To determine whether $\triangle ABC$ is scalene, isosceles, or equilateral, we need to calculate the lengths of the sides $AB$, $BC$, and $AC$ using the distance formula:

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

From the coordinates in the graph:

• Point $A$ is at $(0, 4)$,
• Point $B$ is at $(4, 0)$,
• Point $C$ is at $(0, 0)$.

Step 1: Calculate $AB$

$AB = \sqrt{(4 - 0)^2 + (0 - 4)^2} = \sqrt{4^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

Step 2: Calculate $BC$

$BC = \sqrt{(4 - 0)^2 + (0 - 0)^2} = \sqrt{4^2} = \sqrt{16} = 4$

Step 3: Calculate $AC$

$AC = \sqrt{(0 - 0)^2 + (4 - 0)^2} = \sqrt{4^2} = \sqrt{16} = 4$

Conclusion

Since $AB = 4\sqrt{2}$, $BC = 4$, and $AC = 4$, two sides (BC and AC) are equal, but $AB$ is different. Therefore, $\triangle ABC$ is isosceles.

Answer

The correct choice is:

• A. $\triangle ABC$ is isosceles because $AB = 4\sqrt{2}$, $BC = 4$, and $AC = 4$.

Would you like more details on any step, or do you have any questions?

Additional Questions for Practice:

1. How would you determine if a triangle is a right triangle using side lengths?
2. Can you find the area of $\triangle ABC$ given the vertices?
3. How can you verify the classification of a triangle using slopes?
4. What are the coordinates of the midpoint of $AB$ in $\triangle ABC$?
5. How does the distance formula apply to points in three-dimensional space?

Tip:

For any triangle on a coordinate plane, you can use the distance formula to check for equal side lengths, which helps identify whether the triangle is equilateral, isosceles, or scalene.