In the triangle ABC AC = BC = 25√21, sin BAC = 0.4 find the height AH.

Let’s draw from the vertex C the height CD, which is also the median of the triangle ABC, since AC = BC by condition, which means that the triangle ABC is isosceles.

Then SinA = CD / AC.

СD = АС * SinА = 25 * √21 * 0.4 = 10 * √21 cm.

From the triangle ACD, according to the Pythagorean theorem, we determine the size of the leg AD.

AD ^ 2 = AC ^ 2 – CD ^ 2 = (25 * √21) ^ 2 – (10 * √21) ^ 2 = 13125 – 2100 = 11025.

AD = 105 cm.

Then the base AB = AD * 2 = 105 * 2 = 210 cm.

Determine the area of the triangle using two formulas.

S = CD * AB / 2 = 10 * √21 * 210/2 = 1050 * √21.

S = AH * CB / 2 = AH * 25 * √21 / 2.

1050 * √21 = AH * 25 * √21 / 2.

AH = 2 * 1050 * √21 / 25 * √21.

AH = 84 cm.

Answer: Height AH = 84 cm.