Question:
The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is
Updated On: Jul 29, 2025
- -5
- -6
- -7
- -8
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The Correct Option is D
Solution and Explanation
To solve for the constant \( c \) in the equation of the diagonal \( y = 3x + c \), we start by finding the equation of the line that forms one diagonal of the rectangle. Given the end points of this diagonal are \( (2, 5) \) and \( (6, 3) \), the slope \( m \) of this line is:
\[ m = \frac{3 - 5}{6 - 2} = \frac{-2}{4} = -\frac{1}{2} \]
Using the midpoint formula for the diagonals of a rectangle, the midpoints of both diagonals are the same. Let's find the midpoint of the diagonal with given endpoints:
\[ \left(\frac{2+6}{2}, \frac{5+3}{2}\right) = \left(4, 4\right) \]
Since the diagonals bisect each other, the midpoint \( (4, 4) \) lies on the other diagonal \( y = 3x + c \). Substituting \( (4, 4) \) into this equation:
\[ 4 = 3(4) + c \]
\[ 4 = 12 + c \]
Solving for \( c \):
\[ c = 4 - 12 = -8 \]
Thus, the value of \( c \) is \(-8\).
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