ABCD is a quadrilateral. P (3,7) and Q (7, 3) are the middle points of the diagonals AC and BD respectively. The coordinates of the mean point (or the centre of mean position) of the vertices of the quadrilateral are

Options

A. (0, 0)
B. (3, 3)
C. (5, 5)
D. (7, 7)

Answer

To find the coordinates of the mean point (or the center of mean position) of the vertices of the quadrilateral ABCD, we can follow these steps:

### Step 1: Identify the midpoints of the diagonals
We are given that P (3, 7) is the midpoint of diagonal AC and Q (7, 3) is the midpoint of diagonal BD.

### Step 2: Calculate the midpoint of the segment joining P and Q
The mean point of the vertices of the quadrilateral can be found by calculating the midpoint of the line segment joining points P and Q.

The formula for the midpoint M(x, y) of two points A(x1, y1) and B(x2, y2) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 3: Substitute the coordinates of P and Q into the midpoint formula
Substituting the coordinates of P (3, 7) and Q (7, 3):
\[ M = \left( \frac{3 + 7}{2}, \frac{7 + 3}{2} \right) \]

### Step 4: Simplify the calculations
Calculating the x-coordinate:
\[ \frac{3 + 7}{2} = \frac{10}{2} = 5 \]

Calculating the y-coordinate:
\[ \frac{7 + 3}{2} = \frac{10}{2} = 5 \]

### Step 5: Write the final coordinates of the mean point
Thus, the coordinates of the mean point (or center of mean position) of the vertices of the quadrilateral ABCD are:
\[ (5, 5) \]

### Final Answer:
The coordinates of the mean point of the vertices of the quadrilateral ABCD are (5, 5).

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