In triangle ABC: angle C = 90 (degrees), CD-bisector, angle BDC = 75 (degrees), AC = √3. Find BD.

Since CD is the bisector of the right angle, about the angle BCD = 90/2 = 45.

The sum of the inner angles of the triangle is 180, then in the triangle BCD, the angle CBD = (180 – 75 – 45) = 60.

In a right-angled triangle ABC tgCBA = AC / BC.

ВС = АС / tg60 = √3 / √3 = 1 cm.

In a triangle BCD, by the theorem of sines:

ВС / Sin75 = ВD / Sin45.

ВD = BC * Sin45 / Sin65 = 1 * (√2 / 2) / ((1 + √3) / 2 * √2) = 2 * √2 * √2 / 2 * (1 + √3) = 2 / (1 + √3) see

Answer: The length of the segment BD is 2 / (1 + √3) cm.