In `DeltaABC`, the line parallel to BC Intersects AB and AC at P and Q respectively. If `AB : AP = 5:3 `then `AQ : QC `is :

Options

A. `3 : 2`
B. `2:3`
C. `3:5`
D. `1:2 `

Answer

To solve the problem step by step, we will use the properties of similar triangles and the Basic Proportionality Theorem.

### Step-by-Step Solution:

1. **Draw the Triangle and Parallel Line**:
- Draw triangle ABC with points A, B, and C.
- Draw a line parallel to BC that intersects AB at point P and AC at point Q.

2. **Identify the Given Ratio**:
- We are given that the ratio of AB to AP is 5:3. This means:
\[ \frac{AB}{AP} = \frac{5}{3} \]

3. **Use the Basic Proportionality Theorem**:
- According to the Basic Proportionality Theorem (or Thales' theorem), if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
- Therefore, we can write:
\[ \frac{AP}{AB} = \frac{AQ}{AC} \]

4. **Substitute the Known Ratio**:
- Since we know that:
\[ \frac{AP}{AB} = \frac{3}{5} \]
- We can substitute this into our equation:
\[ \frac{3}{5} = \frac{AQ}{AC} \]

5. **Express AC in Terms of AQ**:
- Rearranging the equation gives us:
\[ AQ = \frac{3}{5} \cdot AC \]

6. **Find the Remaining Segment QC**:
- Since AQ + QC = AC, we can express QC as:
\[ QC = AC - AQ \]
- Substitute AQ from the previous step:
\[ QC = AC - \frac{3}{5} \cdot AC = \frac{2}{5} \cdot AC \]

7. **Find the Ratio AQ to QC**:
- Now, we can find the ratio of AQ to QC:
\[ \frac{AQ}{QC} = \frac{\frac{3}{5} \cdot AC}{\frac{2}{5} \cdot AC} = \frac{3}{2} \]
- Therefore, the ratio AQ : QC is:
\[ AQ : QC = 3 : 2 \]

### Final Answer:
The ratio \( AQ : QC \) is \( 3 : 2 \).