The height BD of the triangle ABC divides the AC side into segments AD, DC, AB = 12 cm

The height BD of the triangle ABC divides the AC side into segments AD, DC, AB = 12 cm, angle A = 60 degrees, angle CBD = 45 degrees. Find the AC side of the triangle.

The height of the ВD forms two right-angled triangles AВD and ВСD.

In a right-angled triangle AВD, the angle AВD = (180 – 90 – 60) = 300, then the leg of the ABP lying opposite it is equal to half the length of AB.

AD = AB / 2 = 12/2 = 6 cm.

Then, by the Pythagorean theorem, ВD ^ 2 = AB ^ 2 – AD ^ 2 = 144 – 36 = 108.

ВD = 6 * √3 cm.

In a right-angled ВСD triangle, one of the acute angles is 450, then the lengths of its legs are equal, СD = ВD = 6 * √3 cm.

Then AC = AD + СD = 6 + 6 * √3 = 6 * (1 + √3) see.

Answer: The length of the speaker side is 6 * (1 + √3) cm.