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Original 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 a...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 12 cm2
cm
2
and the dilated triangle has an area of 36 cm2
cm
2
; 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 12 cm2
cm
2
and the dilated triangle has an area of 108 cm2
cm
2
; 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
(1 point)
Responses
The original triangle has an area of 12 cm2
cm
2
and the dilated triangle has an area of 36 cm2
cm
2
; 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 12 cm2
cm
2
and the dilated triangle has an area of 108 cm2
cm
2
; 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
Answers
GPT-4o mini
To find the area of the original triangle, we use the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values for the original triangle:
\[
\text{Area} = \frac{1}{2} \times 8 \, \text{cm} \times 3 \, \text{cm} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
Next, we dilate the triangle by a scale factor of 3. The new dimensions (base and height) will be:
- New base = \(8 \, \text{cm} \times 3 = 24 \, \text{cm}\)
- New height = \(3 \, \text{cm} \times 3 = 9 \, \text{cm}\)
Now, calculate the area of the dilated triangle:
\[
\text{Area of dilated triangle} = \frac{1}{2} \times \text{new base} \times \text{new height}
\]
\[
\text{Area of dilated triangle} = \frac{1}{2} \times 24 \, \text{cm} \times 9 \, \text{cm} = \frac{1}{2} \times 216 \, \text{cm}^2 = 108 \, \text{cm}^2
\]
The original triangle has an area of \(12 \, \text{cm}^2\) and the dilated triangle has an area of \(108 \, \text{cm}^2\).
To compare the areas:
\[
\text{Factor increase} = \frac{108 \, \text{cm}^2}{12 \, \text{cm}^2} = 9
\]
Thus, 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."
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Substituting the values for the original triangle:
\[
\text{Area} = \frac{1}{2} \times 8 \, \text{cm} \times 3 \, \text{cm} = \frac{1}{2} \times 24 \, \text{cm}^2 = 12 \, \text{cm}^2
\]
Next, we dilate the triangle by a scale factor of 3. The new dimensions (base and height) will be:
- New base = \(8 \, \text{cm} \times 3 = 24 \, \text{cm}\)
- New height = \(3 \, \text{cm} \times 3 = 9 \, \text{cm}\)
Now, calculate the area of the dilated triangle:
\[
\text{Area of dilated triangle} = \frac{1}{2} \times \text{new base} \times \text{new height}
\]
\[
\text{Area of dilated triangle} = \frac{1}{2} \times 24 \, \text{cm} \times 9 \, \text{cm} = \frac{1}{2} \times 216 \, \text{cm}^2 = 108 \, \text{cm}^2
\]
The original triangle has an area of \(12 \, \text{cm}^2\) and the dilated triangle has an area of \(108 \, \text{cm}^2\).
To compare the areas:
\[
\text{Factor increase} = \frac{108 \, \text{cm}^2}{12 \, \text{cm}^2} = 9
\]
Thus, 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."