To find the areas of the original triangle and the dilated triangle, we first calculate the area of the original triangle.
The formula for the area of a triangle is: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle with a base of 4 inches and a height of 6 inches: \[ \text{Area}_{\text{original}} = \frac{1}{2} \times 4 \times 6 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]
Next, we need to determine the dimensions of the dilated triangle. A dilation by a scale factor of 1.5 means that both the base and the height of the triangle will be multiplied by 1.5.
- New base: \[ \text{Base}_{\text{dilated}} = 4 \times 1.5 = 6 \text{ inches} \]
- New height: \[ \text{Height}_{\text{dilated}} = 6 \times 1.5 = 9 \text{ inches} \]
Now we can calculate the area of the dilated triangle: \[ \text{Area}{\text{dilated}} = \frac{1}{2} \times \text{Base}{\text{dilated}} \times \text{Height}_{\text{dilated}} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches} \]
Now, we compare the areas of the dilated triangle and the original triangle. To find how many times the area of the dilated triangle is compared to the area of the original triangle: \[ \text{Factor} = \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{27}{12} = 2.25 \]
Thus, the area of the dilated triangle is 2.25 times the area of the original triangle.
Therefore, the answer is: 2.25
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