Quadrilateral ABCD is a trapezoid. Find its area by making the necessary measurements.

| March 28, 2021 | education

Quadrilateral ABCD is a trapezoid. Find its area by making the necessary measurements. The base is 2cm, the sides are 4cm, the bottom base is 6cm.

Given: Trapezoid ABCD, where AB || CD, AB = 6 cm, AD = BC = 4 cm, CD = 2 cm. It is required to find the area of ​​the trapezoid.
Let’s draw the heights DE ⊥ AB and CF ⊥ AB. By construction, ∠AED = ∠CFB = 90 °. In addition, EF = CD = 2 cm.
Since AD ​​= BC, then AE = FB. Therefore, since AE + EF + FB = AB, then 2 * AE = AB – EF = 6 cm – 2 cm = 4 cm, whence AE = (4 cm): 2 = 2 cm.
Obviously, ΔAED is a right-angled triangle with hypotenuse AD and legs DE and AE. According to the Pythagorean theorem, AD² = DE² + AE², whence DE² = AD² – AE² = (4 cm) ² – (2 cm) ² = (16 – 4) cm² = 12 cm². Therefore, DE = √ (12 cm²) = √ (12) cm = 2√ (3) cm.
Thus, the area of ​​the trapezoid is (6 cm + 2 cm) * (2√ (3) cm) / 2 = 8√ (3) cm².
Answer: 8√ (3) cm².



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