Question

A (-5,2)
B (-3,5)
C (-2,-1)

Use a coordinate proof to show that triangle ABC is scalene. What are the lengths of sides AB, BC, and AC?
a. AB ≈ 10.6, BC ≈ 8.1, and AC ≈ 7.1; therefore, ABC is a scalene triangle
b. AB ≈ 8.5, BC ≈ 6.4, and AC ≈ 7.1; therefore, ABC is a scalene triangle
c. AB ≈ 10.6, BC ≈ 6.4, and AC ≈ 7.1; therefore, ABC is a scalene triangle
d. AB ≈ 3.6, BC ≈ 6.1, and AC ≈ 4.2; therefore, ABC is a scalene triangle

Answer 1

To determine whether triangle ABC is scalene, we need to calculate the lengths of the sides AB, BC, and AC. The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane is:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

**Calculating Lengths:**

1. **Length of AB:**
- A = (-5, 2) and B = (-3, 5)
\[
AB = \sqrt{((-3) - (-5))^2 + (5 - 2)^2}
= \sqrt{(2)^2 + (3)^2}
= \sqrt{4 + 9}
= \sqrt{13}
\approx 3.6
\]

2. **Length of BC:**
- B = (-3, 5) and C = (-2, -1)
\[
BC = \sqrt{((-2) - (-3))^2 + (-1 - 5)^2}
= \sqrt{(1)^2 + (-6)^2}
= \sqrt{1 + 36}
= \sqrt{37}
\approx 6.1
\]

3. **Length of AC:**
- A = (-5, 2) and C = (-2, -1)
\[
AC = \sqrt{((-2) - (-5))^2 + (-1 - 2)^2}
= \sqrt{(3)^2 + (-3)^2}
= \sqrt{9 + 9}
= \sqrt{18}
\approx 4.2
\]

**Summary of Results:**
- \(AB \approx 3.6\)
- \(BC \approx 6.1\)
- \(AC \approx 4.2\)

Since all three sides have different lengths, triangle ABC is indeed scalene.

Thus, the correct answer is:

**d. AB ≈ 3.6, BC ≈ 6.1, and AC ≈ 4.2; therefore, ABC is a scalene triangle.**