In a right-angled triangle, a 12 cm leg is adjacent to an angle of 30 degrees. Find the bisector of the second

In a right-angled triangle, a 12 cm leg is adjacent to an angle of 30 degrees. Find the bisector of the second acute angle of the triangle.

Let a triangle ABC be given, <C = 90 °. ВM – bisector <ABC.
Consider the triangle ABM obtained inside the triangle ABC.

<BAC = <ABM, since <ABC = 90 – <BAC = 90 ° – 30 °. But <ABM = <MВС = 60 ° / 2 = 30 °.

Means <BAC = <ABM = 30 °. And if the angles at the base of AB are equal, then this triangle ABM is isosceles, AM = BM.

From triangle ABC, leg BC = AC * tan (30 °) = AC / √ (3) = 12 / √ (3) = 4 * √ (3).
From the triangle BCM, ВM – hypotenuse, BC – leg adjacent to the angle of 30 °.
Bisector ВМ = cos (30 °) = √ (3) / 2. ВМ = ВС * 2 / √ (3) = 4 * √ (3) * 2 / √ (3) = 8 (cm).