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?

1 Answer
Jul 8, 2016

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

Explanation:

Refer to the figure below
I created this figure using MS ExcelI created this figure using MS Excel

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