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

${\displaystyle (\frac{1-3}{2},\frac{5-1}{2})=(-1,2)}$

Slope of AC = ${\displaystyle \frac{-1-5}{-3-1}=\frac{-6}{-4}=\frac{3}{2}}$

For line AC:

Slope = m = ${\displaystyle \frac{3}{2},(x_1,y_1)=(1,5)}$

Equation of the line AC is

y – y₁ = m(x – x₁)

${\displaystyle y-5=\frac{3}{2}(x-1)}$

2y − 10 = 3x − 3

3x − 2y + 7 = 0

For line BD:

Slope = m = ${\displaystyle \frac{-1}{\text{slope of AC}}=\frac{-2}{3}}$,

(x₁, y₁) = (−1, 2)

Equation of the line BD is

y – y₁ = m(x – x₁)

${\displaystyle y-2=\frac{-2}{3}(x+1)}$

3y − 6 = −2x − 2

2x + 3y = 4