SOLUTION: A road is tangent to a circular lake. Along the road and 12 miles from the point of tangency. Another road opens towards the lake. From the intersection of the two roads to the pe
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Question 1146428: A road is tangent to a circular lake. Along the road and 12 miles from the point of tangency. Another road opens towards the lake. From the intersection of the two roads to the periphery of the lake, the length of the new road is 11 miles. If the new road will be prolonged across the lake, find the length of bridge to be constructed. Answer by greenestamps(13200) (Show Source):
When you draw the figure for this problem, what you have is a circle with a tangent (line from an external point just touching the circle) and a secant (line from the same external point, passing through the circle.)
The operating principle here (which can be proved using similar triangles) is that the square of the length of the tangent is equal to the product of the length of the external part of the secant and the length of the entire secant.
In this problem, the 12 miles is the tangent length, the 11 miles is the length of the external part of the secant. The unknown x is the length of the part of the secant inside the circle -- the bridge that is to be built.
Applying the basic principle with those numbers gives us
