In an isosceles trapezoid, the bases are 3cm and 5cm, and the side is 7cm. Calculate the diagonals and area

univerkov | November 23, 2020 | education

By condition, trapezoid ABCD is isosceles, sides AB and CD are 7 cm, BC = 3 cm, AD = 5 cm, AD and BC are bases.
Let’s find the area of the trapezoid. According to the formula, the area of a trapezoid is equal to the product of half the sum of the bases by the height. Draw heights BK and CM from points B and C. The side KM is equal to the side BC = 3 cm. The side AK is equal to the side MD = 1 cm. Then, by the Pythagorean theorem, we find the height CM: CM ^ 2 = CD ^ 2-MD ^ 2 = 7 ^ 2 – 1 ^ 2 = 48. CD = root of 48 = 4 roots of 3. The area of the trapezoid is S = (5 + 3) / 2 * 4 roots of 3 = 16 roots of 3 cm ^ 2.
Find the diagonal of the trapezoid. The diagonals of an isosceles trapezoid are equal. Let’s pay attention to the triangle ACM. The angle AMC in it is rectangular, so we can find AC by the Pythagorean theorem.
AC ^ 2 = AM ^ 2 + CM ^ 2 = (4 roots of 3) ^ 2 + 4 ^ 2 = 64
AC = root of 64 = 8
Answer: the area is 16 roots of 3
diagonal is 8

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