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Question:

A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, three km north of the north gate, and it can just be seen from a point nine km east of the south gate. What is the diameter of the wall that surrounds the city?

Explanation:

In the figure given below, E is the north gate, A is the south gate and C is the house which can be seen from the point B. AE is the diameter of the wall that surrounds the city.

DB and AB are the tangents to the circle from B.

∴ DB = AB = 9 km

Also, CD² = CE × CA

x2 = 3 × (3 + AE) = 9 + 3AE              ...(i)

Now in ΔABC, (x + 9)² = (3 + AE)² + 9²

∴ x² + 9² + 18x = (3 + AE)² + 9²

∴ x² + 18x = (3 + AE)²                      ...(ii)

Now, start substituting ‘AE’ from the options.

AE = 9, satisfies the equations

Hence, option (b).

Alternatively,

We have 9, AE + 3 and 9 + x, as the sides of a right-angled triangle. The most common Pythagorean triplet is 9, 12, 15. Using this, AE = 9 and x = 6. Substitute these values in the equations and verify.

Hence, option (b).