# The perimeter of an isosceles triangle is 90 cm, and the height dropped to the base is 15 cm.

The perimeter of an isosceles triangle is 90 cm, and the height dropped to the base is 15 cm. Find the area of the triangle.

Let us introduce the designations: b – lateral side, and – half of the base of this isosceles triangle. Then, knowing that its perimeter is 90 cm, we can write:

b + b + a + a = 90;

2 * b + 2 * a = 90;

b + a = 90/2 = 45.

From a right-angled triangle formed by the side, half of the base and the height h, drawn to this base, we can write:

a ^ 2 + h ^ 2 = b ^ 2;

a ^ 2 + 225 = b ^ 2.

We get the system of equations:

1) b + a = 45;

2) a ^ 2 + 225 = b ^ 2.

From the first equation, we express b through a: b = 45 – a.

Substitute this expression into the second equation of the system and solve it with respect to a:

a ^ 2 + 225 = (45 – a) ^ 2;

a ^ 2 + 225 = 2025 – 90a + a ^ 2;

90a = 1800;

a = 1800/90 = 20 cm – half of the base.

We find the area of ​​the triangle as the product of the height and half the base:

S = h * a = 15 * 20 = 300 cm2.

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