# Perpendicular lines

Determine a constant real number k such that the lines AB and CD are perpendicular.
A(1,2), B(4,0), C(k,2), D(1,-3). (answer given is k=-3/2)

If two lines are perpendicular the product of their slopes is -1.

The slope of AB is $$\frac{0-2}{4-1}$$ = $$\frac{-2}{3}$$

The slope of CD is $$\frac{-3-2}{1-k}$$ = $$\frac{-5}{1-k}$$
I set the product of these slopes equal to -1 and solve for k.

$$\frac{-2}{3}$$ x $$\frac{-5}{1-k}$$=-1
10=3(k-1)
10=3k-3
13=3k
13/3=k
This is not the answer given and I am not seeing my error. Any help would be appreciated.
Latex question. The = sign and x symbol don't line up well with the rest of the equations in the first half of my post. How can I correct that?

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Your k value is correct ...

$$\frac{0-2}{4-1}=\frac{-2}{3}$$

Check my LaTeX ...

Last edited:
Thanks for the help.

Verified using an alternate method (direction cosines and perpendicularity) to confirm that your answer is correct.

You are 100% correct, reporting the text book error might be helpful for the rest using the same book...