Find point B of a segment AB if A is located at (-3,7) and the midpoint of AB is located at (1,3).

Options

A. (-2,10)
B. (-1,5)
C. (3,-7)
D. (5,-1)

Answer

To find point B of the segment AB given that point A is located at (-3, 7) and the midpoint of AB is located at (1, 3), we can follow these steps:

### Step 1: Understand the midpoint formula
The midpoint M of a line segment AB, where A has coordinates (x1, y1) and B has coordinates (x2, y2), is given by the formula:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]

### Step 2: Assign the known values
From the problem, we have:
- Point A: \( A(-3, 7) \)
- Midpoint M: \( M(1, 3) \)
- Let point B have coordinates \( B(x, y) \)

### Step 3: Set up the equations using the midpoint formula
Using the midpoint formula, we can write two equations based on the coordinates of the midpoint:
1. For the x-coordinates:
\[ 1 = \frac{-3 + x}{2} \]
2. For the y-coordinates:
\[ 3 = \frac{7 + y}{2} \]

### Step 4: Solve for x
Multiply both sides of the first equation by 2 to eliminate the fraction:
\[ 2 = -3 + x \]
Now, add 3 to both sides:
\[ x = 2 + 3 = 5 \]

### Step 5: Solve for y
Multiply both sides of the second equation by 2:
\[ 6 = 7 + y \]
Now, subtract 7 from both sides:
\[ y = 6 - 7 = -1 \]

### Step 6: State the coordinates of point B
Now we have found the coordinates of point B:
\[ B(5, -1) \]

### Final Answer
The coordinates of point B are \( (5, -1) \).

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