Question

(1, 5) and (–3, –1) are the co-ordinates of vertices A and C respectively of rhombus ABCD. Find the equations of the diagonals AC and BD.

Answer 1

A = (1, 5) and C = (–3, –1)

We know that in a rhombus, diagonals bisect each other at right angle.

Let O be the point of intersection of the diagonals AC and BD.

Co-ordinates of O are

`((1 - 3)/2, (5 - 1)/2) = (-1, 2)`

Slope of AC = `(-1 - 5)/(-3 - 1) = (-6)/-4 = 3/2`

For line AC:

Slope = m = `3/2, (x_1, y_1) = (1, 5)`

Equation of the line AC is

y – y₁ = m(x – x₁)

`y - 5 = 3/2 (x - 1)`

2y − 10 = 3x − 3

3x − 2y + 7 = 0

For line BD:

Slope = m = `(-1)/("slope  of AC") = (-2)/3`,

(x₁, y₁) = (−1, 2)

Equation of the line BD is

y – y₁ = m(x – x₁)

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

3y − 6 = −2x − 2

2x + 3y = 4