Two Triangles, One Height

Part of the Right-Triangle Trigonometry section of Coddy's Trigonometry journey. Lesson 38 of 50.

Some problems need two right triangles that share a side. A tower stands between two observers on level ground. From P, 40 m from its foot, the top is at 50°. From Q, on the other side, the top is at 30°. How far is Q from the tower?

$A tower of height h between P, 40 m away at 50°, and Q, x away at 30°$

Neither triangle alone answers it. The left one gives the height: tan(50°) = h/40, so h = 40tan(50°). The right one links the height to x: tan(30°) = h/x. Put the first into the second and x is the only unknown: x = 40tan(50°)/tan(30°), about 82.57 m, and h is about 47.67 m.

Keep h exact while you carry it across. Any error in h is divided by tan 30°, which is below 1, so it grows; carry the exact value and round once, at the end.

Challenge

A tree stands between two people on level ground. P is 30 m from its foot and sees the top at 40°. Q, on the other side, sees the top at 25°. Find the height h of the tree and Q's distance x from it, both to two decimal places. Then press Check.

Try it yourself

$Use the figure to find x,h$