Question

A point C lies on the perpendicular bisector of the straight-line segment joining the points A(-3,-6) and B(13,-6). If the point C lies in the first quadrant and the distance between the point C and the midpoint of AB is 8 units, then the coordinates of C are

• A. (5,1)
• B. (5,2)
• C. (5,3)
• D. (5,4)
• E. (5,5)

Hint:

Recognizing that a segment is horizontal or vertical can drastically simplify geometry problems. Here, knowing AB is horizontal (\(y_1 = y_2\)) tells you immediately that the bisector is vertical (\(x = \text{const}\)).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The segment AB is horizontal because the y-coordinates of A and B are the same. Its perpendicular bisector will therefore be a vertical line passing through the midpoint of AB.

Step 2: Key Formula or Approach:
1. Midpoint of AB: \(M = \left(\frac{x_A+x_B}{2}, y\right)\).
2. Distance formula for vertical displacement: \(d = |y_C - y_M|\).

Step 3: Detailed Explanation:

Step 1: Find the midpoint \(M\) of A(-3, -6) and B(13, -6).
\[ M = \left(\frac{-3 + 13}{2}, -6\right) = (5, -6) \]

Step 2: Since AB is on the line \(y = -6\), its perpendicular bisector is the vertical line \(x = 5\).
Point C lies on this line, so its coordinates are \((5, y)\).

Step 3: The distance between \(C(5, y)\) and \(M(5, -6)\) is given as 8 units.
\[ \sqrt{(5-5)^2 + (y - (-6))^2} = 8 \]
\[ |y + 6| = 8 \]
This gives two possibilities:
1. \(y + 6 = 8 \implies y = 2\)
2. \(y + 6 = -8 \implies y = -14\)

Step 4: Since C is in the first quadrant, its y-coordinate must be positive. Therefore, \(y = 2\).

Step 4: Final Answer:
The coordinates of C are (5, 2).