Question
Triangle JAM has points J (12, 15), A (3, 9), and M (6, 0).
Triangle J’A’M’ has points J’ (4, 5), A’ (1, 3), and M’ (2, 0).
What was the scale factor?
Triangle J’A’M’ has points J’ (4, 5), A’ (1, 3), and M’ (2, 0).
What was the scale factor?
Answers
GPT-4o mini
To find the scale factor between triangle JAM and triangle J’A’M’, we can calculate the lengths of their corresponding sides and then determine the ratio of those lengths.
First, we calculate the lengths of the sides of triangle JAM:
1. Length of side JA:
\[
JA = \sqrt{(12 - 3)^2 + (15 - 9)^2} = \sqrt{(9)^2 + (6)^2} = \sqrt{81 + 36} = \sqrt{117} \approx 10.82
\]
2. Length of side JM:
\[
JM = \sqrt{(12 - 6)^2 + (15 - 0)^2} = \sqrt{(6)^2 + (15)^2} = \sqrt{36 + 225} = \sqrt{261} \approx 16.155
\]
3. Length of side AM:
\[
AM = \sqrt{(3 - 6)^2 + (9 - 0)^2} = \sqrt{(-3)^2 + (9)^2} = \sqrt{9 + 81} = \sqrt{90} \approx 9.487
\]
Next, we calculate the lengths of the sides of triangle J’A’M’:
1. Length of side J'A':
\[
J'A' = \sqrt{(4 - 1)^2 + (5 - 3)^2} = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.605
\]
2. Length of side J'M':
\[
J'M' = \sqrt{(4 - 2)^2 + (5 - 0)^2} = \sqrt{(2)^2 + (5)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.385
\]
3. Length of side A'M':
\[
A'M' = \sqrt{(1 - 2)^2 + (3 - 0)^2} = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.162
\]
Now we have the lengths of the sides:
- Triangle JAM: \(\sqrt{117}, \sqrt{261}, \sqrt{90}\)
- Triangle J’A’M’: \(\sqrt{13}, \sqrt{29}, \sqrt{10}\)
Next, we find the ratios of corresponding side lengths to determine the scale factor. We will take the first sides as an example:
\[
\text{Scale Factor} = \frac{JA}{J'A'} = \frac{\sqrt{117}}{\sqrt{13}} = \sqrt{\frac{117}{13}} \approx \sqrt{9} = 3
\]
Now verify with other pairs:
\[
\text{Scale Factor} = \frac{JM}{J'M'} = \frac{\sqrt{261}}{\sqrt{29}} = \sqrt{\frac{261}{29}} \approx \sqrt{9} = 3
\]
\[
\text{Scale Factor} = \frac{AM}{A'M'} = \frac{\sqrt{90}}{\sqrt{10}} = \sqrt{\frac{90}{10}} = \sqrt{9} = 3
\]
All corresponding side ratios give us a consistent scale factor of 3.
Therefore, the scale factor between triangle JAM and triangle J’A’M’ is \( \boxed{3} \).
First, we calculate the lengths of the sides of triangle JAM:
1. Length of side JA:
\[
JA = \sqrt{(12 - 3)^2 + (15 - 9)^2} = \sqrt{(9)^2 + (6)^2} = \sqrt{81 + 36} = \sqrt{117} \approx 10.82
\]
2. Length of side JM:
\[
JM = \sqrt{(12 - 6)^2 + (15 - 0)^2} = \sqrt{(6)^2 + (15)^2} = \sqrt{36 + 225} = \sqrt{261} \approx 16.155
\]
3. Length of side AM:
\[
AM = \sqrt{(3 - 6)^2 + (9 - 0)^2} = \sqrt{(-3)^2 + (9)^2} = \sqrt{9 + 81} = \sqrt{90} \approx 9.487
\]
Next, we calculate the lengths of the sides of triangle J’A’M’:
1. Length of side J'A':
\[
J'A' = \sqrt{(4 - 1)^2 + (5 - 3)^2} = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.605
\]
2. Length of side J'M':
\[
J'M' = \sqrt{(4 - 2)^2 + (5 - 0)^2} = \sqrt{(2)^2 + (5)^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.385
\]
3. Length of side A'M':
\[
A'M' = \sqrt{(1 - 2)^2 + (3 - 0)^2} = \sqrt{(-1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.162
\]
Now we have the lengths of the sides:
- Triangle JAM: \(\sqrt{117}, \sqrt{261}, \sqrt{90}\)
- Triangle J’A’M’: \(\sqrt{13}, \sqrt{29}, \sqrt{10}\)
Next, we find the ratios of corresponding side lengths to determine the scale factor. We will take the first sides as an example:
\[
\text{Scale Factor} = \frac{JA}{J'A'} = \frac{\sqrt{117}}{\sqrt{13}} = \sqrt{\frac{117}{13}} \approx \sqrt{9} = 3
\]
Now verify with other pairs:
\[
\text{Scale Factor} = \frac{JM}{J'M'} = \frac{\sqrt{261}}{\sqrt{29}} = \sqrt{\frac{261}{29}} \approx \sqrt{9} = 3
\]
\[
\text{Scale Factor} = \frac{AM}{A'M'} = \frac{\sqrt{90}}{\sqrt{10}} = \sqrt{\frac{90}{10}} = \sqrt{9} = 3
\]
All corresponding side ratios give us a consistent scale factor of 3.
Therefore, the scale factor between triangle JAM and triangle J’A’M’ is \( \boxed{3} \).