Question

Two corners of an isosceles triangle are at `(1 ,5 )` and `(3 ,7 )`. If the triangle's area is `4`, what are the lengths of the triangle's sides?

Answer 1

The lengths of the sides are: `4sqrt2`, `sqrt10`, and `sqrt10`.

Explanation:

Let the given line segment be called `X`. After using the distance formula `a^2+b^2=c^2`, we get `X=4sqrt2`.

Area of a triangle `=1/2bh`
We are given the area is 4 square units, and the base is side length X.

`4=1/2(4sqrt2)(h)`
`4=2sqrt2h`
`h=2/sqrt2`
Now we have the base and the height and the area. we can divide the isosceles triangle into 2 right triangles to find the remaining side lengths, which are equal to each other.
Let the remaining side length = `L`. Using the distance formula:
`(2/sqrt2)^2+(2sqrt2)^2=L^2`
`L=sqrt10`