a triangle?

Question

a triangleis divided by a bisector thus forming two triangles. the length of the base is unknown for both triangles.the first triangle's leg measurement is 20.the other one is 13. the bisector measures 12. what is the measurement of the length of the base?

Answer 1

Even though the only thing we are asked to find is the length of the base, we really have three unknowns to deal with: x, the length of the base of each new triangle (the original triangle had a base of 2x); y, the apex angle of the triangle with a leg of measure 13; and z, the apex angle of the triangle with a leg of measure 20.

One triangle has legs of 12 and 13 and a base of x with an apex angle of y. One triangle has legs of 12 and 20 and a base of x iwth an apex angle of z. The larger triangle has legs of 13 and 20 and a base of 2x with an apex angle of y + z. We can use three applications of the law of cosines to set up a system of equations.

12^2 + 13^2 - 2*12*13*cos(y) = x^2
12^2 + 20^2 - 2*12*20*cos(z) = x^2
13^2 + 20^2 - 2*13*20*cos(y + z) = (2x)^2

We can simplify these equations a bit.

313 - 312cos(y) = x^2
544 - 480cos(z) = x^2
569 - 520[cos(y)cos(z) - sin(y)sin(z)] = 4x^2

It get a bit complicated here, but you can write

cos(y) = (313 - x^2) / 312
cos(z) = (544 - x^2) / 480

And remember that sin(t) = sqrt(1 - cos^2(t)) because sin^2(t) + cos^2(t) = 1, so you can also write expressions for sin(y) and sin(z) that have x as the only variable. That will enable you to rewrite

569 - 520[cos(y)cos(z) - sin(y)sin(z)] = 4x^2

completely in terms of x, and you can then solve for x, which is half the length of the base. The manipulations will be rather complicated, and you may need to use a graphing utility to actually solve for x.

Answer 2

You're funny.

First of all, when you bisect them, you get 2 right triangles.

One of first triangle's leg length, 20

One of second triangles leg length, 13

Second leg of BOTH OF THEM: bisector or length is 12.

The bisector can't be the base of the original triangle so..

The only possible base length is 33.

The question may not be what you're asking for lol, try reposting it.