In an isosceles trapezoid ABCD: AB = 12, angle B = 120 degrees, AD = 16. Find the area of the trapezoid.
| August 6, 2021 | education
From the vertices B and C of the trapezoid, we omit the heights BH and CK.
Consider a right-angled triangle ABH and determine its angles. Angle ABH = 120 – 90 = 30, then angle BАH = 180 – 90 – 60 = 60 cm.
Leg AH lies opposite angle 30, and therefore is equal to half of the hypotenuse AB. AH = AB / 2 = 12/2 = 6 cm.
Cathetus BH = AB * Sin60 = 12 * √3 / 2 = 6 * √3 cm.
Since the trapezoid is isosceles, the heights BH and CK cut off equal segments on a larger base, AH = DK = 6 cm, then HK = BC = 16 – 6 – 6 = 4 cm.
Determine the area of the trapezoid.
S = (AD + BC) * BH / 2 = (16 + 4) * 6 * √3 / 2 = 60 * √3 cm2.
The area of the trapezoid is 60 * √3 cm2.
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