Question

(a) The point A (7, 3) and C (0, - 4) are two opposite vertices of rhombus ABCD. Find the equation of diagonal BD. ​

Answer 1

A( 7,3) and C(0,-4) are two opposite vertices of Rhombus ABCD.

$Mid-point \: of \: AB \:is \: O \\= \Big(\frac{x_{1}+x_{2}}{2} , \frac{x_{1}+x_{2}}{2}\Big)\\= \Big( \frac{7+0}{2}, \frac{3-4}{2}\Big) \\= \Big( \frac{7}{2}, \frac{-1}{2}\Big) \: --(1)$

$Slope \: of \: the \: diagonal\:AB (m) \\= \frac{y_{2}-y_{1}}{x_{2}-x_{1}}\\= \frac{-4-3}{0-7}\\= \frac{-7}{-7} \\= 1 \: --(2)$

$Slope \: of \: the \: diagonal \: BD \\= \frac{-1}{m} \: \blue {( AC \: \perp BD )}$

$= -1 \: --(3)$

$\underline {\blue {Equation \: of \:the \: diagonal \: BD : }}$

$Slope \: of \: BD = -1 \: and \: it \: passes \:through \:the \:point \: O\Big( \frac{7}{2}, \frac{-1}{2}\Big)$

$y - y_{1} = slope \times (x-x_{1})$

$\implies y - \Big(\frac{-1}{2}\Big) = (-1)\Big(x - \frac{7}{2}\Big)$

$\implies y + \Big(\frac{1}{2}\Big) = -x + \frac{7}{2}$

/* Multiplying each term by 2 , we get */

$\implies 2y + 1 = -2x + 7$

$\implies 2x + 2y = 6$

/* Divide each term by 2 , we get */

$\implies x + y = 3$

Therefore.,

$\red { Required \: Equation \: of \: diagonal \:BD}$

$\green { x + y = 3 }$

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