Question

An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from `(5 ,1 )` to `(3 ,2 )` and the triangle's area is `12`, what are the possible coordinates of the triangle's third corner?

Answer 1

Two possibilities: `(8.8,11.1)` or `(-0.8,-8.1)`

Explanation:

Refer to the figure below

It's known that:
`P_2(5,1)`, `P_3(3,2)`
`S=12`
`B=C`
It's asked `P_1`

`A=sqrt((5-3)^2+(2-1)^2)=sqrt(5)`
`S=("base"*"height")/2 => 12=(sqrt5*h)/2 => h=24/sqrt5`
`B^2=(A/2)^2+h^2=(sqrt5/2)^2+(24/sqrt5)^2=5/4+576/5=2329/20 => B~=10.791` (this is the distance `P_1P_2` or `P_1P_3`)

Now we could use the equations of the distance between 2 points to determine `P_1`, but since `P_1` is directly above (or below) `M(4,1.5)` we can find `P_1` in the following easier way:

`k_(P_2P_3)=(Deltay)/(Deltax)=(1-2)/(5-3)=-1/2`
line `P_2P_3`: `(y-1)=(-1/2)(x-5) => 2y-2=-x+5 => x+2y-7=0` [1]
Distance between a point and a line
`d=|ax_0+by_0+c|/sqrt(a^2+b^2)`
So, making `d=h`
`24/cancel(sqrt5)=|x+2y-7|/cancel(sqrt5)`
We get two equations:
(I) `24=x+2y-7 => 2y=-x+31 => y=-x/2+15.5`[2]
(II) `-24=x+2y-7 => 2y= -x-17 => y=-x/2-8.5`[3]

Since `P_1` is also in the bisector line of side A, let's find its equation
`p=-1/k_(P_2P_3)=2`
line `P_1M`: `(y-1.5)=2(x-4) => y=2x-8+1.5 => y=2x-6.5`[4]

Finding the two possible answers (by combining [2] and [4] or [3] and [4]):

(I) `2x-6.5=-x/2+15.5 => 2.5x=22=> x=8.8`
`->y=2*8.8-6.5 => y=11.1`
`=> (8.8,11.1)`

(II) `2x-6.5=-x/2-8.5 => 2.5x=-2 => x=-0.8`
`->y=2*(-0.8)-6.5 => y=-8.1`
`=> (-0.8,-8.1)`

Checking the results:
Notice that `(8.8-0.8)/2=4=x_M`
and `(11.1-8.1)/2=1.5=y_M`
Notice also that
(testing result I)`P_1P_2=sqrt(3.8^2+10.1^2)=sqrt116.45~=10.791`
(testing result I)`P_1P_3=sqrt(5.8^2+9.1^2)=sqrt116.45~=10.791`
(Try result II)
As it should be: all checked