To find the areas of the original triangle and the dilated triangle, we start by calculating the area of the original triangle.
The formula for the area \( A \) of a triangle is:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For the original triangle, the base is 4 inches and the height is 6 inches:
\[ A_{\text{original}} = \frac{1}{2} \times 4 \times 6 = \frac{1}{2} \times 24 = 12 \text{ square inches} \]
Next, we apply the dilation. A dilation by a scale factor of 1.5 will scale both the base and the height of the triangle. Therefore, the new dimensions of the dilated triangle will be:
\[ \text{new base} = 4 \times 1.5 = 6 \text{ inches} \] \[ \text{new height} = 6 \times 1.5 = 9 \text{ inches} \]
Now, we can calculate the area of the dilated triangle:
\[ A_{\text{dilated}} = \frac{1}{2} \times \text{new base} \times \text{new height} = \frac{1}{2} \times 6 \times 9 = \frac{1}{2} \times 54 = 27 \text{ square inches} \]
To find how many times the area of the dilated triangle is compared to the original triangle, we divide the area of the dilated triangle by the area of the original triangle:
\[ \text{Comparison} = \frac{A_{\text{dilated}}}{A_{\text{original}}} = \frac{27}{12} \]
Calculating this:
\[ \frac{27}{12} = 2.25 \]
Therefore, the area of the dilated triangle is \( 2.25 \) times the area of the original triangle.
Final response:
The area of the dilated triangle is 2.25 times the area of the original triangle.
1 answer