In `Delta PQR`, the line drawn from the vertex P intersects QR at a point S. If QR = 4.5 cm and SR = 1.5 cm then the ratios of the area of triangle PQS and triangle PSR is

Options

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

Answer

To solve the problem, we need to find the ratio of the areas of triangles PQS and PSR in triangle PQR, given that QR = 4.5 cm and SR = 1.5 cm.

### Step-by-Step Solution:

1. **Identify the lengths of segments**:
- We know that QR = 4.5 cm and SR = 1.5 cm.
- To find QS, we subtract SR from QR:
\[ QS = QR - SR = 4.5 \, \text{cm} - 1.5 \, \text{cm} = 3 \, \text{cm} \]

2. **Understand the areas of triangles**:
- The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
- For triangles PQS and PSR, the heights from point P to line QR are the same since they share the vertex P.

3. **Set up the ratio of the areas**:
- The area of triangle PQS can be expressed as:
\[ \text{Area of } \triangle PQS = \frac{1}{2} \times QS \times h \]
- The area of triangle PSR can be expressed as:
\[ \text{Area of } \triangle PSR = \frac{1}{2} \times SR \times h \]
- Here, \( h \) is the height from point P to line QR.

4. **Calculate the ratio of the areas**:
- The ratio of the areas of triangles PQS and PSR is:
\[ \frac{\text{Area of } \triangle PQS}{\text{Area of } \triangle PSR} = \frac{\frac{1}{2} \times QS \times h}{\frac{1}{2} \times SR \times h} \]
- The \( \frac{1}{2} \) and \( h \) cancel out:
\[ = \frac{QS}{SR} \]
- Now substitute the values of QS and SR:
\[ = \frac{3 \, \text{cm}}{1.5 \, \text{cm}} = 2 \]

5. **Express the ratio**:
- Therefore, the ratio of the areas of triangles PQS and PSR is:
\[ 2:1 \]

### Final Answer:
The ratio of the area of triangle PQS to the area of triangle PSR is \( 2:1 \).