Using similar triangles

How do you measure something you can't reach, like a tower, a mountain, or the length of a shadow? Find two similar triangles, one you can measure and one you can't, and use the same ratio.

Shadows. At the same moment, the Sun's rays hit everything at the same angle. A 6 m pole casts a 4 m shadow, and a tower casts a 28 m shadow. The triangles are similar (each has a right angle and the same Sun angle), so height/shadow is the same: h/28 = 6/4, and the tower is 42 m tall.

The girl and the lamp-post. A girl 0.9 m tall walks 4.8 m away from a lamp 3.6 m high. Her shadow x makes a small triangle inside a big one: (4.8 + x)/x = 3.6/0.9 = 4, so 4.8 + x = 4x and x = 1.6 m.

A triangle inside a triangle: a line parallel to the base makes a small triangle similar to the big one (AA: the top angle is shared, and the parallel line gives equal corresponding angles).

Proving a product of lengths. If D is on BC and ∠ADC = ∠BAC, then triangles ADC and BAC are similar (that angle plus the common ∠C). Their sides give CA/CB = CD/CA, so CA² = CB · CD.

Medians and bisectors. If ΔABC ~ ΔPQR, their corresponding medians are in the same ratio as the sides (the book's Example 8). The same goes for corresponding angle bisectors (Exercise 6.3, Q10). Prove it with a second pair of similar triangles each time you use it.

RHS: two right triangles are similar if their hypotenuses and one other pair of sides are in the same ratio.

Every time: spot the similar triangles → name the criterion → write the letters in order → set up the ratio.