Find the height of the trapezoid if its diagonals are mutually perpendicular and equal to 15 and 20.
| May 23, 2021 | education
Through the vertex C of the trapezoid ABCD, construct a segment CE parallel to the diagonal BD.
The quadrangle ВСКD is a parallelogram, then DК = ВС, and then AK = АD + DК = АD + ВС.
The area of the triangle ACK is equal to: Sask = (AD + BC) * CH / 2, which is equal to the area of the trapezoid.
The ACK triangle is rectangular, then Sask = AC * CK / 2 = 15 * 20/2 = 150 cm2.
By the Pythagorean theorem, AK ^ 2 = AC ^ 2 + CK ^ 2 = 225 + 400 = 625.
AK = 25 cm.
Then Sask = AK * CH / 2.
150 * 2 = 25 * CH.
CH = 300/25 = 12 cm.
Answer: The length of the height of the trapezoid is 12 cm.
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