In parallelogram ABCD, the bisector of angle A intersects the BC side in a ratio of 1: 2, respectively.

| September 2, 2021 | education

In parallelogram ABCD, the bisector of angle A intersects the BC side in a ratio of 1: 2, respectively. Find AB if the parallelogram perimeter is 40.

The bisector of the angle BAD divides it into two equal angles, the angle BAK = DAK. The BKA angle and the KAD angle are equal as cross-lying angles at the intersection of parallel straight lines ВС and АD of the secant AK.

Then the angle BKA = BAK, and then the triangle ABK is isosceles and AB = BK.

Let the length of the segment CK = X cm, then, by condition, BK = 2 * X cm, and since ABK is isosceles, then AB = BK = 2 * X cm.

The length of the segment BC = BK + CK = 2 * X + X = 3 * X cm.

Since in a parallelogram the opposite sides are equal, then CD = AB = 2 * X cm, AD = BC = 3 * X cm.

The perimeter of the parallelogram is: Ravsd = 2 * (AB + BC) = 2 * (2 * X + 3 * X) = 10 * X cm.

10 * X = 40 cm.

X = CK = 40/10 = 4 cm.

Then AB = 2 * 4 = 8 cm.

Answer: The length of the AB side is 8 cm.



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