In triangle ABC, angle C is 90, AB = 91, sin A = 5/13. Find AC.

If the angle С is equal to 90 ° according to the problem statement, then the triangle ABC will be rectangular. If the triangle is rectangular, then we can use its property as the sin of an angle. In a right-angled triangle, the sine of the angle is equal to the ratio of the opposite leg relative to the angle (in our case, sinA will be equal to the ratio of the BC leg and the hypotenuse AB, BC / AB = sin (A)). Let us find from here the length of the leg BC, since the length of the hypotenuse AB is given to us by condition. BC = AB * sinA = 91 * (5/13) = 7 * 5 = 35. Next, using the Pythagorean theorem for a right-angled triangle, we find the length of the second leg AC. AC ^ 2 + BC ^ 2 = AB ^ 2. It follows that AC ^ 2 = AB ^ 2 – BC ^ 2, AC = (AB ^ 2-BC ^ 2) = √ (8281 – 1225) = √ (7056) = 84
Answer: AC = 84