To find the area of the original triangle, we use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle with a base of 4 inches and a height of 6 inches, the area is:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 4 \times 6 = \frac{24}{2} = 12 \text{ square inches} \]
Next, we consider the effect of dilating the triangle by a scale factor of 1.5. When a triangle is dilated, its area scales by the square of the scale factor. Therefore, the area of the dilated triangle can be calculated as follows:
\[ \text{Area}{\text{dilated}} = \text{Area}{\text{original}} \times (\text{Scale Factor})^2 \]
Calculating the area of the dilated triangle:
\[ \text{Area}_{\text{dilated}} = 12 \times (1.5)^2 = 12 \times 2.25 = 27 \text{ square inches} \]
Now, we compare the area of the dilated triangle to the area of the original triangle by examining the factor by which the area has increased:
\[ \text{Factor} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{27}{12} \]
To convert to a decimal, we can perform the division:
\[ \frac{27}{12} = 2.25 \]
Thus, the area of the dilated triangle is 2.25 times the area of the original triangle.
Answer: The area of the dilated triangle is 2.25 times the area of the original triangle.
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