The point A has coordinates (0,2). The point B has coordinates (-4,-1). Find the coordinates of the midpoint of AB. Work out the gradient of the line AB. Find an equation of the line AB.

Question

The point A has coordinates (0,2).

The point B has coordinates (-4,-1).

Find the coordinates of the midpoint of AB.

Work out the gradient of the line AB.

Find an equation of the line AB.

Answer 1

To find the midpoint, use the midpoint formula: add each of the coordinates together and divide by 2:

MP = ( ((x₁+x₂)/2), ((y₁+y₂)/2) )

so the x coordinate of the midpoint is: (0 + (-4)) / 2 = -4/2 = -2

and the y coordinate of the midpoint is: (2 + (-1)) / 2 = 1/2

Therefore the midpoint is (-2, 1/2)

Gradient is just another word for slope. Slope is defined as change in y over change in x ("rise" over "run")

mathematically, this is (y₂-y₁) / (x₂-x₁)

plugging in our numbers: slope = (-1-2) / (-4-0) = -3/-4 = 3/4

so the slope is 3/4

To find the equation of the line, we start with y = mx + b, where m is the slope and b is the y-intercept.

We just found the slope in the previous part, so m=3/4

the y intercept is the y-coordinate when x is 0. They actually gave that to us for free in the problem statement when they said the point (0, 2) is on the line, so the y-intercept is b=2. Thus, the equation of this line is y = (3/4)x + 2

If the hadn't told us that, we could have plugged in any of the points on the line (-4, -1) or (-2, 1/2) and solved for the intercept. Let's do that with the point (-4, -1):

y = mx + b

we just found that m=3/4

the point (-4, -1) tells us that y=-1 and x=-4, so:

-1 = (3/4) * (-4) + b

-1 = -3 + b

2 = b (which is what we got above)