If the altitude of a triangle is increased by 10% while its area remains the same, its corresponding base will have to be decreased by

Options

A. 0.1
B. 0.09
C. `9 (1)/(11)` %
D. `11 (1)/(9)` %

Answer

To solve the problem step by step, we will analyze the relationship between the altitude, base, and area of the triangle.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the original altitude of the triangle be \( x \) and the original base be \( y \).

2. **Calculate the Original Area:**
The area \( A \) of the triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times y \times x = \frac{xy}{2} \]

3. **Increase the Altitude by 10%:**
The new altitude after a 10% increase will be:
\[ \text{New altitude} = x + 0.1x = 1.1x \]

4. **Set Up the Equation for the New Area:**
Let the new base be \( b \). The area with the new altitude must remain the same, so we have:
\[ A = \frac{1}{2} \times b \times \text{New altitude} = \frac{1}{2} \times b \times 1.1x \]
This can be simplified to:
\[ A = \frac{1.1bx}{2} \]

5. **Equate the Original Area to the New Area:**
Since the area remains the same, we can set the original area equal to the new area:
\[ \frac{xy}{2} = \frac{1.1bx}{2} \]

6. **Solve for the New Base \( b \):**
Canceling \( \frac{1}{2} \) from both sides and \( x \) (assuming \( x \neq 0 \)):
\[ y = 1.1b \]
Rearranging gives:
\[ b = \frac{y}{1.1} = \frac{10y}{11} \]

7. **Calculate the Decrease in Base:**
The original base was \( y \) and the new base is \( \frac{10y}{11} \). The decrease in the base is:
\[ \text{Decrease} = y - \frac{10y}{11} = \frac{11y}{11} - \frac{10y}{11} = \frac{y}{11} \]

8. **Calculate the Percentage Decrease:**
The percentage decrease in the base is given by:
\[ \text{Percentage Decrease} = \left( \frac{\text{Decrease}}{\text{Original Base}} \right) \times 100 = \left( \frac{\frac{y}{11}}{y} \right) \times 100 = \frac{100}{11} \approx 9.09\% \]

### Final Answer:
The corresponding base will have to be decreased by approximately \( 9.09\% \).