The ratio of the areas of two triangles ABC and PQR is 4 : 3 and the ratio of their heights is 5 :3. The ratio of the bases of triangle AHC to that of triangle PQR is:

Options

A. `20:25`
B. `11:15`
C. `15:11`
D. `25:12`

Answer

To solve the problem, we need to find the ratio of the bases of triangles ABC and PQR given the ratio of their areas and heights.

### Step-by-Step Solution:

1. **Understand the Given Ratios**:
- The ratio of the areas of triangles ABC and PQR is given as \( \frac{Area_{ABC}}{Area_{PQR}} = \frac{4}{3} \).
- The ratio of their heights is given as \( \frac{Height_{ABC}}{Height_{PQR}} = \frac{5}{3} \).

2. **Use the Area Formula for Triangles**:
- The area of a triangle can be calculated using the formula:
\[ Area = \frac{1}{2} \times Base \times Height \]
- For triangle ABC:
\[ Area_{ABC} = \frac{1}{2} \times BC \times Height_{ABC} \]
- For triangle PQR:
\[ Area_{PQR} = \frac{1}{2} \times QR \times Height_{PQR} \]

3. **Set Up the Equation Using the Area Ratio**:
- From the area ratio, we can write:
\[ \frac{\frac{1}{2} \times BC \times Height_{ABC}}{\frac{1}{2} \times QR \times Height_{PQR}} = \frac{4}{3} \]
- This simplifies to:
\[ \frac{BC \times Height_{ABC}}{QR \times Height_{PQR}} = \frac{4}{3} \]

4. **Substitute the Height Ratios**:
- Let \( Height_{ABC} = 5x \) and \( Height_{PQR} = 3x \) (based on the height ratio \( \frac{5}{3} \)).
- Substitute these values into the equation:
\[ \frac{BC \times 5x}{QR \times 3x} = \frac{4}{3} \]
- The \( x \) cancels out:
\[ \frac{BC \times 5}{QR \times 3} = \frac{4}{3} \]

5. **Cross Multiply to Solve for the Base Ratio**:
- Cross multiplying gives:
\[ 3 \times BC \times 5 = 4 \times QR \times 3 \]
\[ 15BC = 12QR \]
- Dividing both sides by \( QR \) gives:
\[ \frac{BC}{QR} = \frac{12}{15} = \frac{4}{5} \]

6. **Conclusion**:
- The ratio of the bases of triangle ABC to triangle PQR is \( \frac{BC}{QR} = \frac{4}{5} \).

### Final Answer:
The ratio of the bases of triangle ABC to triangle PQR is \( 4 : 5 \).