Question

The angles of elevation of the top of a tower from two points at a distances a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is $square root of ab$ metres.

Solution

Let AB be the tower of height h metres, D and C are two points on the horizontal line, which are at distances a and b metres respectively from the base of the tower. It is also given that the angles of elevation of the top of a tower from two points D and C be complementary i.e.,
∠ADB = Ս then ∠ACB = (90 -ө)

In right triangle ADB, we have
$tan space straight theta space equals space AB over BD rightwards double arrow space space space tan space straight theta space equals space straight h over straight a space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis straight i right parenthesis$
In right triangle ACB, we have
$tan space left parenthesis 90 degree space minus space straight theta right parenthesis space equals space AB over BC rightwards double arrow space space space space space cot space straight theta space space space space equals space straight h over straight b space space space space space space space space space space space space space space space space space space space space space... left parenthesis ii right parenthesis$
Multiplying (i) and (ii), we get
$tan space straight theta space straight x space cot space equals space straight h over straight a straight x straight h over straight b rightwards double arrow space space 1 space equals space straight h squared over ab rightwards double arrow space space space straight h squared space space equals space ab rightwards double arrow space space space straight h space equals space plus-or-minus space square root of ab$
But height can't be negative.
$apostrophe therefore space space space space space space space space space space space space space space straight h space equals space square root of ab$
Hence the height of the tower is $square root of ab space$ mts.

For Daily Free Study Material Join wiredfaculty

Circles

The angles of elevation of the top of a tower from two points at a distances a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is $square root of ab$ metres.

Some More Questions From Circles Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

Circles

The angles of elevation of the top of a tower from two points at a distances a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is metres.

Some More Questions From Circles Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

The angles of elevation of the top of a tower from two points at a distances a and b metres from the base and in the same straight line with it are complementary. Prove that the height of the tower is $square root of ab$ metres.