In parallelogram ABCD, the bisector of angle B intersects side CD at point T and line AD at point M.

| June 1, 2021 | education

In parallelogram ABCD, the bisector of angle B intersects side CD at point T and line AD at point M. Find the perimeter of triangle CBT if AB = 21, BM = 35, MD = 9.

The bisector of angle B cuts off the isosceles triangle ABM, then AB = AM = 21 cm.

Then the side length AD = AM + DМ = 21 + 9 = 30 cm.

Angle AMB = CBM = ABM. The angle DMT = AMB as vertical angles, then the angle CBM = DMT, and therefore the triangle CBT is isosceles, BC = TC.

Since the opposite sides of the parallelogram are equal, then AD = BC = 30 cm, and then CT = 30 cm.

Triangles CBT and DMT are similar in two angles then CT / DT = BC / MD.

30 / DТ = 30/9.

DТ = 30 * 9/30 = 9 cm.Then CT = 21 + 9 = 30 cm.

Let the segment MT = X cm, then BT = 35 + X cm.

ВТ / ВМ = ВС / DМ.

(35 + X) / X = 30/9.

315 + 9 * X = 30 * X.

21 * X = 315.

X = MT = 315/21 = 15 cm.

Then BT = BM + MT = 35 + 15 = 50 cm.

Determine the perimeter of the triangle BCT. Рвст = ВС + CT + ВТ = 30 + 30 + 50 = 110 cm.

Answer: The perimeter of the BСT is 110 cm.



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