Question

The points B(1, 3) and D(6, 8) are two opposite vertices of a square ABCD. Find the equation of the diagonal AC.

Answer 1

The diagonals of a square bisect each other, so the midpoint of BD is also on AC,

Given B(1, 3) and D(6, 8):

Using the midpoint formula:

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

`M = ((1 + 6)/2, (3 + 8)/2)`

∴ `M = (7/2, 11/2)` or (3.5, 5.5)

Using the slope formula with the diagonal BD:

`m_1 = (y_2 - y_1)/(x_2 - x_1)`

`m_1 = (8 - 3)/(6 - 1)`

∴ `m_1 = 5/5`

Since AC is perpendicular to BD, the product of their slopes is −1:

m_AC × 1 = −1

∴ m_AC = −1

Using the point-slope formula with the midpoint `M = (7/2, 11/2)` and slope m = −1:

y − y₁ = m(x − x₁)

`y - 11/2 = -1(x - 7/2)`

Multiplying by 2 to clear fractions:

2y − 11 = −2x + 7

Let’s rearrange into the general form (Ax + By + C = 0):

2x + 2y − 18 = 0

x + y − 9 = 0     ...[Divided by 2]

Hence, the equation of the diagonal AC is x + y − 9 = 0.