Question

The angle of elevation of the top of a tower observed from each of the three points A, B, and C on the ground forming a triangle is same angle β. If R is the circumradius of triangle ABC, then show that the height of the tower is (R tanβ).

Answer 1

Let

OP be the tower.

Do check the above diagram for your reference ⤴

We know,

Tower makes an equal angle at the vertices of the triangles.

We conclude,

Foot of the tower lies on circumcenter.

We have,

In triangle (DAP),

tan(Beta) = OP/OA,

So,

OP = OA*tan(Beta).... (1)

Given,

OA = R..... (2)

From (1) and (2),

OP = R*tan(Beta)

Hence proved!

In these questions, assumptions are the key point for a better picturizing of the situation,

Like in this question assuming the height of the tower in the figure did a pretty good job for our understanding,

A student should also be aware of the important properties of mathematical topics, especially circles.

Answer 2

Refer to the attachment .

β is the angle marked in the figure .

Let our circum-radius be R .

A , B and C are the vertices of the triangle .

Let OH be the tower .

Hence we have to prove that OH = R tan β

In Δ OAH ,

tan β = height/base

⇒ tan β = OH/R

⇒ R tan β = OH

Hence the height of the tower is R tan β .

Hence proved !