# The value of one of the angles of the parallelogram is 60 degrees, and the smaller diagonal is 2√31 cm

The value of one of the angles of the parallelogram is 60 degrees, and the smaller diagonal is 2√31 cm. The length of the perpendicular drawn from the point of intersection of the diagonals to the larger side is 0.5√75 cm. Find the lengths of the sides and the larger diagonal of the parallelogram.

Let’s draw the height of the ВC parallelogram.

The triangle DBK and DOH are similar, since both are rectangular and the angle D is common to them.

Then BD / BK = OD / OH. Since at point O the diagonals are divided in half, then ОD = ВD / 2 = 2 * √31 / 2 = √31 cm.

ВK = BD * OH / OD = 2 * √31 * 0.5 * √75 / √31 = √75 cm.

In a right-angled triangle ABK, the angle ABK = 180 – 90 – 60 = 30, then the leg AK is equal to half the length of AB. AB = 2 * AK.

In the triangle ABK, according to the Pythagorean theorem, AB ^ 2 = AK ^ 2 + BK ^ 2.

(2 * AK) ^ 2 = AK ^ 2 + VK ^ 2.

3 * AK ^ 2 = ВK ^ 2 = 75.

AK ^ 2 = 25.

AK = 5 cm, then AB = 2 * 5 = 10 cm.

In a right-angled triangle BKD, according to the Pythagorean theorem, we find the length of the leg KD.

KD ^ 2 = BD ^ 2 – ВK ^ 2 = (2 * √31) ^ 2 – (√75) ^ 2 = 124 – 75 = 49.

KD = 7 cm.Then AD = AK + KD = 5 + 7 = 12 cm.

From the top of C, draw the perpendicular CP to the base of AD. Triangles ABK and DСР are equal in hypotenuse and leg, then DP = AK = 5 cm, and AR = AD + AK = 12 + 5 = 17 cm.

In a right-angled triangle ACP, according to the Pythagorean theorem, AC ^ 2 = AP ^ 2 + CP ^ 2 = 17 ^ 2 + (√75) ^ 2 = 364.

AC = √364 = 2 * √91 cm.

Answer: The sides of the parallelogram are 10 cm, 12 cm, the large diagonal is 2 * √91 cm.

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