To determine the effect on the area of the triangle when its base is increased by 2 inches and its height is decreased by 1 inch, we need to first calculate the initial area of the triangle and then calculate the new area after these changes. Finally, we can find the percentage change in the area.
Let's start by finding the initial area of the triangle.
The area of a triangle is given by the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
The initial base (\(b_i\)) is 4 inches, and the initial height (\(h_i\)) is 5 inches.
So the initial area (\(A_i\)) is:
\[ A_i = \frac{1}{2} \times b_i \times h_i \]
\[ A_i = \frac{1}{2} \times 4 \times 5 \]
\[ A_i = 2 \times 5 \]
\[ A_i = 10 \text{ square inches} \]
Now, we increase the base by 2 inches, which means the new base (\(b_n\)) is 4 inches + 2 inches = 6 inches. We decrease the height by 1 inch, which means the new height (\(h_n\)) is 5 inches - 1 inch = 4 inches.
The new area (\(A_n\)) is calculated in the same way:
\[ A_n = \frac{1}{2} \times b_n \times h_n \]
\[ A_n = \frac{1}{2} \times 6 \times 4 \]
\[ A_n = 3 \times 4 \]
\[ A_n = 12 \text{ square inches} \]
With both areas calculated, we can find the percentage change. The formula for finding percentage change is:
\[ \text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\% \]
Applying the formula for percentage change in the area:
\[ \text{Percentage Change} = \frac{A_n - A_i}{A_i} \times 100\% \]
\[ \text{Percentage Change} = \frac{12 - 10}{10} \times 100\% \]
\[ \text{Percentage Change} = \frac{2}{10} \times 100\% \]
\[ \text{Percentage Change} = 0.2 \times 100\% \]
\[ \text{Percentage Change} = 20\% \]
The area of the triangle increases by 20%. Therefore, the correct answer is:
It increases by 20%.
