Question
If ΔABC is dilated by a scale factor of 2 with a dilation center at vertex A, by what factor does the area of the triangle increase?
Responses
A 22
B 44
C 88
D 16
Responses
A 22
B 44
C 88
D 16
Answers
GPT-4o mini
When a triangle is dilated by a scale factor \( k \), the area of the triangle increases by a factor of \( k^2 \).
In this case, the scale factor is \( 2 \). Therefore, the area increases by a factor of:
\[
k^2 = 2^2 = 4
\]
Hence, the area of triangle \( ABC \) after dilation increases by a factor of \( 4 \).
Looking at the options, it seems none of them directly correspond to the correct answer of \( 4 \). If you are only considering the increase in area (and not the resultant area), it is simply a \( 4 \) factor increase. The area of the dilated triangle is four times the area of the original triangle.
Thus, confirming the factor increase that the options do not seem to encompass, the increase in area is by a factor of \( 4 \).
If the question intends to account for the original amount leading to a multiplication, please review the question details. Otherwise, the correct increase factor is \( 4 \) (Area remains in context of \( \text{original} \times 4 \)), and if constrained to options, please verify the context where noted could be misinterpreted as a growth example.
In this case, the scale factor is \( 2 \). Therefore, the area increases by a factor of:
\[
k^2 = 2^2 = 4
\]
Hence, the area of triangle \( ABC \) after dilation increases by a factor of \( 4 \).
Looking at the options, it seems none of them directly correspond to the correct answer of \( 4 \). If you are only considering the increase in area (and not the resultant area), it is simply a \( 4 \) factor increase. The area of the dilated triangle is four times the area of the original triangle.
Thus, confirming the factor increase that the options do not seem to encompass, the increase in area is by a factor of \( 4 \).
If the question intends to account for the original amount leading to a multiplication, please review the question details. Otherwise, the correct increase factor is \( 4 \) (Area remains in context of \( \text{original} \times 4 \)), and if constrained to options, please verify the context where noted could be misinterpreted as a growth example.