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

Answer 1

The Correct Option is A

Step 1: Find the midpoint of the given line segment 

Midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

Given points: \( (10,0) \) and \( (0,-4) \)

\( M = \left( \frac{10 + 0}{2}, \frac{0 + (-4)}{2} \right) = \left( \frac{10}{2}, \frac{-4}{2} \right) = (5, -2) \)

Step 2: Find the slope of the given line segment

Slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

\( m = \frac{-4 - 0}{0 - 10} = \frac{-4}{-10} = \frac{2}{5} \)

Step 3: Find the slope of the perpendicular bisector

The perpendicular slope is the negative reciprocal of \( \frac{2}{5} \):

\( m' = -\frac{5}{2} \)

Step 4: Find the equation of the perpendicular bisector

Using point-slope form: \( y - y_1 = m(x - x_1) \)

\( y - (-2) = -\frac{5}{2} (x - 5) \)

\( y + 2 = -\frac{5}{2}x + \frac{25}{2} \)

Multiplying everything by 2 to eliminate fractions:

\( 2y + 4 = -5x + 25 \)

\( 5x + 2y = 21 \)

Thus, the correct answer is:

\( 5x + 2y = 21 \)