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 ,9 )` and the triangle's area is `18`, what are the possible coordinates of the triangle's third corner?

Answer 1

There are two possible points at `color(blue)((1377/178"," 625/89))` and
`color(red)((225/178"," 265/89))`

Explanation:

The solution makes use of the formula for the area of a triangle

Area `=1/2[(x_1, x_2, x_3, x_1),(y_1, y_2, y_3, y_1)]`

positive area for counterclockwise arrangement of points
negative area for clockwise arrangements of points

Let (x_1, y_1) be the unknown point and Let `P_2(2, 9)` and `P_3(7, 1)` and P_1(x_1, y_1) and Area`=18`

Area `=1/2[(x_1, x_2, x_3, x_1),(y_1, y_2, y_3, y_1)]`

`+18=1/2[(x_1, 2, 7, x_1),(y_1, 9, 1, y_1)]`

`18(2)=(9x_1+2+7y_1-2y_1-63-x_1)`

`36=8x_1+5y_1-61`

`8x_1+5y_1=97" "`first equation

For the equal sides because the triangle is isosceles, we have

`(x_1-2)^2+(y_1-9)^2=(x_1-7)^2+(y_1-1)^2`
`x_1^2-4x_1+4+y_1^2-18y_1+81=x_1^2-14x_1+49+y_1^2-2y_1+1`
Simplify
`-4x_1+4-18y_1+81=-14x_1+49-2y_1+1`
`10x_1-16y_1+85-50=0`
`10x_1-16y_1+35=0" "`second equation

Simultaneous solution using first and second equations results to
`color(red)((x_1, y_1)=(1377/178, 625/89))`

There is another point which may be solved by using the negative area

Area `=1/2[(x_1, x_2, x_3, x_1),(y_1, y_2, y_3, y_1)]`
`-18=1/2[(x_1, 2, 7, x_1),(y_1, 9, 1, y_1)]`

`-18(2)=(9x_1+2+7y_1-2y_1-63-x_1)`

`-36=8x_1+5y_1-61`

`8x_1+5y_1=25" "`third equation

Simultaneous solution using second and third equations results to
`color(red)((x_1, y_1)=(225/178, 265/89))`

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