Consider the following points.

(−2,8),(1,5)(−2,8),(1,5) and (1,8)(1,8)

Step 1 of 2 :

Determine whether or not the given points form a right triangle. If the triangle is **not** a right triangle, determine if it is isosceles or scalene.

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.

Consider the following points: (−2,8), (1,5), and (1,8). Determine whether or not the given points form a right triangle. If the triangle is not a right triangle, determine if it is isosceles or scalene.

Learn how to determine if the triangle formed by the points (-2,8), (1,5), and (1,8) is a right triangle or isosceles using the distance formula and the Pythagorean theorem. This geometry problem helps in understanding triangle classification, ideal for students in Grades 9-10.

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