The perimeter of an isosceles triangle is 216 cm, and the lateral side is 78 cm. Find the area of the triangle.

An isosceles triangle is a triangle in which opposite sides are equal:

AB = BC = 78 cm.

The perimeter of a triangle is the sum of all its sides:

P = AB + BC + AC;

AC = P – AB – BC;

AC = 216 – 78 – 78 = 60 cm.

The area of ​​a triangle is equal to the half-product of its base by its height:

S = 1/2 a h, where:

S is the area of ​​the triangle;

a – the base of the AU;

h – HV height.

To calculate the height of the VN, consider the triangle AVN. This triangle is right-angled, so let’s use the Pythagorean theorem:

AB ^ 2 = BH ^ 2 + AH ^ 2;

BH ^ 2 = AB ^ 2 – AH ^ 2.

Since the height of an isosceles triangle, lowered to the base, divides it in half:

AH = BH = AC / 2;

AH = HC = 60/2 = 30 cm.

BH ^ 2 = 78 ^ 2 – 30 ^ 2 = 6084 – 900 = 5184;

BH = √5184 = 72 cm.

S = 1/2 * 60 * 72 = 2160 cm2.

Answer: the area of ​​the triangle is 2160 cm2.