# geometry

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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 -

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= (7-4x/6)/(4-x)
slope BC= (8-4x/6)/(1-x)

these are neg reciprocals, so this leads to

(7-4x/6)(8-4x/6)=-(4-x)(1-x)
solve for x. It looks a little messy.
56-8x/6+16x^2/36=-4+5x-x^2
combine terms, simplify, solve for x with the binomial equation.

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