In which quadrant does the mid-point of the line segment joining the points (-1, 2) and (3, 4) Lies?

Options

A. I
B. II
C. Ill
D. IV

Answer

To find the quadrant in which the midpoint of the line segment joining the points (-1, 2) and (3, 4) lies, we will follow these steps:

### Step 1: Identify the coordinates
The two points given are:
- Point A: (-1, 2) which we can denote as (x1, y1)
- Point B: (3, 4) which we can denote as (x2, y2)

### Step 2: Use the midpoint formula
The formula for the midpoint (M) of a line segment joining two points (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]

### Step 3: Substitute the values
Now, we will substitute the values of the coordinates into the midpoint formula:
- x1 = -1, y1 = 2
- x2 = 3, y2 = 4

Calculating the x-coordinate of the midpoint:
\[ M_x = \frac{-1 + 3}{2} = \frac{2}{2} = 1 \]

Calculating the y-coordinate of the midpoint:
\[ M_y = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]

### Step 4: Determine the coordinates of the midpoint
Thus, the midpoint M is:
\[ M = (1, 3) \]

### Step 5: Identify the quadrant
Now, we need to determine in which quadrant the point (1, 3) lies.

- The x-coordinate (1) is positive.
- The y-coordinate (3) is also positive.

In the Cartesian coordinate system:
- Quadrant I is where both x and y coordinates are positive.

### Conclusion
Therefore, the midpoint (1, 3) lies in the **first quadrant**.

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