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:

Check which option is satisfied by the midpoint $(5, -2)$.
(A): $5(5) + 2(-2) = 25 - 4 = 21$. Correct.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The perpendicular bisector passes through the midpoint of the segment and has a slope that is the negative reciprocal of the segment's slope.
Step 2: Key Formula or Approach:
Midpoint $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$
Slope $m_{\perp} = -\frac{1}{m_{\text{line}}}$
Step 3: Detailed Explanation:
Midpoint of $(10, 0)$ and $(0, -4)$ is $M = \left(\frac{10+0}{2}, \frac{0-4}{2}\right) = (5, -2)$.
Slope of segment $= \frac{-4 - 0}{0 - 10} = \frac{2}{5}$.
Slope of perpendicular bisector $= -\frac{5}{2}$.
Equation: $y - (-2) = -\frac{5}{2}(x - 5)$
$2(y + 2) = -5(x - 5) \Rightarrow 2y + 4 = -5x + 25 \Rightarrow 5x + 2y = 21$.
Step 4: Final Answer:
The equation is $5x + 2y = 21$.