Question

Use the following graph and answer the given questions:

1. Write the co-ordinates of points A, B and C.
2. Find the equation of a line passing through the mid-point of AC and parallel to AB.

Answer 1

(a) From the graph,

Coordinates of: A(4, 8), B(−1, 2), C(6, 2)

Hence, A(4, 8), B(−1, 2), C(6, 2).

(b) Midpoint of AC,

By formula,

Midpoint = `((x_1+x_2)/2,(y_1+y_2)/2)`

Substituting values, we get:

Mid-point of AC = `((4 + 6)/2, (8 + 2)/6)`

= (5, 5)

By formula,

Slope =`(y_2 - y_1)/(x_2 - x_1)`

Thus,

Slope of AB = `(8−2)/(4−(−1))`

= `6/5`

Since the slopes of parallel lines are equal, the slope of the required line (m) = `6/5`

By point-slope form,

Equation: y − y₁ = m(x − x₁)

The equation of a line parallel to AB and passing through the midpoint of AC is given by,

⇒ y − 5 = `6/5 (x - 5)`

⇒ 5(y − 5) = 6(x − 5)

⇒ 5y − 25 = 6x − 30

⇒ 5y − 25 + 30 = 6x − 30

⇒ 5y − 5 = 6x

⇒ 6x − 5y − 5 = 0

Hence, the equation of the required line = 6x − 5y − 5 = 0.