To compare the areas of the dilated triangle and the original triangle, we start by calculating the area of the original triangle.
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle, with a base of 4 inches and a height of 6 inches, the area is:
\[ A_{\text{original}} = \frac{1}{2} \times 4 \times 6 = \frac{24}{2} = 12 \text{ square inches} \]
Next, we dilate the triangle by a scale factor of 1.5. When a triangle is dilated by a scale factor \( k \), the area of the dilated triangle is scaled by \( k^2 \).
Calculating the scale factor for the area:
\[ k^2 = (1.5)^2 = 2.25 \]
Thus, the area of the dilated triangle \( A_{\text{dilated}} \) is:
\[ A_{\text{dilated}} = A_{\text{original}} \times k^2 = 12 \times 2.25 = 27 \text{ square inches} \]
Now, we compare the areas:
The area of the dilated triangle is \( \frac{A_{\text{dilated}}}{A_{\text{original}}} \):
\[ \text{Comparison} = \frac{27}{12} = 2.25 \]
Therefore, the area of the dilated triangle is 2.25 times the area of the original triangle.
So, the final answer is:
The area of the dilated triangle is 2.25 times the area of the original triangle.