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प्रश्न
(1, 5) and (–3, –1) are the co-ordinates of vertices A and C respectively of rhombus ABCD. Find the equations of the diagonals AC and BD.
उत्तर
A = (1, 5) and C = (–3, –1)
We know that in a rhombus, diagonals bisect each other at right angle.
Let O be the point of intersection of the diagonals AC and BD.
Co-ordinates of O are
`((1 - 3)/2, (5 - 1)/2) = (-1, 2)`
Slope of AC = `(-1 - 5)/(-3 - 1) = (-6)/-4 = 3/2`
For line AC:
Slope = m = `3/2, (x_1, y_1) = (1, 5)`
Equation of the line AC is
y – y1 = m(x – x1)
`y - 5 = 3/2 (x - 1)`
2y − 10 = 3x − 3
3x − 2y + 7 = 0
For line BD:
Slope = m = `(-1)/("slope of AC") = (-2)/3`,
(x1, y1) = (−1, 2)
Equation of the line BD is
y – y1 = m(x – x1)
`y - 2 = (-2)/3 (x + 1)`
3y − 6 = −2x − 2
2x + 3y = 4
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