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Rectangle and Circle

Suppose we have a circle inscribed within a square. Now suppose a rectangle extends from one corner of the square to a point on the circle (see diagram).

If this rectangle is 6 inches by 12 inches, what is the radius of the circle in inches?

Solution

If we put the circle at the origin its equation would be

ReplyDelete$latex x^2+y^2=r^2$

the coordinate of the corner of the rectangle touching the circle would then be.

$latex -r + 12, r-6$

If we now substitute those coordinates in the circle equation we get

$latex (r-6)^2+(12-r)^2=r^2$

expanding and solving for r we get

$latex r^2-36 r+180=0$

giving r = 6 or 30 and clearly it can't be 6, so we get r=30 inches

Paul.