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Find the exact area of the given trapezoid in the figure:
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| Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure above
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| Photo by Math Principles in Everyday Life |
Since the given figure is an isosceles trapezoid, then it follows that ∠A ≅ ∠B, ∠C ≅ ∠D, and AD ≅ BC. If a circle is inscribed in an isosceles trapezoid, then its radius is tangent to the sides of an isosceles trapezoid. Let's analyze and label further the given figure as follows
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| Photo by Math Principles in Everyday Life |
Line segment EG that passes the center of a circle bisects the two bases of an isosceles trapezoid. Line segment OB bisects ∠B and line segment OC bisects ∠C.
Consider rt. ∆OGB and rt. ∆OFG. If OG ≅ OF and OB ≅ OB, then it follows that BG ≅ BF. In this case, BF = 9.
Consider rt. ∆OEC and rt. ∆OFC. If OE ≅ OF and OC ≅ OC, then it follows that EC ≅ CF. In this case, CF = 4.
Hence, the length of CB = CF + FB = 9 + 4 = 13.
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| Photo by Math Principles in Everyday Life |
The value of x is
By Pythagorean Theorem, the altitude of an isosceles trapezoid is
Therefore, the area of an isosceles trapezoid is
