To determine the area of the new triangle after scaling the sides of the original triangle by a factor of 6, follow these steps:
1. Identify the Area of the Original Triangle:
The area of the original right triangle is given as 72 square feet.
2. Understand the Effect of Scaling on Area:
When you multiply the lengths of the sides of a triangle by a scale factor, the area of the triangle changes by the square of that scale factor. This is because the area is a two-dimensional measure, and scaling each dimension (side) affects the area by the square of the scale factor.
3. Calculate the New Area:
The scale factor given is 6. Therefore, you need to square this scale factor to determine how much the area will increase:
[tex]\[
\text{Scale Factor}^2 = 6^2 = 36
\][/tex]
4. Multiply the Original Area by the Squared Scale Factor:
The new area of the triangle is calculated by multiplying the original area by 36:
[tex]\[
\text{New Area} = 72 \, \text{square feet} \times 36
\][/tex]
5. Determine the Final New Area:
[tex]\[
\text{New Area} = 2592 \, \text{square feet}
\][/tex]
Thus, the area of the new triangle, after multiplying the lengths of the sides by 6, is [tex]\( 2592 \)[/tex] square feet.
