# In parallelogram ABCD, a perpendicular dropped from the apex of an obtuse angle B to side AD divides it in half.

In parallelogram ABCD, a perpendicular dropped from the apex of an obtuse angle B to side AD divides it in half. Find the diagonal BD if the perimeter of the parallelogram is 20 cm and the perimeter of the triangle ABD is 16 cm.

Let the sides of the parallelogram AB = СD = X cm, and BC = AD = Y cm.
Then:
2 * X + 2 * Y = 20 cm.
X + Y = 10 cm. (1).
Since, according to the condition, the height of the VN divides the base of the blood pressure in half, the AВD triangle is isosceles, AB = ВD.
Then Ravd = AB + ВD + AD = X + X + Y = 16 cm.
2 * X + Y = 16 cm. (2).
Let us solve the system of equations 1 and 2 by the substitution method.
X = 10 – Y.
2 * (10 – Y) + Y = 16.
20 – 2 * Y + Y = 16.
Y = BC = AD = 20 – 16 = 4 cm.
X = AB = СD = ВD = 10 – 4 = 6 cm.
Answer: The length of the ВD diagonal is 6 cm.

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