Question

Two corners of an isosceles triangle are at `(2 ,1 )` and `(7 ,5 )`. If the triangle's area is `4`, what are the lengths of the triangle's sides?

Answer 1

There are three possibilities:
`color(white)("XXX"){6.40,3.44,3.44}`
`color(white)("XXX"){6.40, 6.40, 12.74}`
`color(white)("XXX"){6.40, 6.40, 1.26}`

Explanation:

Note the distance between `(2,1)` and `(7,5)` is `sqrt(41)~~6.40`
(using the Pythagorean Theorem)

Case 1
If the side with length `sqrt(41)` is not one of the equal length sides

then using this side as a base the height `h` of the triangle can be calculated from the area as
`color(white)("XXX")((hsqrt(41))/2=4) rArr (h=8/sqrt(41))`

and the two equal length sides (using Pythagorean Theorem) have lengths
`color(white)("XXX")sqrt((sqrt(41)/2)^2+(8/sqrt(41))^2) ~~3.44`

Case 2
If the side with length `sqrt(41)` is one of the sides of equal length

then if the other side has a length of `a`, using Heron's Formula
`color(white)("XXX")`the semiperimeter, `s` equals `a/2+sqrt(41)`
and
`color(white)("XXX")"Area" = 4 = sqrt((a/2+sqrt(41))(a/2)(a/2)(sqrt(41)-a/2))`
`color(white)("XXXXXXXXX")=a/2sqrt(41-a^2)`
which can be simplified as
`color(white)("XXX")a^4-164a^2+256=0`
then substituting `x=a^2` and using the quadratic formula
we get:
`color(white)("XXX")a=12.74 or a=1.26`