The lateral side of the trapezoid is equal to five roots of 2 cm, it forms an angle of 45 degrees with a large base. The bases are 12cm, 20cm. Calculate the area of the trapezoid.
Consider a right-angled triangle formed by this lateral side, its projection onto the larger base and the height of the trapezoid. The lateral side is the hypotenuse, its projection and height are the legs. The leg, equal to the height of the trapezium, lies opposite the angle formed by the lateral side and the large base. This angle is 45 °. The ratio of the opposite leg to the hypotenuse is the sine of the angle. Means:
h / 5√2 = sin 45 °.
Hence, h = sin 45 ° * 5√2 = (√2 / 2) * 5√2 = 5 cm – the height of the trapezoid.
The area of the trapezoid is equal to the product of the half-sum of the bases and the height:
S = h * (a + b) / 2 = 5 * (12 + 20) / 2 = 5 * 16 = 80 cm2 – the required area of the trapezoid.
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