If the base of triangle is increased by 10% and height is decreased by 20%, then what will be the percentage change in the area of a triangle ?

Options

A. 30
B. 20
C. 22
D. 12

Answer

To find the percentage change in the area of a triangle when the base is increased by 10% and the height is decreased by 20%, we can follow these steps:

### Step 1: Understand the formula for the area of a triangle.
The area \( A \) of a triangle is given by the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

### Step 2: Let the original base and height be defined.
Let the original base be \( b \) and the original height be \( h \).

### Step 3: Calculate the new base and height after the changes.
- The new base after a 10% increase:
\[ \text{New base} = b + 0.10b = 1.10b \]
- The new height after a 20% decrease:
\[ \text{New height} = h - 0.20h = 0.80h \]

### Step 4: Calculate the original area of the triangle.
The original area \( A_{\text{original}} \) is:
\[ A_{\text{original}} = \frac{1}{2} \times b \times h \]

### Step 5: Calculate the new area of the triangle.
The new area \( A_{\text{new}} \) is:
\[ A_{\text{new}} = \frac{1}{2} \times (1.10b) \times (0.80h) \]
\[ A_{\text{new}} = \frac{1}{2} \times 1.10 \times 0.80 \times b \times h \]
\[ A_{\text{new}} = \frac{1}{2} \times 0.88 \times b \times h \]
\[ A_{\text{new}} = 0.44 \times b \times h \]

### Step 6: Calculate the percentage change in area.
To find the percentage change, we use the formula:
\[ \text{Percentage Change} = \left( \frac{A_{\text{new}} - A_{\text{original}}}{A_{\text{original}}} \right) \times 100 \]
Substituting the areas:
\[ \text{Percentage Change} = \left( \frac{0.44 \times b \times h - \frac{1}{2} \times b \times h}{\frac{1}{2} \times b \times h} \right) \times 100 \]
Since \( A_{\text{original}} = \frac{1}{2} \times b \times h \):
\[ \text{Percentage Change} = \left( \frac{0.44 - 0.5}{0.5} \right) \times 100 \]
\[ \text{Percentage Change} = \left( \frac{-0.06}{0.5} \right) \times 100 \]
\[ \text{Percentage Change} = -12\% \]

### Conclusion:
The area of the triangle decreases by 12%.