Coordinates of midpoint of line joining two points `((5)/(7),(-3)/(2)) and ((-5)/(7),(3)/(2))` are :

Options

A. `(5, -3)`
B. `((5)/(14),(3)/(4))`
C. `(0,0)`
D. None of these

Answer

To find the coordinates of the midpoint of the line joining the two points \((\frac{5}{7}, -\frac{3}{2})\) and \((-\frac{5}{7}, \frac{3}{2})\), we will use the midpoint formula. The midpoint \(M\) of a line segment connecting two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 1: Identify the coordinates of the points
Let:
- Point \(A\) be \((x_1, y_1) = \left(\frac{5}{7}, -\frac{3}{2}\right)\)
- Point \(B\) be \((x_2, y_2) = \left(-\frac{5}{7}, \frac{3}{2}\right)\)

### Step 2: Calculate the x-coordinate of the midpoint
Using the formula for the x-coordinate of the midpoint:

\[ x = \frac{x_1 + x_2}{2} = \frac{\frac{5}{7} + \left(-\frac{5}{7}\right)}{2} \]

Calculating the numerator:

\[ \frac{5}{7} - \frac{5}{7} = 0 \]

So,

\[ x = \frac{0}{2} = 0 \]

### Step 3: Calculate the y-coordinate of the midpoint
Now, using the formula for the y-coordinate of the midpoint:

\[ y = \frac{y_1 + y_2}{2} = \frac{-\frac{3}{2} + \frac{3}{2}}{2} \]

Calculating the numerator:

\[ -\frac{3}{2} + \frac{3}{2} = 0 \]

So,

\[ y = \frac{0}{2} = 0 \]

### Step 4: Combine the results
Thus, the coordinates of the midpoint \(M\) are:

\[ M = (0, 0) \]

### Final Answer
The coordinates of the midpoint of the line joining the two points are \((0, 0)\).

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