A circle is inscribed in a rectangular trapezoid, the radius of which is 3 dm.

| March 11, 2021 | education

A circle is inscribed in a rectangular trapezoid, the radius of which is 3 dm. Find the perimeter of the trapezoid if the difference between the larger and smaller base is 8 dm.

A circle can be inscribed into a quadrilateral only if the sums of the lengths of its opposite sides are equal. Therefore, if a circle is inscribed in a trapezoid, then the sum of the lengths of its bases is equal to the sum of the lengths of the lateral sides. The height of the trapezoid, and in this case the side perpendicular to the bases, is equal to the diameter of the inscribed circle: h = 2r = 6 dm.
The difference in the lengths of the larger and smaller bases is the projection of the inclined lateral side onto the larger base.
Consider a right-angled triangle in which the oblique side of the trapezium is the hypotenuse, the legs are the height of the trapezium and the projection of the side. The sum of the squares of the legs is equal to the square of the hypotenuse, which means that the square of the side is 6 ^ 2 + 8 ^ 2 = 36 + 64 = 100, the side is √100 = 10 dm.
The sum of the lengths of the sides: 6 + 10 = 16 dm. This means that the sum of the lengths of the bases is also 16 dm, respectively, the perimeter of the trapezoid is 16 + 16 = 32 dm.



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