The base of a triangle is one-fifth of the base of a parallelogram having the same area as that of the triangle . The ratio of the corresponding heights of the triangle and the parallelogram will be

Options

A. `5 : 1`
B. `1 : 10`
C. `1 : 5`
D. `10 : 1`

Answer

To solve the problem, we need to find the ratio of the heights of a triangle and a parallelogram given that the base of the triangle is one-fifth of the base of the parallelogram, and both shapes have the same area.

### Step-by-Step Solution:

1. **Define Variables:**
- Let the base of the parallelogram be \( B \).
- Then, the base of the triangle \( b \) will be \( \frac{B}{5} \).

2. **Area of Triangle:**
- The area \( A_T \) of the triangle can be expressed as:
\[ A_T = \frac{1}{2} \times b \times h_T \]
- Substituting for \( b \):
\[ A_T = \frac{1}{2} \times \frac{B}{5} \times h_T = \frac{B \times h_T}{10} \]

3. **Area of Parallelogram:**
- The area \( A_P \) of the parallelogram is given by:
\[ A_P = B \times h_P \]

4. **Equating Areas:**
- Since the areas of the triangle and the parallelogram are equal:
\[ \frac{B \times h_T}{10} = B \times h_P \]

5. **Simplifying the Equation:**
- We can cancel \( B \) from both sides (assuming \( B \neq 0 \)):
\[ \frac{h_T}{10} = h_P \]

6. **Finding the Ratio of Heights:**
- Rearranging gives:
\[ h_T = 10 \times h_P \]
- Therefore, the ratio of the heights of the triangle to the parallelogram is:
\[ \frac{h_T}{h_P} = 10:1 \]

### Conclusion:
The ratio of the corresponding heights of the triangle and the parallelogram is \( 10:1 \).