# How would you solve this problem: The angle of elevation of the top of a tree from point P due west of the tree is 40 degrees. From a second point Q due east of the tree, the angle of elevation is 32 degrees. If the distance between P and Q is 200m, find the height of the tree, correct to four significant figures?

One angle is 40 degrees, the other angle is 32 degrees, therefore the third angle opposite the base PQ is 180 - (32 + 40) = 108 degrees.

You know one side of the triangle has length PQ = 200 m

A right angled triangle is formed between point P, the top of the tree and its base and also point Q, the top of the tree and its base.

The best way to solve is to find the hypotenuse of one of the triangles.

So use the triangle with vertex P.

Call the point at the top of the tree T

Call the height of the tree H

The angle formed between sides PT and QT was worked out as 108 degrees.

Using the Sine Rule, PQ / Sin(108) = PT/ Sin(32)

So for the right angled triangle we choose

PT the hypotenuse is PQSin(32) / Sin(108)

Sin(40) = H / hypotenuse

So H = Hypotenuse x Sin(40)

Substituting the value for the hypotenuse we calculated above gives

H = (PQSin(32) / Sin(108)) x Sin(40)

= PQSin(32)Sin(40)/Sin(108)

= 71.63 m

Updated on May 31, 2018

### Original Article:

By Eugene Brennan
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