In parallelogram ABCD, the bisector of acute angle A intersects side BC at point F.

| May 23, 2021 | education

In parallelogram ABCD, the bisector of acute angle A intersects side BC at point F. Calculate the length of the segment AF if the angle BCD = 30 and DC = 6 cm.

Since the opposite sides of the parallelogram are equal, AB = CD = 6 cm.

AF is the bisector of the angle BAD, then the angle BAF is equal to FAD. The angle BFA is equal to the angle FAD as the intersecting angles at the intersection of parallel lines AD and BC of the secant AF. Then the angle BFA is equal to FAD, and the triangle ABF is isosceles, AB = BF = 6 cm.

The sum of the adjacent angles of the parallelogram is 180, then the angle ABC = 180 – BCD = 180 – 30 = 150.

From the triangle ABF, by the cosine theorem, we determine the size of the side AF.

AF ^ 2 = AB ^ 2 + BF ^ 2 – 2 * AB * BF * Cos150.

AF ^ 2 = 36 + 36 – 2 * 6 * 6 * (-√3 / 2) = 72 + 36 * √3 = 36 * (2 + √3) cm.

AF = 6 * √ (2 + √3) cm.

Answer: The length of the bisector AF is 6 * √ (2 + √3) cm.



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