To compare the areas of the original triangle and the dilated triangle, we first need to calculate the area of the original triangle.
The formula for the area \( A \) of a triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle:
- Base = 4 inches
- Height = 6 inches
Plugging in these values:
\[ A_{original} = \frac{1}{2} \times 4 \times 6 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]
Next, Marisol dilates the triangle by a scale factor of 1.5. The dimensions of the dilated triangle will be:
- New base = \( 1.5 \times 4 = 6 \) inches
- New height = \( 1.5 \times 6 = 9 \) inches
Now, we can compute the area of the dilated triangle:
\[ A_{dilated} = \frac{1}{2} \times \text{new base} \times \text{new height} \] \[ A_{dilated} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches} \]
Now we can compare the areas of the original and dilated triangles:
- Area of the original triangle: \( 12 \) square inches
- Area of the dilated triangle: \( 27 \) square inches
To determine the ratio of the areas, we can use the fact that the area of a triangle is proportional to the square of the scale factor. The scale factor is \( 1.5 \), so the area of the dilated triangle will be:
\[ \text{Area scale factor} = (1.5)^2 = 2.25 \]
Thus, we can verify the relationship:
\[ A_{dilated} = A_{original} \times 2.25 \] \[ A_{dilated} = 12 \times 2.25 = 27 \text{ square inches} \]
In conclusion, the areas of the original triangle and the dilated triangle are:
- The original triangle's area is \( 12 \) square inches.
- The dilated triangle's area is \( 27 \) square inches.
This means the dilated triangle has an area that is \( \frac{27}{12} = 2.25 \) times larger than the area of the original triangle.
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