Question

Similar Figures and Area Quick Check 1 of 51 of 5 Items Question Use the image to answer the question. 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 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. The original triangle has an area of 24 cm squared and the dilated triangle has an area of 36 cm squared ; therefore, the dilated triangle has an area 1.5 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 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. Skip to navigation page 19 of 19 Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.Formula keypad has been closed. Press Control + Backslash to open it again.

Answer 1

To solve the problem, we first calculate the area of the original triangle and then the area of the dilated triangle.

1. Original Triangle Area Calculation: The formula for the area of a triangle is: $$\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$$ Given the base $b=8$ cm and height $h=3$ cm: $$\text{Area}=\frac{1}{2}\times 8\times 3=\frac{1}{2}\times 24=12{\text{ cm}}^2$$

2. Dilated Triangle Dimensions: The triangle is dilated by a scale factor of 3, which means:

   • New base = $3\times 8=24$ cm
   • New height = $3\times 3=9$ cm

3. Dilated Triangle Area Calculation: Using the same area formula: $$\text{Area}=\frac{1}{2}\times \text{new base}\times \text{new height}$$ $$\text{Area}=\frac{1}{2}\times 24\times 9=\frac{1}{2}\times 216=108{\text{ cm}}^2$$

4. Comparison of Areas:

   • Area of the original triangle: $12$ cm²
   • Area of the dilated triangle: $108$ cm²

To find how many times larger the area of the dilated triangle is compared to the original triangle: $$\text{Factor}=\frac{108}{12}=9$$

Thus, 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 correct response is: 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.