In the trapezoid, the bases are 24 and 12 and the edges are 10.A circle of radius 1 is inscribed in each corner

| June 3, 2021 | education

In the trapezoid, the bases are 24 and 12 and the edges are 10.A circle of radius 1 is inscribed in each corner of the trapezoid. find the area of a quadrilateral with vertices at the centers of these circles

Let us omit the height CH of the trapezoid ABCD, and in the right-angled triangle CHD we determine the length of CH.

НD = (АD – ВС) / 2 = (24 – 12) / 2 = 6 cm.

CH ^ 2 = CD ^ 2 – HD ^ 2 = 10 ^ 2 – 6 ^ 2 = 100 – 36 = 64.

CH = 8 cm.

Determine the area of ​​the trapezoid ABCD.

S1 = (AD + BC) * CH / 2 = (24 + 12) * 8/2 = 144 cm2.

The figure formed by the centers of the circles is a trapezoid.

Let us omit the height С1Н1 of the trapezium A1В1С1D1, the length of which is less than the height СН by two radii of the inscribed circles. C1H1 = CH – R – R = 8 – 1 – 1 = 6 cm.

Trapezium ABCD is similar to trapezoid A1B1C1D1, the coefficient of similarity of which is equal to:

K = C1H1 / CH = 6/8 = 3/4.

The ratio of the areas of similar figures is equal to the square of the similarity coefficient.

S2 / S1 = (3/4) 2 = 9/16.

S2 = S1 * 9/16 = 144 * 9/16 = 81 cm2.

Answer: The area of ​​the figure is A1B1C1D1 = 81 cm2.



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