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Question 11
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, 3 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?
Solution
B is the south gate. A is the house and BC = 9 km.
In triangle ABC and triangle AMO
ABC = OMA = 90 degree
BAC = OAM = Common angle
Triangle ABC is similar to triangle AMO.
$$\frac{(2r+3)}{\sqrt{(r+3)^2-r^2}} = \frac{9}{r}$$
r = 9/2
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