In triangle ABC, the height BD divides the AC side into segments AD and DC, BC = 6cm, angle A = 30 degrees, angle CBD = 45 degrees. Find the AC side of the triangle.
The height BD forms two right-angled triangles BCD and ABD.
The right-angled triangle ВСD is isosceles, ВD = СD, since its acute angle is 45.
Then SinСВD = СD / ВС.
СD = ВD = ВС * Sin45 = 6 * √2 / 2 = 3 * √2 cm.
In a right-angled triangle ABD,
tg30 = BD / AD.
АD = ВD / tg30 = 3 * √2 / (1 / √3) = 3 * √6 cm.
Side length АС = АD + СD = 3 * √6 + 3 * √2 = 3 * (√6 + √2) cm.
Answer: The length of the AC segment is 3 * (√6 + √2) cm.
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