Find the length of the side of an isosceles triangle if its area is 420 and the length of the height drawn to the base is 35.

The area of ​​an isosceles triangle can be found as half of the product of the base and the height drawn to the base: S = 0.5 * h * a. From here we find the base of this isosceles triangle: a = 2 * S / h = 2 * 420/35 = 840/35 = 24 cm.

In an isosceles triangle, the height drawn to the base divides it in half.

Consider a right-angled triangle, in which the side of this triangle is the hypotenuse, and the height and half of the base are legs. The square of the hypotenuse is equal to the sum of the squares of the legs, so the square of the side is 352 + 122 = 1225 + 144 = 1369.

The desired side is √1369 = 37.