Question: In the given figure, ABCD is a rhombus. Find the equation of diagonals AC and BD if diagonal AC passes through the point (8,-4).

Solution:



We know,

Diagonals of a rhombus intersect at right angle.

Given,
ABCD is a rhombus.
AC and BD are the diagonals.
Coordinates of A are (2,- 4) and the coordinates of B are (4,-3).
P is any point in AC whose coordinates are (8,-4).

Since, AC is a straight line, slope of AC = slope of AP.
Let A(2,- 4) be ($x_1,y_1$) and P(8,- 4) be ($x_2,y_2$)
Slope of AP (or AC) ($m_1$) = $\dfrac{y_2 - y_1}{x_2 - x_1}$
$= \dfrac{-4 -(-4)}{8 -2}$
$= \dfrac{0}{6}$
$= 0$

Now, equation of diagonal AC when A(2,-4) = ($x_1,y_1$) and $m_1$ = 0,
$or, y - y_1 = m_1 (x - x_1)$
$or, y -(-4) = 0(x -2)$
$or, y +4 = 0$
$or, y = -4$ is the required equation of AC.

Since, AC and BD are perpendicular.
$m_1 × m_2 = -1$
$or, 0 × m_2 = -1$
$or, m_2 = \dfrac{-1}{0}$

And, equation of diagonal BD when B(4,-3) = ($x_1,y_1$) and $m=2$ = \frac{-1}{0}$,
$or, y - y_1 = m_2(x - x_1)$
$or, y -(-3) = \dfrac{-1}{0} (x -4)$
$or, 0×(y +3) = -(x -4)$
$or, 0 = -x +4$
$or, x = 4$ is the required equation of BD.

Therefore, the required equation of AC and BD are (y= -4) and (x = 4), respectively.