Question

Q1. Help...!!
The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m. 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.

Answer 1

Explanation:

Question :-

The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m. 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.

Solution :-

Let AQ be the tower and R, S are the points 4m, 9m away from the base of the tower respectively.

The angles are complementary. Therefore, if one angle is θ, the other will be 90 - θ.

1st Equation :-

In ΔAQR,

$\large \sf \frac{AQ}{QR} </p><p></p><p></p><p> .</p><p> = tanθ</p><p></p><p></p><p>$

2nd Equation :-

In ∆AQS,

$\large \sf \frac{AQ}{SQ} </p><p></p><p> = tan(90 - tanθ)$

$\large \sf \frac{AQ}{9} </p><p></p><p> </p><p> = cotθ</p><p>$

Next...

On multiplying equations (1st) and (2nd), we obtain;

$\large \sf (\frac{AQ}{4})(\frac{AQ}{9})</p><p> </p><p> ) = (tanθ) · (cotθ)</p><p>$

$\large \sf \frac{AQ²}{36} </p><p></p><p> </p><p> = 1$

AQ² = 36

AQ = √36 ± 6

However, height cannot be negative.

Therefore, the height of the tower is 6 m.

Thank You*

Answer 2

Height of Tower = 6 metre

Given Terms:

• Angles of elevation at two points are complementary.
• Distance of two points from the base of tower is 4 and 9 m.

Need To Prove:

• Height of tower is 6 metre.

Proof: Let the height of tower be x metre. In the attached diagram we have

• AB is tower of 6 m.
• BC is first point = 4 m.
• BD is second point = 9 m.

• ∠ACB = θ
• ∠ADB = 90° – θ

We'll use three formulae here

1. Tanθ = Perpendicular/ Base
2. Tan(90 – θ) = cotθ
3. tanθ × cotθ = 1

In triangle ACB

➥ tanθ = AB/BC

➥ tanθ = h/4

➥ 4tanθ = h equation (i)

In triangle ADB

➥ tan(90 – θ) = AB/BD

➥ cotθ = h/9

➥ 9cotθ = h equation (ii)

Now multiply both the equations , we will get

➼ 4tanθ × 9cotθ = h × h

➼ 36 = h²

➼ √36 = h

➼ 6 = h

Here it is proved that the height of tower is 6 metre.