For what value of 'r' will the line y=1/2x + r bisect the interval which joines the points (-3,5),(1,-7)?

Question

i need full working out and explanation

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ANSWER:

r=-1/2

Answer 1

Step 1 Find the gradient of the line joining the two points.

m = (-7-5)/(1--3)

    = (-12)/(4) = -3

Step 2 Take a point (x,y) on the line.

Step 3 Find the equation of the line

The point (x,y) should give you the same gradient of -3 with any of the two points used above.

Let's choose the point (-3,5)

(y-5)/(x--3) = -3

y-5 = -3(x+3)

y-5 = -3x-9

y = -3x-4

Equation of line : y = -3x-4

Step 5 Find the midpoint of the line.

Midpoint = ( (-3+1)/2 , (5-7)/2 )

             = (-1,-1)

Step 6 If the lines intersect, they must be equal for some value of y.

(1/2)x+r = -3x-4

x+2r = -6x-8

2r = -7x-8

Replace the x-coordinate of the midpoint.

2r = -7(-1)-8 = -1

r = -1/2

Hope it is clear in your mind now !

  

Answer 2

find the midpoint of the line that joins the 2 points.

the midpoint is (-1,-1)

substitute (-1,-1) into the equation.

-1 = 1/2(-1) + r

so you have r = -1/2.

:)

Answer 3

Let A be the mid point of the interval joining the points (-3,5) and (1,-7)

==>The coordinators of A=(-1,-1)

The line y=1/2x+r goes through A

<=> -1 = 1/2* (-1)+r

<=> r=-1/2