Geometry Question

Question

In an isosceles triangle, the base is 60% longer than than each of the other sides. The area of the triangle is 432 cm2 .

how do I find the Other sides of the triangle , the height and the perimeter ? Please help

Answer 1

As this is geometry, not trig, we will not use sin and cos.

Mike's idea about bisecting the base is great. We have two similar triangles created with base .8x and hypotenuse x. We can either use the Pythagorean theorem to determine that the altitude is .6x, or notice that we have a right triangle with one side : hypotenuse in the ratio 4 : 5, which makes us immediately think of 3-4-5 right triangles.

Then, we can calculate the area of the triangle by summing the area of the individual triangles with base .8x and height .6x or just use the original triangle with base 1.6x and height .6x.

The area of the triangle is 1/2(1.6 x)(.6x) = .48x².

Then, if .48x² = 432, x² = 900, or x = 30. Then, the base is 1.6x or 48. The height is .6x = 18

The perimeter is 48 + 2(30) = 108

Answer 2

Jaden

If the 2 sides are x, then the base will be 1.6x using the information given.

Half of the base will be 0.8 x

If you draw a perpendicular from the top of the triangle to the base, you will get a right triangle with base 0.8 x, hypotenuse x.

That means the cosine of the base angle is 0.8 x / x = 0.8

If h is the height then h/x = sin θ, if θ is the base angle, so h = x sin θ

Area of the triangle will be 1/2 base x height = 0.8 x h = 0.8 x . x sin θ = 0.8 x² sin θ = 432

You can find θ , θ = cos ^-1 (0.8), and knowing that you can solve this equation for x.

(the right triangle is actually going to be a 3-4-5 triangle, with base 4, so the sin is going to be 0.6)

So 0.8 x² 0.6 = 432, giving x = 30

So then the base is 48, the two other sides are 30, and the perimeter = 30 + 30 + 48 = 108

The height is 18

Mike