A straight line intersects X and Y-axes at P and Q respectively . If (3 , 5) is the middle point of PQ , then what is the area of the `Delta OPQ ` ?

Options

A. 12 sq units
B. 15 sq units
C. 20 sq units
D. 30 sq units

Answer

To find the area of triangle OPQ where P and Q are the points where the line intersects the X-axis and Y-axis respectively, and given that (3, 5) is the midpoint of PQ, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the coordinates of points P and Q:**
- Let the coordinates of point P (intersection with the X-axis) be (x, 0).
- Let the coordinates of point Q (intersection with the Y-axis) be (0, y).

2. **Use the midpoint formula:**
- The midpoint M of a line segment joining two points (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- For points P and Q, the midpoint M is:
\[ M = \left( \frac{x + 0}{2}, \frac{0 + y}{2} \right) = \left( \frac{x}{2}, \frac{y}{2} \right) \]
- Given that the midpoint is (3, 5), we can set up the equations:
\[ \frac{x}{2} = 3 \quad \text{and} \quad \frac{y}{2} = 5 \]

3. **Solve for x and y:**
- From the first equation:
\[ x = 3 \times 2 = 6 \]
- From the second equation:
\[ y = 5 \times 2 = 10 \]

4. **Identify the coordinates of points P and Q:**
- Therefore, the coordinates of point P are (6, 0) and the coordinates of point Q are (0, 10).

5. **Calculate the area of triangle OPQ:**
- The area \( A \) of triangle OPQ can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
- Here, the base is the length of OP (which is 6) and the height is the length of OQ (which is 10):
\[ A = \frac{1}{2} \times 6 \times 10 = \frac{60}{2} = 30 \]

6. **Conclusion:**
- The area of triangle OPQ is \( 30 \) square units.