Points P and Q lie on sides AB and AC of triangle ABC respectively such that segment PQ is parallel to side BC. If the ratio of areas of triangle APQ: triangle ABC is 25 : 36, then the ratio PA : PB is

Options

A. `5:6`
B. `1:5`
C. `6:5`
D. `5:1`

Answer

To solve the problem, we will use the properties of similar triangles and the relationship between the areas of triangles.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have triangle ABC with points P and Q on sides AB and AC, respectively, such that line segment PQ is parallel to side BC. We are given that the ratio of the areas of triangle APQ to triangle ABC is 25:36.

2. **Using Area Ratio:**
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Since PQ is parallel to BC, triangles APQ and ABC are similar.

\[ \frac{\text{Area of } \triangle APQ}{\text{Area of } \triangle ABC} = \left(\frac{AP}{AB}\right)^2 \]

Given that:
\[ \frac{\text{Area of } \triangle APQ}{\text{Area of } \triangle ABC} = \frac{25}{36} \]

3. **Setting Up the Equation:**
We can set up the equation based on the area ratio:
\[ \left(\frac{AP}{AB}\right)^2 = \frac{25}{36} \]

4. **Taking the Square Root:**
To find the ratio of the sides, we take the square root of both sides:
\[ \frac{AP}{AB} = \frac{5}{6} \]

5. **Finding the Ratios of PA and PB:**
Since \(AB = PA + PB\), we can express \(PB\) in terms of \(PA\):
\[ PA = \frac{5}{6} AB \quad \text{and} \quad PB = AB - PA = AB - \frac{5}{6} AB = \frac{1}{6} AB \]

6. **Calculating the Ratio PA:PB:**
Now we can find the ratio \(PA : PB\):
\[ PA : PB = \frac{5}{6} AB : \frac{1}{6} AB = 5 : 1 \]

### Final Answer:
The ratio \(PA : PB\) is \(5 : 1\).

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