Question

Similar Figures and Area Quick Check 3 of 53 of 5 Items Question Kierra draws a triangle with a base of 10 cm and a height of 15 cm. She dilates the triangle using a scale factor of 45 . Then, Kierra finds the difference between the two areas by subtracting. How much greater is the area of the original triangle than the area of the dilated triangle?(1 point) Responses 54 cm2 54 cm squared 48 cm2 48 cm squared 27 cm2 27 cm squared 75 cm2 75 cm squared 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 find the areas of the original and dilated triangles and the difference between them, we can follow these steps:

1. Calculate the area of the original triangle using the formula: $$\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$$ For the original triangle with a base of 10 cm and a height of 15 cm: $${\text{Area}}_{\text{original}}=\frac{1}{2}\times 10,\text{cm}\times 15,\text{cm}=\frac{1}{2}\times 150,{\text{cm}}^2=75,{\text{cm}}^2$$

2. Determine the new dimensions of the dilated triangle. Since Kierra dilates the triangle using a scale factor of 45, the base and height of the dilated triangle will be: \[ \text{Base}{\text{dilated}} = 10 , \text{cm} \times 45 = 450 , \text{cm} \] \[ \text{Height}{\text{dilated}} = 15 , \text{cm} \times 45 = 675 , \text{cm} \]

3. Calculate the area of the dilated triangle: \[ \text{Area}{\text{dilated}} = \frac{1}{2} \times \text{Base}{\text{dilated}} \times \text{Height}{\text{dilated}} \] \[ \text{Area}{\text{dilated}} = \frac{1}{2} \times 450 , \text{cm} \times 675 , \text{cm} = \frac{1}{2} \times 303750 , \text{cm}^2 = 151875 , \text{cm}^2 \]

4. Find the difference between the two areas: \[ \text{Difference} = \text{Area}{\text{dilated}} - \text{Area}{\text{original}} \] $$\text{Difference}=151875,{\text{cm}}^2-75,{\text{cm}}^2=151800,{\text{cm}}^2$$

However, based on the response options given (54 cm², 48 cm², 27 cm², 75 cm²), it seems there is a misunderstanding in the question as we're looking for how much greater the area of the original triangle is compared to the dilated triangle.

Given that the dilated triangle is actually much larger than the original, the final response is:

The area of the dilated triangle (151875 cm²) is greater than the area of the original triangle (75 cm²).

To answer the question as stated: The original triangle is not greater than the dilated triangle. Rather, the dilated triangle's area is much greater than the original one.

For further simplification, should the scale factor truly be considered in a different context, or if the scale factor was a mistake, we would need to revisit that to select an option congruent with the calculated areas. Hence, I suggest you recheck the parameters regarding scale factors or identify the essence of the area calculations.