# What is an equation of the line that passes through the points (-5, 6) and (7, 6)?

###### Question:

## Answers

(a) The equation of AD is y = 4.5·x -7.5

The equation of BE is y = -0.3·x + 0.5

The equation of CF is y = -1.5·x + 2.5.

(b) The three equations representing the three lines AD, BE and CF, pass through the same point [tex](1\frac{2}{3} , 0)[/tex] (which can be written as (1.667, 0)).

Step-by-step explanation:

The given triangle coordinates are;

A = (3, 6)

B = (-5, 2)

C = (7, -8)

The midpoint of BC = D

The midpoint of AC = E

The midpoint of AB = F

Therefore, we have;

The coordinates of point D = (B + C)/2 = (((-5) + 7 )/2, (2 + (-8))/2)

The coordinates of point D = (1, -3)

The coordinates of point E = (A + C)/2 = ((3 + 7 )/2, (6 + (-8))/2)

The coordinates of point E = (5, -1)

The coordinates of point F = (A + B)/2 = ((3 + (-5) )/2, (6 + 2)/2)

The coordinates of point F = (-1, 4)

The equation of AD in slope and intercept form is therefore;

[tex]Slope, \, m =\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

A = (3, 6), D = (1, -3)

[tex]m =\dfrac{(-3)-6}{1-3} = 4.5[/tex]

Therefore, we have;

y - 6 = 4.5×(x - 3)

y - 6 = 4.5·x -13.5

y = 4.5·x -13.5 + 6 = 4.5·x -7.5

y = 4.5·x -7.5

The equation of BE in slope and intercept form is found as follows;

B = (-5, 2), E = (5, -1)

Slope = ((-1)-2)/(5 - (-5)) = -0.3

The point slope equation is y - 2 = -0.3×(x - (-5))

y - 2 = -0.3·x - 1.5

y = -0.3·x - 1.5 + 2 = -0.3·x + 0.5

y = -0.3·x + 0.5

The equation of CF in slope and intercept form is found as follows;

C = (7, -8), F = (-1, 4)

Slope = (4 - (-8))/((-1) - 7) = -1.5

The point slope equation is y - (-8) = -1.5×(x - 7)

y + 8 = -1.5·x + 10.5

y = -1.5·x + 10.5 - 8

y = -1.5·x + 2.5.

Where CF and BE intersect, we have;

-0.3·x + 0.5 = -1.5·x + 2.5.

1.5·x -0.3·x = 2.5 - 0.5 = 2.0

x = 2.0/1.2 = 5/3

y = -1.5×5/3+ 2.5 = 0

We check for line AD, where y = 4.5·x -7.5

When x = 5/3, we get;

y = 4.5×(5/3) -7.5 = 0

Therefore, the three equations pass through the point (5/3, 0) which is the same point.

Step-by-step explanation:

[tex]What is an equation of the line that passes through the points (-5, 6) and (7, 6)?[/tex]