points and line segments

Question

Given the points A(-1, 6) and B(2, -3), find the following??
a. The slope of AB
b. Length of AB
c. The co-ordinates of the midpoint of AB
d. The equation of a line perpendicular to AB that passes through the point (4, 7). Leave your answer in standard form

Answer 1

A(-1,6), B(2,-3)

 

slope = (change in y)/(change in x)

m= (-3-6)/(2-(-1)) = -9/3 = -3

 

Length is distance from A to B

d = √[(x₂-x₁)²+(y₂-y₁)²]

d = √[(2-(-1))²+(-3-6)²]

d = √[(3)²+(-9)²]

d = √90

d = 3√10

 

Midpoint = [(x₁+x₂)/2 , (y₁+y₂)/2]

= ((-1+2)/2 , (6-3)/2)

= (1/2 , 3/2)

 

Perpendicular lines have negative reciprocal

slopes... m⊥ = -1/m

m⊥ = -1/-3 = 1/3

Equation of line with slope 1/3 through (4,7)

y -y₁ = m(x-x₁)

y - 7 = (1/3)(x-4)

3y - 21 = x-4

-x + 3y = 17  or  x - 3y = -17