# 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.