Question

ABCD is a rhombus. The co-ordinates of A and C are (3, 6) and (−1, 2) respectively. Write down the equation of BD.

Answer 1

In a rhombus, the diagonals bisect each other,

Therefore, the midpoint of the diagonal AC is also a point on the diagonal BD,

Using the midpoint formula for A(3, 6) and C(−1, 2):

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

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

`M = (2/2, 8/2)`

∴ M = (1, 4)

Using the slope formula:

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

`m_(AC) = (2 - 6)/(-1 - 3)`

`m_(AC) = (-4)/-4`

∴ m = 1

The diagonals of a rhombus are perpendicular to each other,

Thus, the slope of BD (m_BD) is the negative reciprocal of the slope of AC,

`m_(BD) = - 1/m_(AC)`

`m_(BD) = - 1/1`

m_BD = −1

Using the point-slope formula with the midpoint (1, 4) and slope m = −1:

y − y₁ = m(x − x₁)

y − 4 = −1(x − 1)

y − 4 = −x + 1

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

x + y − 4 − 1 = 0

x + y − 5 = 0

Hence, the equation of the diagonal BD is x + y − 5 = 0.