In trapezoid ABCD (BC is parallel to AD), AB is perpendicular to BD. BD = 2√5, AD = 2√10.

| January 20, 2021 | education

In trapezoid ABCD (BC is parallel to AD), AB is perpendicular to BD. BD = 2√5, AD = 2√10. CE is the height ΔBCD, and tg ECD = 3. Find BE.

From the right-angled triangle AВD, by the Pythagorean theorem, we determine the length AB.

AB ^ 2 = AD ^ 2 – ВD ^ 2 = 40 – 20 = 20.

AB = 2 * √5 cm.

AB = ВD, then the AВD triangle is rectangular and isosceles. Then the angle ВAD = ADВ = 450.

The triangle is ВСЕ rectangular, in which the angle CBЕ = ADВ as criss-crossing angles at the intersection of parallel straight lines, then the angle CBE = ADВ = 450. Then the triangle is ВСЕ rectangular and isosceles, BE = CE.

Let the length of the segment BE = X cm, then DE = (2 * √5 – X) cm.

From the right-angled triangle СDE, tgEСD = 3 = DE / CE = DE / BE = (2 * √5 – X) / X.

3 * X = 2 * √5 – X.

4 * X = 2 * √5.

X = BE = 2 * √5 / 4 = √5 / 2 cm.

Answer: The length of the segment BE is equal to √5 / 2 cm.



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