tree is. She asks Susie to help her out by standing next to the tree so her shadow and the shadow of the tree coincide. Using similar triangles, and the fact that Susie is five feet tall, Julie figures that the ratio of the lengths of their shadows on the ground is the same ratio as their heights. (Add the 10 + 6 to get the tree's shadow length)

Julie measures Susie's shadow as six feet as she stands ten feet away from the tree. She creates the proportional equation h/16 = 5/6 and concludes that the maple tree is 13.3 feet high. Do you agree?