To determine if the given points form a right triangle, we need to calculate the distances between each pair of points. The points given are:

• $A(-2, 8)$
• $B(1, 5)$
• $C(1, 8)$

Step 1: Calculate the distances between the points

We will use the distance formula:

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

Distance between $A(-2, 8)$ and $B(1, 5)$:

$AB = \sqrt{(1 - (-2))^2 + (5 - 8)^2} = \sqrt{(1 + 2)^2 + (5 - 8)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$

Distance between $B(1, 5)$ and $C(1, 8)$:

$BC = \sqrt{(1 - 1)^2 + (8 - 5)^2} = \sqrt{0 + 3^2} = \sqrt{9} = 3$

Distance between $A(-2, 8)$ and $C(1, 8)$:

$AC = \sqrt{(1 - (-2))^2 + (8 - 8)^2} = \sqrt{(1 + 2)^2 + 0} = \sqrt{3^2} = 3$

Step 2: Check the triangle type

• $AB = 3\sqrt{2}$
• $BC = 3$
• $AC = 3$

Right Triangle Check:

To check if this is a right triangle, we use the Pythagorean theorem: $AB^2 = BC^2 + AC^2$

However: $(3\sqrt{2})^2 = 18, \quad BC^2 = 9, \quad AC^2 = 9$ $18 \neq 9 + 9 \implies \text{This is not a right triangle.}$

Isosceles or Scalene Check:

Since $BC = AC = 3$, the triangle is isosceles, because it has two equal sides.

Conclusion:

The triangle formed by the points $A(-2, 8)$, $B(1, 5)$, and $C(1, 8)$ is isosceles, but it is not a right triangle.

Would you like more details or have any questions?

Here are 5 related questions:

1. How can you generalize checking if any triangle is isosceles or scalene?
2. What other conditions could classify a triangle besides right, isosceles, and scalene?
3. Can you use a different method to check for a right triangle (like slopes)?
4. How does the distance formula work for points in 3D space?
5. What is the significance of the Pythagorean theorem in real-life applications?

Tip: Always check if two sides are equal first; this helps quickly determine if a triangle is isosceles.