P(4,9) ,Q(3,2),R(13,-3) and S(11,13) are four points in a plane and L is the mid point of RS . the line through Q perpendicular to RS meets PR at M find :
1 the equation of the lines QL and PR
2 the coordinates of the point of intersection N ,of QL and PR
3 the length of QM
4 the area of triangle PQN
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Expert's answer
2019-09-03T11:02:30-0400
1. Find the coordinates of L:
x=213+11=224=12y=2−3+13=210=5
Find the slope of the line QL:
mQL=x2−x1y2−y1=12−35−2=93=1/3
Substitute the slope and point Q in the slope intercept form of the equation (y=mx+b ) and solve for b:
2=31⋅3+b2=1+bb=1
The equation of the line QL is: y=31x+1.
Find the slope of the line PR:
mPR=13−4−3−9=9−12=−34
Substitute the slope and point P in the slope intercept form of the equation and solve for b:
9=−34⋅4+b9=−316+bb=343
The equation of the line PR is: y=−34x+343.
2. If N is the point of intersection of QL and PR, then its coordinates satisfy the system of equations:
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