To find the areas of the original triangle and the dilated triangle, we can follow these steps:
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Calculate the area of the original triangle. The formula for the area \( A \) of a triangle is: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the given base and height: \[ A = \frac{1}{2} \times 4 , \text{inches} \times 6 , \text{inches} = \frac{1}{2} \times 24 , \text{square inches} = 12 , \text{square inches} \]
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Apply the dilation. When a triangle is dilated by a scale factor \( k \), the area of the new triangle is scaled by \( k^2 \). Here, the scale factor is \( 1.5 \). Therefore, the area of the dilated triangle will be: \[ \text{Area of dilated triangle} = k^2 \times \text{Area of original triangle} \] Calculating \( k^2 \): \[ k^2 = (1.5)^2 = 2.25 \] Now, substituting this into the area equation: \[ \text{Area of dilated triangle} = 2.25 \times 12 , \text{square inches} = 27 , \text{square inches} \]
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Compare the areas.
- Area of the original triangle: \( 12 , \text{square inches} \)
- Area of the dilated triangle: \( 27 , \text{square inches} \)
So, the area of the dilated triangle is greater than the area of the original triangle. In decimal terms, you can say:
- The area increased from \( 12.0 , \text{square inches} \) to \( 27.0 , \text{square inches} \).
Response Summary: The area of the original triangle is \( 12.0 \) square inches, and the area of the dilated triangle is \( 27.0 \) square inches.