A triangle has a base of 8 cm and a height of 3 cm. The triangle is dilated by a scale factor of 3. Calculate and compare the areas of the dilated triangle and the original triangle. (1 point) Responses The original triangle has an area of 12 cm2 and the dilated triangle has an area of 36 cm2 ; therefore, the dilated triangle has an area 3 times the size of the original triangle. The original triangle has an area of 12 cm squared and the dilated triangle has an area of 36 cm squared ; therefore, the dilated triangle has an area 3 times the size of the original triangle. The original triangle has an area of 24 cm2 and the dilated triangle has an area of 108 cm2 ; therefore, the dilated triangle has an area 4.5 times the size of the original triangle. The original triangle has an area of 24 cm squared and the dilated triangle has an area of 108 cm squared ; therefore, the dilated triangle has an area 4.5 times the size of the original triangle. The original triangle has an area of 12 cm2 and the dilated triangle has an area of 108 cm2 ; therefore, the dilated triangle has an area 9 times the size of the original triangle. The original triangle has an area of 12 cm squared and the dilated triangle has an area of 108 cm squared ; therefore, the dilated triangle has an area 9 times the size of the original triangle. The original triangle has an area of 24 cm2 and the dilated triangle has an area of 36 cm2 ; therefore, the dilated triangle has an area 1.5 times the size of the original triangle.

Question

GPT-4o mini · Answer

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.

Base = 8 cm
Height = 3 cm

\[ \text{Area}_{\text{original}} = \frac{1}{2} \times 8 \times 3 = 12 \text{ cm}^2 \]

Step 2: Determine the dimensions of the dilated triangle. Since the triangle is dilated by a scale factor of 3:

New base = \(3 \times 8 = 24\) cm
New height = \(3 \times 3 = 9\) cm

Step 3: Calculate the area of the dilated triangle.

\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 24 \times 9 \]

Calculating that:

\[ \text{Area}_{\text{dilated}} = \frac{1}{2} \times 216 = 108 \text{ cm}^2 \]

Step 4: Compare the areas.

Area of the original triangle = 12 cm²
Area of the dilated triangle = 108 cm²

To find how many times larger the dilated triangle's area is compared to the original:

\[ \frac{\text{Area}{\text{dilated}}}{\text{Area}{\text{original}}} = \frac{108}{12} = 9 \]

Conclusion:

The original triangle has an area of 12 cm², and the dilated triangle has an area of 108 cm²; therefore, the dilated triangle has an area 9 times the size of the original triangle.