Question

Point A lies at (-4, 4) and point B lies at (-3, 6). Which point is needed so that triangle ABC is scalene?
A. C (-2, 4)
B. C (-1, 4)

Answer 1

To determine which point C would create a scalene triangle ABC with A(-4, 4) and B(-3, 6), we need to find a point C such that all the sides of the triangle AB, AC, and BC have different lengths.

1. **Calculate the distance AB**:
Using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For points A(-4, 4) and B(-3, 6):
\[
AB = \sqrt{((-3) - (-4))^2 + (6 - 4)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

2. **Check point C (-2, 4)**:
- For AC:
\[
AC = \sqrt{((-2) - (-4))^2 + (4 - 4)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2
\]
- For BC:
\[
BC = \sqrt{((-2) - (-3))^2 + (4 - 6)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
\]

The sides are:
\( AB = \sqrt{5}, AC = 2, BC = \sqrt{5} \)
Here, \( AB \) and \( BC \) are equal, so triangle ABC is not scalene.

3. **Check point C (-1, 4)**:
- For AC:
\[
AC = \sqrt{((-1) - (-4))^2 + (4 - 4)^2} = \sqrt{(3)^2 + (0)^2} = \sqrt{9} = 3
\]
- For BC:
\[
BC = \sqrt{((-1) - (-3))^2 + (4 - 6)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]

The sides are:
\( AB = \sqrt{5}, AC = 3, BC = 2\sqrt{2} \)
Since \( \sqrt{5}, 3, \text{ and } 2\sqrt{2} \) are all different, triangle ABC is scalene.

Thus, the point needed so that triangle ABC is scalene is **B. C (-1, 4)**.