Question
Use a coordinate proof to show that triangle ABC is scalene.
What are the lengths of sides AB,BC and AC?
ABC coordinates: A(-5,2) B(-3,5) C(-2,-1)
What are the lengths of sides AB,BC and AC?
ABC coordinates: A(-5,2) B(-3,5) C(-2,-1)
Answers
GPT-4o mini
To prove that triangle ABC is scalene, we need to calculate the lengths of the three sides: AB, BC, and AC. We will use the distance formula to find the lengths of the sides. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Step 1: Calculate lengths of sides
**Length of AB:**
Using points A \((-5, 2)\) and B \((-3, 5)\):
\[
AB = \sqrt{((-3) - (-5))^2 + (5 - 2)^2}
\]
\[
= \sqrt{(-3 + 5)^2 + (5 - 2)^2}
\]
\[
= \sqrt{(2)^2 + (3)^2}
\]
\[
= \sqrt{4 + 9} = \sqrt{13}
\]
**Length of BC:**
Using points B \((-3, 5)\) and C \((-2, -1)\):
\[
BC = \sqrt{((-2) - (-3))^2 + (-1 - 5)^2}
\]
\[
= \sqrt{(-2 + 3)^2 + (-1 - 5)^2}
\]
\[
= \sqrt{(1)^2 + (-6)^2}
\]
\[
= \sqrt{1 + 36} = \sqrt{37}
\]
**Length of AC:**
Using points A \((-5, 2)\) and C \((-2, -1)\):
\[
AC = \sqrt{((-2) - (-5))^2 + (-1 - 2)^2}
\]
\[
= \sqrt{(-2 + 5)^2 + (-1 - 2)^2}
\]
\[
= \sqrt{(3)^2 + (-3)^2}
\]
\[
= \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
### Step 2: Compare the lengths
We have:
- Length \( AB = \sqrt{13} \)
- Length \( BC = \sqrt{37} \)
- Length \( AC = 3\sqrt{2} \)
### Step 3: Show that triangle ABC is scalene
A triangle is scalene if all three sides have different lengths. We need to check if \(\sqrt{13}\), \(\sqrt{37}\), and \(3\sqrt{2}\) are all different.
1. Compare \( \sqrt{13} \) and \( 3\sqrt{2} \):
\[
\sqrt{13} \approx 3.605 \quad \text{and} \quad 3\sqrt{2} \approx 4.243
\]
Thus, \( \sqrt{13} < 3\sqrt{2} \).
2. Compare \( \sqrt{37} \) and \( 3\sqrt{2} \):
\[
\sqrt{37} \approx 6.083 \quad \text{and} \quad 3\sqrt{2} \approx 4.243
\]
Thus, \( 3\sqrt{2} < \sqrt{37} \).
3. Compare \( \sqrt{13} \) and \( \sqrt{37} \):
\[
\sqrt{13} \approx 3.605 \quad \text{and} \quad \sqrt{37} \approx 6.083
\]
Thus, \( \sqrt{13} < \sqrt{37} \).
### Conclusion
Since all three lengths are different (\(\sqrt{13} \neq \sqrt{37} \neq 3\sqrt{2}\)), triangle ABC is a scalene triangle. The lengths of the sides are:
- AB = \(\sqrt{13}\)
- BC = \(\sqrt{37}\)
- AC = \(3\sqrt{2}\)
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Step 1: Calculate lengths of sides
**Length of AB:**
Using points A \((-5, 2)\) and B \((-3, 5)\):
\[
AB = \sqrt{((-3) - (-5))^2 + (5 - 2)^2}
\]
\[
= \sqrt{(-3 + 5)^2 + (5 - 2)^2}
\]
\[
= \sqrt{(2)^2 + (3)^2}
\]
\[
= \sqrt{4 + 9} = \sqrt{13}
\]
**Length of BC:**
Using points B \((-3, 5)\) and C \((-2, -1)\):
\[
BC = \sqrt{((-2) - (-3))^2 + (-1 - 5)^2}
\]
\[
= \sqrt{(-2 + 3)^2 + (-1 - 5)^2}
\]
\[
= \sqrt{(1)^2 + (-6)^2}
\]
\[
= \sqrt{1 + 36} = \sqrt{37}
\]
**Length of AC:**
Using points A \((-5, 2)\) and C \((-2, -1)\):
\[
AC = \sqrt{((-2) - (-5))^2 + (-1 - 2)^2}
\]
\[
= \sqrt{(-2 + 5)^2 + (-1 - 2)^2}
\]
\[
= \sqrt{(3)^2 + (-3)^2}
\]
\[
= \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
### Step 2: Compare the lengths
We have:
- Length \( AB = \sqrt{13} \)
- Length \( BC = \sqrt{37} \)
- Length \( AC = 3\sqrt{2} \)
### Step 3: Show that triangle ABC is scalene
A triangle is scalene if all three sides have different lengths. We need to check if \(\sqrt{13}\), \(\sqrt{37}\), and \(3\sqrt{2}\) are all different.
1. Compare \( \sqrt{13} \) and \( 3\sqrt{2} \):
\[
\sqrt{13} \approx 3.605 \quad \text{and} \quad 3\sqrt{2} \approx 4.243
\]
Thus, \( \sqrt{13} < 3\sqrt{2} \).
2. Compare \( \sqrt{37} \) and \( 3\sqrt{2} \):
\[
\sqrt{37} \approx 6.083 \quad \text{and} \quad 3\sqrt{2} \approx 4.243
\]
Thus, \( 3\sqrt{2} < \sqrt{37} \).
3. Compare \( \sqrt{13} \) and \( \sqrt{37} \):
\[
\sqrt{13} \approx 3.605 \quad \text{and} \quad \sqrt{37} \approx 6.083
\]
Thus, \( \sqrt{13} < \sqrt{37} \).
### Conclusion
Since all three lengths are different (\(\sqrt{13} \neq \sqrt{37} \neq 3\sqrt{2}\)), triangle ABC is a scalene triangle. The lengths of the sides are:
- AB = \(\sqrt{13}\)
- BC = \(\sqrt{37}\)
- AC = \(3\sqrt{2}\)