The base of an isosceles triangle is 12 cm. The lateral side is 2 cm higher than the height. Find the area.

In an isosceles triangle, the median is the bisector and height. This means that the height of the triangle divides the base in half. Half of the base of the triangle, the side and the height form a right-angled triangle. In this triangle, one leg is 12: 2 = 6 (cm), the second leg (aka the height of an isosceles triangle) is taken as x cm, the hypotenuse (it is the lateral side of an isosceles triangle) is equal to (x + 2) cm. Let’s apply the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs.

(x + 2) ^ 2 = x ^ 2 + 6 ^ 2;

x ^ 2 + 4x + 4 – x ^ 2 = 36;

4x = 36 – 4;

4x = 32;

x = 32: 4;

x = 8 (cm) – the height of the isosceles triangle.

The area of ​​a triangle is equal to half the product of the base by the height drawn to the given base.

S = 1/2 ah;

S = 1/2 * 12 * 8 = 6 * 8 = 48 (cm ^ 2).

Answer. 48 cm ^ 2.