In parallelogram ABCD, side AB is equal to 2√5, side BC is equal to 5√2. Point M is the middle of AD

| June 16, 2021 | education

In parallelogram ABCD, side AB is equal to 2√5, side BC is equal to 5√2. Point M is the middle of AD, segment BM is perpendicular to the AC diagonal. Find the diagonal BD

Consider a triangle ABD, which BM has a median, since it divides the AD side in half, AO has a median, like half the diagonal of a parallelogram.

Let the segment OH = X cm, then by the property of the medians, AH = 2 * X cm, MH = Y m, then BH = 2 * Y cm.

Triangles ABN and AMH are rectangular, since BM is perpendicular, according to the condition, to the AC.

Let us express AH in both triangles by the Pythagorean theorem.

AH ^ 2 = AB ^ 2 – BH ^ 2 = (2 * √5) ^ 2 – (2 * Y) ^ 2 = 20 – 4 * Y ^ 2.

AH ^ 2 = AM ^ 2 – MH ^ 2 = (2.5 * √2) ^ 2 – (Y) ^ 2 = 12.5 – Y ^ 2.

Then 20 – 4 * Y ^ 2 = 12.5 – Y ^ 2.

3 * Y ^ 2 = 7.5.

Y ^ 2 = 2.5.

In a right-angled triangle ABH, according to the Pythagorean theorem, AH ^ 2 = AB ^ 2 – BH ^ 2.

4 * X² = 20 – 4 * Y² = 20 – 10 = 10.

X ^ 2 = 2.5.

From the right-angled triangle BOH, according to the Pythagorean theorem, BO ^ 2 = BH ^ 2 + OH ^ 2 = 4 * Y ^ 2 + X ^ 2 = 4 * 2.5 + 2.5 = 10 + 2.5 = 12.5.

BО = √12.5 = 2.5 * √2 cm.

Then BD = 2 * BO = 5 * √2 cm.

Answer: The length of the diagonal is 5 * √2 cm.



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