Question

If R (5, 6) is the midpoint of the line segment AB joining the points A (6, 5) and B (4, y), then y equals to

• A. 5
• B. 7
• C. 6
• D. 12

Hint:

The coordinates of the endpoints and the midpoint form an arithmetic progression.
For the y-coordinates: the sequence from \(A\) to \(R\) to \(B\) is \(5 \to 6 \to y\).
Since the step from \(5\) to \(6\) is \(+1\), the next step from \(6\) to \(y\) must also be \(+1\).
Thus, \(y = 6 + 1 = 7\) instantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from Coordinate Geometry, focusing on the midpoint of a line segment.
We are given the coordinates of one endpoint \(A\), the midpoint \(R\), and the coordinates of the other endpoint \(B\) with an unknown y-coordinate. We need to find \(y\).

Step 2: Key Formula or Approach:
The Midpoint Formula states that the coordinates of the midpoint \((x_m, y_m)\) of the segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) are given by:
\[ x_m = \frac{x_1 + x_2}{2} \]
\[ y_m = \frac{y_1 + y_2}{2} \]

Step 3: Detailed Explanation:
Here, the coordinates are:
Endpoint \(A(x_1, y_1) = (6, 5)\)
Endpoint \(B(x_2, y_2) = (4, y)\)
Midpoint \(R(x_m, y_m) = (5, 6)\)
Let us apply the midpoint formula for the y-coordinate:
\[ y_m = \frac{y_1 + y_2}{2} \]
Substitute the known values:
\[ 6 = \frac{5 + y}{2} \]
Multiply both sides by 2 to eliminate the denominator:
\[ 12 = 5 + y \]
Subtract 5 from both sides to solve for \(y\):
\[ y = 12 - 5 \]
\[ y = 7 \]
We can also verify the x-coordinate:
\[ x_m = \frac{6 + 4}{2} = \frac{10}{2} = 5 \]
This matches the given midpoint x-coordinate of 5, confirming our calculation.

Step 4: Final Answer:
The value of \(y\) is \(7\).