Question

An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from `(7 ,1 )` to `(2 ,5 )` and the triangle's area is `32`, what are the possible coordinates of the triangle's third corner?

Answer 1

Possible coordinates are`(881/82, 443/41)` and `(-143/82, -197/41)`

Explanation:

We use the formula for the area A

`A=1/2[(x_a, x_b, x_c, x_a),(y_a, y_b, y_c, y_a)]`

`A=1/2[x_a*y_b+x_b*y_c+x_c*y_a-x_b*y_a-x_c*y_b-x_a*y_c]`

We have to consider two points on each side of side `a`

For the first point `A`

`A=1/2[(x_a, x_b, x_c, x_a),(y_a, y_b, y_c, y_a)]`

`A=1/2[x_a*y_b+x_b*y_c+x_c*y_a-x_b*y_a-x_c*y_b-x_a*y_c]`

`32=1/2[(x_a, 2, 7, x_a),(y_a, 5, 1, y_a)]`

`32=1/2[5x_a+2+7y_a-2y_a-35-x_a]`

`64-2+35=4x_a+5y_a`

`97=4x_a+5y_a`

This is an isosceles triangle so that the third vertex `A(x_a, y_a)` passes through the perpendicular bisector of side `a`

Perpendicular bisector equation is passing through the midpoint of side `a` which `M(9/2, 3)` with slope `m_p=5/4`

`y-3=5/4(x-9/2)`

`10x-8y=21`

Also this is

`10x_a-8y_a=21`

Solving Vertex `A` with equation `97=4x_a+5y_a`

`4x_a+5y_a=97`
`10x_a-8y_a=21`

Simultaneous solution results to

`color(red)(A(x_a, y_a)=(881/82, 443/41))`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For the second point `A` on the other side of side `a`

`A=1/2[(x_a, x_c, x_b, x_a),(y_a, y_c, y_b, y_a)]`

`A=1/2[x_a*y_b+x_b*y_c+x_c*y_a-x_b*y_a-x_c*y_b-x_a*y_c]`

`32=1/2[(x_a, 7, 2, x_a),(y_a, 1, 5, y_a)]`

`32=1/2[5x_a+2+7y_a-2y_a-35-x_a]`

`64+2-35=-4x_a-5y_a`

`4x_a+5y_a=-31`

This is an isosceles triangle so that the other third vertex `A(x_a, y_a)` passes through the perpendicular bisector of side `a`

Perpendicular bisector equation is passing through the midpoint of side `a` which `M(9/2, 3)` with slope `m_p=5/4`

`y-3=5/4(x-9/2)`

`10x-8y=21`

Also this is

`10x_a-8y_a=21`

Solving the other Vertex `A` with equation `4x_a+5y_a=-31`

`4x_a+5y_a=-31`
`10x_a-8y_a=21`

Simultaneous solution results to

`color(red)(A(x_a, y_a)=(-143/82, -197/41))`

God bless....I hope the explanation is useful.