Question
What is the area of a triangle similar to one with an area of 12 square cm and a linear scale factor of 2:3?
Answers
GPT-4o mini
The area of similar triangles is proportional to the square of their corresponding linear dimensions. Given that the linear scale factor between the triangles is \( \frac{2}{3} \), we can calculate the ratio of their areas.
Let \( A_1 \) be the area of the original triangle and \( A_2 \) be the area of the similar triangle. The relationship between the areas can be expressed as:
\[
\frac{A_1}{A_2} = \left(\frac{\text{side of triangle 1}}{\text{side of triangle 2}}\right)^2 = \left(\frac{2}{3}\right)^2
\]
Calculating \( \left(\frac{2}{3}\right)^2 \):
\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]
This means the ratio of the areas is:
\[
\frac{12}{A_2} = \frac{4}{9}
\]
To find \( A_2 \), we can set up the equation:
\[
12 \cdot 9 = 4 \cdot A_2
\]
\[
108 = 4 A_2
\]
Now, divide both sides by 4:
\[
A_2 = \frac{108}{4} = 27
\]
Thus, the area of the triangle similar to the one with an area of 12 square cm is
\[
\boxed{27} \text{ square cm.}
\]
Let \( A_1 \) be the area of the original triangle and \( A_2 \) be the area of the similar triangle. The relationship between the areas can be expressed as:
\[
\frac{A_1}{A_2} = \left(\frac{\text{side of triangle 1}}{\text{side of triangle 2}}\right)^2 = \left(\frac{2}{3}\right)^2
\]
Calculating \( \left(\frac{2}{3}\right)^2 \):
\[
\left(\frac{2}{3}\right)^2 = \frac{4}{9}
\]
This means the ratio of the areas is:
\[
\frac{12}{A_2} = \frac{4}{9}
\]
To find \( A_2 \), we can set up the equation:
\[
12 \cdot 9 = 4 \cdot A_2
\]
\[
108 = 4 A_2
\]
Now, divide both sides by 4:
\[
A_2 = \frac{108}{4} = 27
\]
Thus, the area of the triangle similar to the one with an area of 12 square cm is
\[
\boxed{27} \text{ square cm.}
\]