Question

There is a line defined by two end-points, \( (11, -5) \) and \( (a, b) \). The midpoint between these two points is \( (-6, -21) \). What is the value of the point \( (a, b) \)?

• A. \( (4, -19) \)
• B. \( (12, -14) \)
• C. \( (-14, -25) \)
• D. \( (-23, -37) \)
• E. \( (5, -26) \)

Hint:

Always use the midpoint formula carefully: average the \(x\)-coordinates and \(y\)-coordinates separately.

Answer 1

The Correct Option is D

Step 1: Apply midpoint formula.
For two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint is: \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Step 2: Substitute given values.
\[ \left( \frac{11 + a}{2}, \frac{-5 + b}{2} \right) = (-6, -21) \] Step 3: Solve for \(a\).
\[ \frac{11 + a}{2} = -6 \quad \Rightarrow \quad 11 + a = -12 \quad \Rightarrow \quad a = -23 \] Step 4: Solve for \(b\).
\[ \frac{-5 + b}{2} = -21 \quad \Rightarrow \quad -5 + b = -42 \quad \Rightarrow \quad b = -37 \] Final Answer: \[ \boxed{(-23, -37)} \]