a. The points A and B are (-2,1) and (6,-5) respectively. Calculate (i) the gradient of the line AB,?

Question

(ii) the equation of the line through A and B.

b. The points C and D are (4,5) and (p,q) respectively.
(i) Write down, in terms of p and q, the coordinates of the midpoint of CD.

(ii) Given that the midpoint of CD is (7,1), find the coordinates of D.

Answer 1

a. i) the gradient is the difference between y-coordinates/ the difference between the x-coordinates = ( -5-1) / ( 6 +2)
= -6/8 = -3/4

ii) y = mx +c where m is the gradient and c is the y-intercept
take the coordinates of one point, A for example:
1 = -2( -3/4) +c 1 = 6/4 +c
1 - 6/4 = c = -1/2
so y= (-3/4)x - 1/2 = -3x/4 -2/4 , y = ( -3x -2)/4
so 4y = -3x -2

b. i) midpoint is the half of the sum of each coordinate
midpoint = { (p+4)/2 , (q+5)/2 }

ii) 7 = (p+4)/2 , 14 = p+4 , p = 14 -4 =10
1 = (q+5)/2 , 2 = q+5 , q= 2-5 = -3
D= ( 10, -3)

Answer 2

i.
gradient = slope = rise/run = (-5 - 1)/(6 - (-2)) = -6/8 = -3/4

y - y1 = slope(x - x1)
y - 1 = -3/4(x + 2)
4y - 4 = -3x - 6
3x + 4y = -2

b.

i.
midpoint (x1/2 + x2/2,y1/2 + y2/2) = (2 + p/2,2 1/2 + q/2)

ii.
(2 + p/2,2 1/2 + q/2) = (7,1)
2 + p/2 = 7
p/2 = 5
p = 10
2 1/2 + q/2 = 1
q/2 = -1 1/2
q = -3
D(p,q) = D(10,-3)

^_^

Answer 3

a.
1. gradient= -3/4
2. 3x+4y+2=0

b.
1. [(p+4)/2;(q+5)/2]
2. (10,-3)