Posted by Emily on .
triangle ABC is an isosceles right triangle. The endpoints of the hypotenuse of ABC are A(4,7)B(8,1). Find the possible coordinates for the vertex C

geometry 
bobpursley,
The C willbe on the line which is the perpendicular bisector of the hypotenuse.
slope of AB= (6/4)
slope of bisector= 4/6
but a point on the bisector is (6,4)
y= 4/6 x + b
4= 4/6 *6+b
B=0
equation of bisector is y= 4/6 x
now, there is a point on that line which forms to line segments which are 90 deg to each other.
C (x,y)=(x,4/6x)
Now, if AC and BC are perpendicular the slopes are neg reciprocals.
slope AC= (74x/6)/(4x)
slope BC= (84x/6)/(1x)
these are neg reciprocals, so this leads to
(74x/6)(84x/6)=(4x)(1x)
solve for x. It looks a little messy.
568x/6+16x^2/36=4+5xx^2
combine terms, simplify, solve for x with the binomial equation.