The points (3, 7), (6, 2) and (2,k) are the vertices of a triangle. For how many real values of k is the triangle a right triangle?
(3,7) (6, 2) ( 2, k)
We need to find the slope between (3,7) and (6,2)
(7 - 2) / ( 3 - 6) = -5/3
Now we need to find two lines with reciprocal slopes ( perpendicular to AB) that pass through (3,7) and (6,2)
So...one line has the equation
y = (3/5) ( x - 3) + 7
y = (3/5)x - 9/5 + 35/5
y = (3/5)x + 26/5
So one value of k is
k = (3/5)(2) + 26/5
k = 6/5 + 26/5
k = 32/5
And the other line has the equation
y = (3/5)( x - 6) + 2
y = (3/5)x - 18/5 + 10/5
y = (3/5)x -8/5
So the other value of k is
k = (3/5)(2) - 8/5
k = 6/5 - 8/5
k = -2/5