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Answer :
The correct answer is:
a. √ab meters
The angles of elevation of the top of a tower from two points at distances 'a' and 'b' meters from the base and in the same straight line with it are complementary.
This means that the sum of the angles of elevation from these two points to the top of the tower is 90 degrees. When two angles are complementary, the sum of their measures is 90 degrees.
Let's denote the height of the tower as 'h' meters.
From the first point at a distance 'a' meters, the angle of elevation to the top of the tower is \( \theta \). From the second point at a distance 'b' meters, the angle of elevation to the top of the tower is \( 90 - \theta \) since the angles are complementary.
Now, we can set up two right triangles with the height 'h' being the opposite side, and the distances 'a' and 'b' being the adjacent sides with the respective angles.
For the first right triangle:
[tex]\[ \tan(\theta) = \frac{h}{a} \][/tex]
For the second right triangle:
[tex]\[ \tan(90 - \theta) = \cot(\theta) = \frac{h}{b} \][/tex]
Using the trigonometric identity [tex]\( \cot(\theta) = \frac{1}{\tan(\theta)} \)[/tex], we can rewrite the second equation as:
[tex]\[ \frac{1}{\tan(\theta)} = \frac{h}{b} \][/tex]
[tex]\[ \tan(\theta) = \frac{b}{h} \][/tex]
Now, we can equate the two expressions for [tex]\( \tan(\theta) \)[/tex] from both triangles:
[tex]\[ \frac{h}{a} = \frac{b}{h} \][/tex]
[tex]\[ h = \sqrt{ab} \][/tex]
Therefore, the height of the tower is [tex]\( \sqrt{ab} \)[/tex] meters.
The complete question is:The angles of elevation of the top of a tower from two points at distances ‘a’ and ‘b’ metres from the base and in the same straight line with it, are complementary. The height of the tower is:
a. √ab metres
b. ab metres
c. a/b metres
d. (a+b) metres
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