Question

The equation of perpendicular bisector of the line segment joining the points (10, 0) and (0, -4) is

• A. \( 5x + 2y = 21 \)
• B. \( 5x + 2y = 0 \)
• C. \( 2x - 5y = 21 \)
• D. \( 5x - 2y = 21 \)
• E. \( 2x + 3y = 21 \)

Hint:

The perpendicular bisector of a line segment passes through the midpoint of the segment and has a slope that is the negative reciprocal of the slope of the line segment.

Answer 1

The Correct Option is A

Step 1: Find the midpoint of the segment.
The midpoint of the line segment joining the points \( (10, 0) \) and \( (0, -4) \) is calculated using the midpoint formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of the points: \[ \text{Midpoint} = \left( \frac{10 + 0}{2}, \frac{0 + (-4)}{2} \right) = \left( 5, -2 \right) \]
Step 2: Find the slope of the line.
The slope of the line joining the points \( (10, 0) \) and \( (0, -4) \) is given by the formula: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{0 - 10} = \frac{-4}{-10} = \frac{2}{5} \]
Step 3: Find the slope of the perpendicular bisector.
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line. Since the slope of the line is \( \frac{2}{5} \), the slope of the perpendicular bisector is \( -\frac{5}{2} \).

Step 4: Use the point-slope form of the equation of the line.
The equation of the perpendicular bisector can be written in point-slope form as: \[ y - y_1 = m(x - x_1) \] Where \( m \) is the slope and \( (x_1, y_1) \) is the midpoint. Substituting \( m = -\frac{5}{2} \), \( x_1 = 5 \), and \( y_1 = -2 \): \[ y + 2 = -\frac{5}{2}(x - 5) \] Simplifying the equation: \[ y + 2 = -\frac{5}{2}x + \frac{25}{2} \] Multiply through by 2 to eliminate the fraction: \[ 2y + 4 = -5x + 25 \] Rearranging to get the equation in standard form: \[ 5x + 2y = 21 \]
Step 5: Conclusion.
The equation of the perpendicular bisector is \( 5x + 2y = 21 \), which corresponds to option (A).
\[ \boxed{5x + 2y = 21} \]
Final Answer:} \( 5x + 2y = 21 \)