To find the areas of the original triangle and the dilated triangle, we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Step 1: Calculate the area of the original triangle
Given:
- Base = 8 cm
- Height = 3 cm
Calculating the area:
\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 8 \times 3 = \frac{24}{2} = 12 , \text{cm}^2 \]
Step 2: Find the dimensions of the dilated triangle
The scale factor of dilation is 3. This means that both the base and height of the triangle will be multiplied by 3.
- New base = \( 8 , \text{cm} \times 3 = 24 , \text{cm} \)
- New height = \( 3 , \text{cm} \times 3 = 9 , \text{cm} \)
Step 3: Calculate the area of the dilated triangle
Using the new dimensions:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 24 \times 9 \]
Calculating the area:
\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 216 = 108 , \text{cm}^2 \]
Step 4: Compare areas of the dilated triangle and the original triangle
Now we can summarize the areas:
- Area of the original triangle = \( 12 , \text{cm}^2 \)
- Area of the dilated triangle = \( 108 , \text{cm}^2 \)
Ratio of the Areas
The ratio of the area of the dilated triangle to the original triangle is given by the square of the scale factor, which is:
\[ \text{Scale factor} = 3 \]
\[ \text{Area ratio} = \text{Scale factor}^2 = 3^2 = 9 \]
This means the area of the dilated triangle is 9 times larger than that of the original triangle, confirming that:
\[ \text{Area}{\text{dilated}} = 9 \times \text{Area}{\text{original}} = 9 \times 12 , \text{cm}^2 = 108 , \text{cm}^2 \]
Conclusion
- Area of the original triangle: \( 12 , \text{cm}^2 \)
- Area of the dilated triangle: \( 108 , \text{cm}^2 \)
- The area of the dilated triangle is 9 times the area of the original triangle.
4 answers