Question

use a coordinate proof to show that triangle ABC is scalene using coordinates (-5, 2), (-3, 5), and (-2, -1). what are the lengths of sides AB, BC, and AC?

Answer 1

To show that triangle \( ABC \) is scalene, we need to find the lengths of the sides \( AB \), \( BC \), and \( AC \) using the distance formula. The coordinates of the points are:

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

The distance formula is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 1: Calculate \( AB \)

Using the coordinates of points \( A \) and \( B \):
\[
AB = \sqrt{((-3) - (-5))^2 + (5 - 2)^2}
\]
\[
= \sqrt{(-3 + 5)^2 + (5 - 2)^2}
\]
\[
= \sqrt{(2)^2 + (3)^2}
\]
\[
= \sqrt{4 + 9}
\]
\[
= \sqrt{13}
\]

### Step 2: Calculate \( BC \)

Using the coordinates of points \( B \) and \( C \):
\[
BC = \sqrt{((-2) - (-3))^2 + ((-1) - 5)^2}
\]
\[
= \sqrt{(-2 + 3)^2 + (-1 - 5)^2}
\]
\[
= \sqrt{(1)^2 + (-6)^2}
\]
\[
= \sqrt{1 + 36}
\]
\[
= \sqrt{37}
\]

### Step 3: Calculate \( AC \)

Using the coordinates of points \( A \) and \( C \):
\[
AC = \sqrt{((-2) - (-5))^2 + ((-1) - 2)^2}
\]
\[
= \sqrt{(-2 + 5)^2 + (-1 - 2)^2}
\]
\[
= \sqrt{(3)^2 + (-3)^2}
\]
\[
= \sqrt{9 + 9}
\]
\[
= \sqrt{18}
\]

### Conclusion

Now we have the lengths of the sides:
- \( AB = \sqrt{13} \)
- \( BC = \sqrt{37} \)
- \( AC = \sqrt{18} \)

To prove that triangle \( ABC \) is scalene, we need to show that all three side lengths are different:
- \( \sqrt{13} \) is approximately \( 3.61 \)
- \( \sqrt{37} \) is approximately \( 6.08 \)
- \( \sqrt{18} \) is approximately \( 4.24 \)

Since \( \sqrt{13} \), \( \sqrt{37} \), and \( \sqrt{18} \) are all distinct, triangle \( ABC \) is indeed a scalene triangle.