If the perimeter of an isosceles triangle is 36 cm and the altitude to its base is 12 cm, what is the triangle's area?

Question

If the perimeter of an isosceles triangle is 36 cm and the altitude to its base is 12 cm, what is the triangle's area?

Answer 1

Base angles are 45 each.

sin 45 = sqrt(2)/2 = 12 / y where y is the measure of the equal sides

y = 24*sqrt(2)/2 = 12*sqrt(2)

this forces the base to be 24, or 12 in each half

the area therefore is (1/2)*24*12 = 144

Answer 2

An isosceles triangle has two equivalent sides and one non-equivalent side.

2a + b = 36cm

h = 12cm

We can use the pythagorean theorem to find the lengths of each side.

Below is an image of an isosceles triangle: https://www.google.com/search?q=isosceles+triangle&sxsrf=ACYBGNRj9AFAlV1CcqksHfy9XlQhwW3Vpg:1574331320008&source=lnms&tbm=isch&sa=X&ved=2ahUKEwinytmYifvlAhUSWBoKHb0kAPgQ_AUoAXoECBAQAw&biw=1800&bih=986#imgrc=QdPfV3l3TeCrEM:

As described earlier, the equivalent sides are a. The nonequivalent side b is 36 - 2a.

The pythagorean theorem states that a² + b² = c².

Below is an image of an isosceles triangle with the height drawn:

https://images.app.goo.gl/8sbGmFU5KN6xzuas5

If we use the pythagorean theorem, we will only be concerned with b/2 because this is the length that forms a triangle with the altitude, or height given.

So:

a² = 0.5((36 - 2a)/2)² + 12²

a² = (18-a)² + 144

a² = 324 -36a + a² + 144

0 = -36a + 468

36a = 468

a = 13

b = 36 - 2(13) = 36 - 26 = 10

The triangles area will be:

Area = 0.5(b*h)

Area = 0.5(10*12cm) = 60cm²