Find the area of an isosceles trapezoid whose diagonal is 8√2 and makes an angle of 45 degrees with the base.
| August 7, 2021 | education
Through the vertex C we draw a straight line СK parallel to the diagonal ВD.
Angle CKA = BDA = 45, then triangle ACK is rectangular.
By the Pythagorean theorem, we determine the length of the hypotenuse AK.
AK ^ 2 = AC ^ 2 + CK ^ 2 = 128 + 128 = 256.
AK = 16 cm.
The quadrangle ВСКD is a parallelogram, then DК = ВС, and AK = АD + DК = АD + ВС = 16 cm.
In a right-angled triangle ACH, Sin45 = CH / AC.
CH = AC * Sin45 = 8 * √2 * √2 / 2 = 8 cm.
Determine the area of the trapezoid.
Savsd = (ВС + АD) * СН / 2 = 16 * 8/2 = 64 cm2.
Answer: The area of the trapezoid is 64 cm2.
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