OPQR is a square with M and N as the middle points of the sides PQ and QR, respectively. The ratio of the areas of the square and the triangle OMN is

Options

A. ` 4: 1`
B. ` 2 : 1`
C. ` 8 : 3`
D. ` 4 : 3`

Answer

To solve the problem of finding the ratio of the areas of square OPQR and triangle OMN, we will follow these steps:

### Step 1: Define the square and its vertices
Let the square OPQR be positioned in the coordinate plane with:
- O (0, 0) as the origin,
- P (A, 0) on the x-axis,
- Q (A, A) on the y-axis,
- R (0, A) on the y-axis.

### Step 2: Identify the midpoints M and N
- M is the midpoint of side PQ. The coordinates of M can be calculated as:
\[ M = \left(\frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2}\right) = \left(\frac{A + A}{2}, \frac{0 + A}{2}\right) = \left(A, \frac{A}{2}\right) \]

- N is the midpoint of side QR. The coordinates of N can be calculated as:
\[ N = \left(\frac{x_Q + x_R}{2}, \frac{y_Q + y_R}{2}\right) = \left(\frac{A + 0}{2}, \frac{A + A}{2}\right) = \left(\frac{A}{2}, A\right) \]

### Step 3: Calculate the area of the square OPQR
The area of square OPQR is given by the formula:
\[ \text{Area of square} = \text{side} \times \text{side} = A \times A = A^2 \]

### Step 4: Calculate the area of triangle OMN
The area of triangle OMN can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Where:
- \( O(0, 0) \)
- \( M(A, \frac{A}{2}) \)
- \( N(\frac{A}{2}, A) \)

Substituting the coordinates into the formula:
\[ \text{Area} = \frac{1}{2} \left| 0\left(\frac{A}{2} - A\right) + A\left(A - 0\right) + \frac{A}{2}\left(0 - \frac{A}{2}\right) \right| \]
This simplifies to:
\[ = \frac{1}{2} \left| A^2 - \frac{A^2}{4} \right| = \frac{1}{2} \left| \frac{4A^2}{4} - \frac{A^2}{4} \right| = \frac{1}{2} \left| \frac{3A^2}{4} \right| = \frac{3A^2}{8} \]

### Step 5: Find the ratio of the areas
Now, we can find the ratio of the area of the square to the area of triangle OMN:
\[ \text{Ratio} = \frac{\text{Area of square}}{\text{Area of triangle OMN}} = \frac{A^2}{\frac{3A^2}{8}} = \frac{A^2 \cdot 8}{3A^2} = \frac{8}{3} \]

### Final Answer
The ratio of the areas of the square OPQR and triangle OMN is:
\[ \frac{8}{3} \]