In a triangle ABC AC = BC = 25√21, sin of angle BAC = 0.4. Find the height of AH.

Let’s build the height CK.

In a right-angled triangle ACK, CK = AC * SinA = 25 * √21 * 0.4 = 10 * √21 cm.

By the Pythagorean theorem, AK ^ 2 = AC ^ 2 – CK ^ 2 = 13125 – 2100 = 11025.

AK = 105 cm.

Since triangle ABC is isosceles, the height of CK is also the median of the triangle, then AB = AK * 2 = 2 * 105 = 210 cm.

Determine the area of the triangle ABC.

Savs = AC * AB * SinA / 2 = 25 * √21 * 210 * 0.4 / 2 = 1050 * √21 cm2.

Also Savs = BC * AН / 2.

AH = 2 * Savs / BC = 2 * 1050 * √21 / 25 * √21 = 84 cm.

Answer: The length of the height AH is 84 cm