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

To find the equation of the perpendicular bisector of the line segment joining the points \((10, 0)\) and \((0, -4)\), we need to follow these steps:

1. Identify the midpoint of the line segment:
   • The midpoint formula for a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).
   • Substituting the given points: \(\left( \frac{10 + 0}{2}, \frac{0 - 4}{2} \right) = (5, -2)\).
2. Find the slope of the line segment:
   • The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
   • Substituting the given points: \(m = \frac{-4 - 0}{0 - 10} = \frac{-4}{-10} = \frac{2}{5}\).
3. Determine the slope of the perpendicular bisector:
   • The slope of a line perpendicular to another with slope \(m\) is given by: \(m' = -\frac{1}{m}\).
   • For the given slope \( \frac{2}{5} \), the perpendicular slope is: \(m' = -\frac{1}{\frac{2}{5}} = -\frac{5}{2}\).
4. Use the point-slope form to find the equation of the perpendicular bisector:
   • The point-slope form of a line is: \(y - y_1 = m'(x - x_1)\).
   • Using the midpoint \((5, -2)\) and perpendicular slope \(-\frac{5}{2}\), \(y + 2 = -\frac{5}{2}(x - 5)\).
   • Expanding and simplifying gives: \(y + 2 = -\frac{5}{2}x + \frac{25}{2}\), \(2y + 4 = -5x + 25\), \(5x + 2y = 25 - 4\), \(5x + 2y = 21\).

Thus, the equation of the perpendicular bisector of the line segment joining the points \((10, 0)\) and \((0, -4)\) is \(5x + 2y = 21\).