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

Options

A. (a) `10%`
B. (b) `9%`
C. (c) `9(1)/(11)%`
D. (d) `11(1)/(9)%`

Answer

To solve the problem, we need to understand the relationship between the altitude, base, and area of a triangle. Let's go through the solution step by step.

### Step 1: Understand the Area of a Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Let the original base be \( b \) and the original height (altitude) be \( h \). Therefore, the area can be expressed as:
\[ A = \frac{1}{2} \times b \times h \]

### Step 2: Increase the Altitude by 10%
If the altitude is increased by 10%, the new altitude \( h' \) can be calculated as:
\[ h' = h + 0.1h = 1.1h \]

### Step 3: Area Remains the Same
Since the area remains the same, we can set up the equation using the new altitude:
\[ A = \frac{1}{2} \times b' \times h' \]
Where \( b' \) is the new base we need to find. Since the area remains unchanged, we have:
\[ \frac{1}{2} \times b \times h = \frac{1}{2} \times b' \times (1.1h) \]

### Step 4: Cancel Out Common Terms
We can cancel \( \frac{1}{2} \) and \( h \) (assuming \( h \neq 0 \)):
\[ b = b' \times 1.1 \]

### Step 5: Solve for the New Base
Rearranging the equation gives:
\[ b' = \frac{b}{1.1} \]

### Step 6: Calculate the Decrease in Base
To find out how much the base has decreased, we calculate the decrease:
\[ \text{Decrease} = b - b' = b - \frac{b}{1.1} \]
This can be simplified as:
\[ \text{Decrease} = b \left(1 - \frac{1}{1.1}\right) = b \left(\frac{1.1 - 1}{1.1}\right) = b \left(\frac{0.1}{1.1}\right) = \frac{b}{11} \]

### Step 7: Calculate the Percentage Decrease
The percentage decrease in the base is given by:
\[ \text{Percentage Decrease} = \left(\frac{\text{Decrease}}{b}\right) \times 100 = \left(\frac{\frac{b}{11}}{b}\right) \times 100 = \frac{100}{11} \approx 9.09\% \]

### Final Answer
Thus, the corresponding base will have to be decreased by approximately \( \frac{100}{11}\% \).

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