The height AD of triangle ABC divides the sides BC into segments BD and CD so that BD = 15 cm

The height AD of triangle ABC divides the sides BC into segments BD and CD so that BD = 15 cm, CD = 5 cm. Find the sides AC if the angle B = 30 degrees.

angle ADB = 90 (AD – height), angle B = 30.

Hence the angle DAB = 180 – ADB – B = 180 – 90 – 30 = 60.

Let us calculate the length of side AD along side BD and two adjacent corners B and DAB.

We get:

AD = (BD * sinB) / sinDAB = (15 * sin30) / sin60 = 5√3.

By the Pythagorean theorem we have: AC = √ (AD ^ 2 + CD ^ 2).

We get:

AC = √ ((5√3) ^ 2 + 5 ^ 2) = √ (75 + 25) = √100 = 10.

Answer: AC = 10cm.