Question

A = (7, −2) and C = (−1, −6) are the vertices of square ABCD. Find the equations of diagonals AC and BD.

Answer 1

We know that in a square, 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

`((7 - 1)/2, (-2 - 6)/2) = (3, -4)`

Slope of AC = `(-6 + 2)/(-1 - 7) = (-4)/-8 = 1/2`

For line AC: 

Slope = m = `1/2`, (x₁, y₁) = (7, –2)

Equation of the line AC is

y – y₁ = m(x – x₁)

`y + 2 =1/2 (x - 7)`

2y + 4 = x – 7

2y = x – 11

For line BD:

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

(x₁, y₁) = (3, −4)

Equation of the line BD is

y − y₁ = m(x − x₁)

y + 4 = −2(x − 3)

y + 4 = −2x + 6

2x + y = 2