Question:
If \(A(2,-3)\) and \(C(-6,7)\) are opposite vertices of a rhombus ABCD, then the equation of diagonal BD is
If \(A(2,-3)\) and \(C(-6,7)\) are opposite vertices of a rhombus ABCD, then the equation of diagonal BD is
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The diagonals of a rhombus bisect each other at right angles.
Updated On: Oct 1, 2026
- \(5x-4y+2 = 0\)
- \(5x+4y+2 = 0\)
- \(4x-5y+18 = 0\)
- \(4x+5y+18 = 0\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
In a rhombus, diagonals bisect each other perpendicularly. So BD passes through the midpoint of AC and is perpendicular to AC.
Step 2: Key Formula or Approach
Midpoint of AC, slope of AC, then slope of BD is the negative reciprocal.
Step 3: Detailed Explanation
Midpoint of \(A(2,-3)\) and \(C(-6,7)\): \((-2,2)\).
Slope of AC: \(\dfrac{7-(-3)}{-6-2}=-\dfrac{5}{4}\), so the slope of BD is \(\dfrac45\).
Line through \((-2,2)\): \(y-2=\dfrac45(x+2)\), so \(5y-10=4x+8\), which gives
\[ 4x-5y+18=0 \]
Final Answer:
The diagonal BD is \(4x-5y+18=0\), option (C). \[ \boxed{4x-5y+18=0\ \text{(C)}} \]
In a rhombus, diagonals bisect each other perpendicularly. So BD passes through the midpoint of AC and is perpendicular to AC.
Step 2: Key Formula or Approach
Midpoint of AC, slope of AC, then slope of BD is the negative reciprocal.
Step 3: Detailed Explanation
Midpoint of \(A(2,-3)\) and \(C(-6,7)\): \((-2,2)\).
Slope of AC: \(\dfrac{7-(-3)}{-6-2}=-\dfrac{5}{4}\), so the slope of BD is \(\dfrac45\).
Line through \((-2,2)\): \(y-2=\dfrac45(x+2)\), so \(5y-10=4x+8\), which gives
\[ 4x-5y+18=0 \]
Final Answer:
The diagonal BD is \(4x-5y+18=0\), option (C). \[ \boxed{4x-5y+18=0\ \text{(C)}} \]
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