An isosceles triangle has sides A, B, and C with sides B and C being equal in length. If side A goes from #(7 ,5 )# to #(8 ,1 )# and the triangle's area is #15 #, what are the possible coordinates of the triangle's third corner?
We can find the length of 'a' by finding the distance between the two points:
Let side 'a' be the base of the triangle.
Using the area, we can compute the height:
The height must lie on the line that is the perpendicular bisector of side 'a'. Let's find the equation of that line:
Side 'a' goes from left to right 1 unit and down 4 units (For later use, remember this is slope, -4), therefore, the midpoint goes from left to right
The midpoint is
A perpendicular line will have a slope that is the negative reciprocal of -4:
Using the point-slope form of the equation of a line,
Using the equation for a circle we write an equation where the radius is
I used Wolframalpha to solve this:
The corresponding y coordinates are: