Question

Write your question here (Keep it simple and clear to get the best answer)striangle has vertices A(2;5); B(1;-2) and C(-5;1). Determine:(a) the equation of the line BC. (b) The equation of the perpendicular line from A to Bc

Answer 1

(a) y =
`(-1)/(2)`x -
`(3)/(2)`

(b) y = 2x + 3

Explanation:

(a) The equation of a line given by points M(x₁, y₁) and N(x₂, y₂) is given by:

y - y₁ = m(x - x₁) -------------------(i)

Where;

m =
`(y_2 - y_1)/(x_2 - x_1)` = slope or gradient of the line ---------------(ii)

Given points on the triangle are:

A(2,5)

B(1,-2)

C(-5,1)

To find the equation of line BC, we use the formulas in equations (i) and (ii) where the points of the line are B(1,-2) and C(-5,1) and;

x₁ = 1

y₁ = -2

x₂ = -5

y₂ = 1

==> First get the gradient using equation (ii) as follows;

m =
`(y_2 - y_1)/(x_2 - x_1)` =
`(1 - (-2))/(-5 -1)`

m =
`(3)/(-6)`

m =
`(-1)/(2)`

==> Now, use equation (i) to find the equation of the line as follows;

y - (-2) =
`(-1)/(2)`(x - 1)

y + 2 =
`(-1)/(2)`(x - 1)

Multiply both sides by 2

2(y+2) = -1 ( x - 1 )

2y + 4 = -x + 1

2y = -x + 1 - 4

2y = - x - 3

y =
`(-1)/(2)`x -
`(3)/(2)`

Therefore, the equation of the line is y =
`(-1)/(2)`x -
`(3)/(2)`

(b) To find the perpendicular line from A to BC, note that

i. two lines are perpendicular if they meet at 90°

ii. the general equation of a line could also be written as y = mx + c where m is the slope and c is the intercept.

iii. when one line has a slope of m, then a perpendicular line to that line will have a slope of
`(-1)/(m)`

The equation of line BC is y =
`(-1)/(2)`x -
`(3)/(2)`.

This means that BC has a slope of
`(-1)/(2)`

A perpendicular line from A to BC will have a slope of 2.

Now to get the equation of this perpendicular line from A(2, 5) to BC, we use the general equation of a line given in equation (i)

where;

m = 2

x₁ = 2

y₁ = 5

Substitute these values into equation (i)

y - 5 = 2(x - 2)

Solving by simplification gives;

y - 5 = 2x - 2

y = 2x + 3

Therefore, the equation of the perpendicular line is y = 2x + 3