applet for HL Problem #99

Area of triangle around 2 circles

HL Problem-of-the-Day #99

Consider the diagram at right.  The larger circle has a radius that is twice the smaller circle’s radius.  The larger circle is tangent to the two legs and hypotenuse of a right triangle, and the smaller circle is tangent to one leg and the hypotenuse.  The distance between the points where the circles are tangent to the same leg is d, and the radius of the smaller circle is r.  Show that the area of the right triangle is equal to \(4dr\left( {\frac{{d + r}}{{d - r}}} \right)\).

Geogebra applet

In the Geogebra applet below, you can change the radius, \(r\), of the smaller circle, and you can also change the location of either circle's point of tangency on the horizontal leg of the right triangle (the endpoints of the segment of length \(d\) - the horizontal distance between the centers of the circles).  Some questions to investigate: (1) Can the two circles overlap?  (2) What happens when  \(d = r\)?

click here for the two-page document for HL Problem-of-the-Day #99 that includes a worked solution

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