In triangle ABC, angle B is twice the angle A, and the length of the side BC is 200.

| April 28, 2021 | education

In triangle ABC, angle B is twice the angle A, and the length of the side BC is 200. Find the bisector BD of this triangle if DC = 125.

Let the angle BAC = X0, then, by condition, the angle ABC = 2 * X0. Since BD is a bisector, the angle ABD = CBD = 2 * X / X = X0.

Then, in triangle ABD, the angles at the base of AB are equal, then ABD is isosceles.

The angle BDC is the outer angle of the triangle ABD, then the angle BDC is equal to the sum of two angles that are not adjacent to it. Angle BDC = ABD + BAD = X + X = 2 * X.

In triangle ABC, angle A = X0, B = 2 * X0, in triangle BCD, angle B = X0, angle D = 2 * X0, then triangles ABC and BCD are similar in two angles.

BC / AC = CD / BC.

AC = BC * BC / CD = 200 * 200/125 = 320 cm.

АD = АС – СD = 320 – 125 = 195 cm.

Since ABD is isosceles, then BD = AD = 195 cm.

Answer: The length of the bisector is 195 cm.



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