Question

Two corners of an isosceles triangle are at `(8 ,5 )` and `(6 ,7 )`. If the triangle's area is `15`, what are the lengths of the triangle's sides?

Answer 1

Sides:`{2.8284, 10.7005,10.7005}`

Explanation:

Side `color(red)(a)` from `(8,5)` to `(6,7)`
has a length of
`color(red)(abs(a))=sqrt((8-6)^2+(5-7)^2)=2sqrt(2)~~2.8284`

Not that `color(red)(a)` can not be one of the equal length sides of the equilateral triangle since the maximum area such a triangle could have would be `(color(red)(2sqrt(2)))^2/2` which is less than `15`

Using `color(red)(a)` as the base and `color(blue)(h)` as the height relative to that base, we have
`color(white)("XXX")(color(red)(2sqrt(2))*color(blue)(h))/2 = color(brown)(15)`

`color(white)("XXX")rarr color(blue)(h) = 15/sqrt(2)`

Using the Pythagorean Theorem:
`color(white)("XXX")color(red)(b)=sqrt((15/sqrt(2))^2+((2sqrt(2))/2)^2) ~~10.70047`

and since the triangle is isosceles
`color(white)("XXX")c=b`