SOLUTION: The coordinates of 3 points are A(1,5), B(9,7) and C(k,-6). M is the midpoint of AB and MC is perpendicular to AB. Find the value of k.

Algebra ->  Equations -> SOLUTION: The coordinates of 3 points are A(1,5), B(9,7) and C(k,-6). M is the midpoint of AB and MC is perpendicular to AB. Find the value of k.      Log On


   

Question 1141044: The coordinates of 3 points are A(1,5), B(9,7) and C(k,-6).
M is the midpoint of AB and MC is perpendicular to AB.
Find the value of k.
Answer by MathLover1(20850) About Me  (Show Source):
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given:
The coordinates of 3 points are A(1,5), B(9,7) and C(k,-6).
M is the midpoint of AB and MC is perpendicular to AB.
Find the value of k.



first find equation of the line that passes through A+and B:
y=mx%2Bb........use A(1,5), B(9,7) to find a slope m
m=%287-5%29%2F%289-1%29=2%2F8=1%2F4
y=%281%2F4%29x%2Bb......use one point to find b
5=%281%2F4%291%2Bb
5-1%2F4=b
b=19%2F4
equation is: y=%281%2F4%29x%2B19%2F4

now, if M is the midpoint of AB, coordinates of the midpoint+M+ are:
(%281%2B9%29%2F2,%285%2B7%29%2F2)=(5,6)

if MC is perpendicular to AB, find the equation of the line perpendicular to AB that passes through point+M
a slope of the line perpendicular to AB is negative reciprocal to m+=1%2F4, and it is
m%5Bp%5D=-1%2F%281%2F4%29=-4
so, y=-4x%2Bb.....use the point M to find b
6=-4%2A5%2Bb
6=-20%2Bb
b=6%2B20
b=26
equation the line perpendicular to+AB is:
y=-4x%2B26......use the point C(k,-6) to find k
-6=-4k%2B26
4k=26%2B6
4k=32
k=32%2F4
k=8
so, C(8,-6)