Find the value of k if lines AB and CD are perpendicular where A=(3,7), B=(k+1,-3), C=(2,-5), and D=(6,7).

To find the value of k, we first need to find the slopes of lines AB and CD.

The slope of line AB is given by:
m_AB = (y2 - y1) / (x2 - x1)
m_AB = (-3 - 7) / (k + 1 - 3)
m_AB = -10 / (k - 2)

Since AB is perpendicular to CD, the product of their slopes is -1:
m_AB * m_CD = -1
(-10 / (k - 2)) * m_CD = -1
m_CD = (k - 2) / 10

The slope of line CD is given by:
m_CD = (y2 - y1) / (x2 - x1)
m_CD = (7 + 5) / (6 - 2)
m_CD = 12 / 4
m_CD = 3

Now, we can substitute the slope of line CD into the equation we found earlier:
(k - 2) / 10 = 3
k - 2 = 30
k = 32

Therefore, the value of k is 32.

The slope of AB is (1-(-3))/(k-3) = 4/(k-3)
The slope of CD is (7-(-5))/(6-2) = 12/4 = 3
Since the lines are perpendicular, their slopes are negative reciprocals, so
4/(k-3) = -1/3
Cross-multiplying, we get
12 = -4/(k-3)
-48 = k-3
k = -45