Question

A village has a circular wall around it, and the wall has four gates pointing north, south,
east and west. A tree stands outside the village. 16 m north of the north gate, and it can be just
seen appearing on the horizon from a point 48 m east of the south gate. What is the diameter,
in meters, of the wall that surrounds the village?​

Answer 1

Answer:

Diameter of Wall = 48 m

Explanation:

A village has a circular wall

Let say Diameter of Wall =  2R

Then Radius = R

Let say Tree is at G = 16 m North Of North Gate   NG = 16 m

& point of Observation = P  48m east of the south gate PS =  48m

PS is  Tangent

now Tree appearing on the horizon  so Let say PQ would be Tangent which on extend goes upto G

PQ = PS = 48  ( equal Tangent)

Let say O is center of wall

then  OG² = QG² + OQ²

OQ = Radius = R

let say QG = X

OG = R + 16

(R + 16)² = X² + R²

=> X² = 32R + 256

=> X² = 32(R + 8)   - eq 1

in ΔPSG

PG² =  PS²  + SG²

=> (48 + X)²  =  48²  + (2R + 16)²

=> 48² + X²  + 96X =  48²  + 4R² + 256 + 64R

=> 32R + 256 + 96X = 4R² + 256 + 64R

=> 96X = 4R² + 32R

=> 24X = R² + 8R

=> 24X = R(R + 8)  

Squaring both Sides

=> 24²X² = R²(R + 8)²

R²(R + 8)² = 24² * 32(R + 8)

=> R² (R + 8) = 24² * (24 + 8)

=> R = 24

Diameter of Wall = 2R = 2 * 24 = 48 m

Diameter of Wall = 48 m