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Try this beautiful problem from the Pre-RMO, 2019 based on Diameter of a circle.

## Diameter of a circle – PRMO 2019

A village has a circular wall around it, and the wall has four gates pointing north, southeast 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. Find the diameter in meters of the wall that surrounds the village.

• is 107
• is 48
• is 840
• cannot be determined from the given information

### Key Concepts

Pythagoras Theorem

Equations

Integer

But try the problem first…

Source

PRMO, 2019, Question 25

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

or,$AB=\sqrt{AO^{2}-OB^{2}}=\sqrt{(16+r)^{2}-r^{2}}$

$=\sqrt{256+32r}$

or, $AD^{2}+DC^{2}=CA^{2}$

Second Hint

or, $48^{2}+(2r+16)^{2}$

$=(48+\sqrt{256+32r})^{2}$

or, $r^{2}+8r=24\sqrt{256+32r}$

or, $r(r+8)=24(4\sqrt{2})(\sqrt{r+8})$

Final Step

or, r=24

or, 2r=diameter=48.