To find the area of the enlarged triangle after a dilation by a scale factor of 4.3, we need to follow these steps:
1. Find the area of the original right triangle.
2. Apply the scale factor to find the area of the enlarged triangle.
Step 1: Find the area of the original right triangle.
The formula to find the area (A) of a right triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given that the side lengths of the original right triangle are 5, 12 (these can be considered as base and height), and 13 (which is the hypotenuse), the area can be calculated as:
\[ A_{\text{original}} = \frac{1}{2} \times 5 \times 12 \]
\[ A_{\text{original}} = \frac{1}{2} \times 60 \]
\[ A_{\text{original}} = 30 \]
So, the area of the original right triangle is 30 square units.
Step 2: Apply the scale factor to find the area of the enlarged triangle.
When a two-dimensional figure is dilated, the area is affected by the square of the scale factor. Thus, if the linear dimensions are increased by a factor of 4.3, the area of the enlarged triangle is the original area multiplied by the square of the scale factor (4.3^2).
So the new area (A_{enlarged}) is calculated as:
\[ A_{\text{enlarged}} = A_{\text{original}} \times (\text{scale factor})^2 \]
\[ A_{\text{enlarged}} = 30 \times (4.3)^2 \]
\[ A_{\text{enlarged}} = 30 \times 18.49 \]
\[ A_{\text{enlarged}} = 554.7 \]
Therefore, the area of the enlarged triangle is 554.7 square units.
