1988 AHSME Problems/Problem 24
Problem
An isosceles trapezoid is circumscribed around a circle. The longer base of the trapezoid is , and one of the base angles is . Find the area of the trapezoid.
Solution
Let the trapezium have diagonal legs of length and a shorter base of length . Drop altitudes from the endpoints of the shorter base to the longer base to form two right-angled triangles, which are congruent since the trapezium is isosceles. Thus using the base angle of gives the vertical side of these triangles as and the horizontal side as . Now notice that the sides of the trapezium can be seen as being made up of tangents to the circle, and thus using the fact that "the tangents from a point to a circle are equal in length" gives . Also, using the given length of the longer base tells us that . Solving these equations simultaneously gives and , so the height of the trapezium is . Thus the area is , which is .
See also
1988 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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