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# The angle of elevation of the top of a tower from two points at a distance of 4 m and 9m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.

Let AB be the tower. Let C and D be the two points at distances 9 m and 4 m respectively from the base of the tower. Then AC = 9 m, AD = 4m.

Let $$∠ \ ACB \ = \ \theta$$ and $$∠ \ ADB \ = \ 90º \ - \ \theta$$

Let h be the height of the tower AB.

In $$∆ \ ACB$$, we have

$$\frac{AB}{AC} \ = \ tan \theta \$$
$$=> \ \frac{h}{9} \ = \ tan \theta$$ ...... (1)

In $$∆ \ ADB$$, we have

$$\frac{AB}{AD} \ = \ tan(90º \ - \ \theta) \$$
$$=> \ \frac{h}{4} \ = \ cot \theta$$ .. (2)

From (1) and (2), we have

$$\frac{h}{9} \ × \ \frac{h}{4} \ = \ tan \theta \ × \ cot \theta \$$
$$=> \ \frac{h^2}{36} \ = \ 1$$
$$=> \ h^2 \ = \ 36 \$$
$$=> \ h \ = \ 6$$

∴ the height of the tower is 6 metres.