Question

ABCD is a rhombus the coordinates of A and C are (3,6) and (-1,2) write the equation of BD

Answer 1

Please refer to the attachment below. I hope it is helpful. (PS - Excuse my handwriting)

Answer 2

Answer:

$\boxed{ x + y = 5 }$

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We know that the diagonals of a rhombus bisect at right angles.

AC ⊥ BD

Hence the slope of AC is $\frac{ - 1 }{\bf{ slope\: of\: BD}}$

Slope of AC

$\implies \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

$\implies \frac{2 - 6}{-1 - (3)}$

$\implies \frac{-4}{-4}$

$\implies 1$

Slope of BD

= $\frac{-1}{\bf{slope\:of\: AC}}$

$\implies \frac{-1}{1}$

$\implies-1$

O passes through BD

O is also the midpoint of A and C

Let O be x,y

Midpoint formula:

$x=\frac{3-1}{2}$

$x=\frac{2}{2}$

$\implies x=1$

$y=\frac{6+2}{2}$

$\implies y=\frac{8}{2}$

$\implies y=4$

Equation of BD

$\implies y - y_{1} = m ( x - x_{1} )$

$y_{1} = 4$

$x_{1} = 1$

$\implies y - 4 = -1( x - 1)$

$\implies y - 4 = 1 - x$

$\implies x + y = 5$

The equation of BD is : $\boxed{ x + y = 5 }$

Hope it helps you

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