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. III
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 of the points
Let the first point be \( P_1(-1, 2) \) and the second point be \( P_2(3, 4) \).

### Step 2: Use the midpoint formula
The formula for the midpoint \( M \) of a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 3: Substitute the coordinates into the formula
Here, \( x_1 = -1 \), \( y_1 = 2 \), \( x_2 = 3 \), and \( y_2 = 4 \). Plugging these values into the midpoint formula gives:
\[ M = \left( \frac{-1 + 3}{2}, \frac{2 + 4}{2} \right) \]

### Step 4: Calculate the x-coordinate of the midpoint
Calculating the x-coordinate:
\[ M_x = \frac{-1 + 3}{2} = \frac{2}{2} = 1 \]

### Step 5: Calculate the y-coordinate of the midpoint
Calculating the y-coordinate:
\[ M_y = \frac{2 + 4}{2} = \frac{6}{2} = 3 \]

### Step 6: Determine the coordinates of the midpoint
Thus, the midpoint \( M \) is:
\[ M(1, 3) \]

### Step 7: Identify the quadrant
Since both coordinates \( M_x = 1 \) and \( M_y = 3 \) are positive, the midpoint lies in the **first quadrant**.

### Final Answer:
The midpoint of the line segment joining the points (-1, 2) and (3, 4) lies in the **first quadrant**.

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