Q.2 : The line represented by the equation y = -2x + 6 is the perpendicular bisector of the line segment AB. If A has the coordinates (7, 2), what are the coordinates for B?
Solution :
The line represented by the equation y = -2x + 6 is the perpendicular bisector of the line segment AB. So, The line segment AB and the line y = -2x + 6 should have a common point. That common point is the point of intersection of the line y = -2x + 6 and the line segment AB. To find that common point, we have to solve the 2 line equations. We have one line equation -> y = -2x + 6. To find the other we follow these steps :
1.Slope : Since, these two lines are perpendicular, the product of their slopes = -1. The line, y = -2x + 6, has a slope = -2 [y=mx+c]. So, the perpendicular line should have a slope = (1/2). Hence the product is -1.
Line Equ 2 = y=(1/2)x+c
2.y-intercept(c) : Substitute the point A(7, 2) in y=(1/2)x+c.
2=(1/2)7+c
2=(7/2)+c
2-(7/2)=c
(4-7)/2=c
c=(-3/2)
Line equ 2 = y=(1/2)x-(3/2)
Now, we have the 2 equations. We have to solve them.
y=-2x+6 ------ 1
y=(1/2)x-(3/2) --------- 2
-2x+6=(1/2)x-(3/2)
-4x+12=x-3
15=5x
x=3
Sub x=3 in 1
y=-2(3)+6
y=-6+6
y=0
We 've now got the midpoint of AB [The common point] = (3,0)
To find B. Since they are equally spaced. We can follow this
A(7,2) B(x,y) common point(3,0)
(7+x)/2 = 3
7+x=6
x=-1
Similarly,
(2+y)/2=0
2+y=0
y=-2
Hence, the point B is (-1,-2).
Hope this clears the doubt
