3. Show that the points (-2,-1), (8,3), and (1,6) are vertices of a right triangle.
(Tip: search what is the characteristics of a right riangle)


To show that the points (-2, -1), (8, 3), and (1, 6) form the vertices of a right triangle, we can use the property of a right triangle that its sides must satisfy the **Pythagorean theorem**. Specifically, the square of the length of the hypotenuse (the longest side) must be equal to the sum of the squares of the other two sides.

### Step-by-Step Solution

1. **Calculate the distance between each pair of points** using the distance formula:
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   This will give us the lengths of the three sides of the triangle.

2. **Step 1: Distance between (-2, -1) and (8, 3)**:
   \[
   d_1 = \sqrt{(8 - (-2))^2 + (3 - (-1))^2} = \sqrt{(8 + 2)^2 + (3 + 1)^2} = \sqrt{10^2 + 4^2} = \sqrt{100 + 16} = \sqrt{116}
   \]
   So, \( d_1 = \sqrt{116} \approx 10.77 \).

3. **Step 2: Distance between (8, 3) and (1, 6)**:
   \[
   d_2 = \sqrt{(1 - 8)^2 + (6 - 3)^2} = \sqrt{(-7)^2 + (3)^2} = \sqrt{49 + 9} = \sqrt{58}
   \]
   So, \( d_2 = \sqrt{58} \approx 7.62 \).

4. **Step 3: Distance between (-2, -1) and (1, 6)**:
   \[
   d_3 = \sqrt{(1 - (-2))^2 + (6 - (-1))^2} = \sqrt{(1 + 2)^2 + (6 + 1)^2} = \sqrt{3^2 + 7^2} = \sqrt{9 + 49} = \sqrt{58}
   \]
   So, \( d_3 = \sqrt{58} \approx 7.62 \).

### Step 5: Check the Pythagorean theorem
We now check if one of these sides satisfies the Pythagorean theorem:
- The longest side is \( d_1 = \sqrt{116} \).
- Check if \( d_1^2 = d_2^2 + d_3^2 \).

Let's calculate:
\[
d_1^2 = 116, \quad d_2^2 = 58, \quad d_3^2 = 58
\]
Since \( d_1^2 = d_2^2 + d_3^2 \):
\[
116 = 58 + 58
\]
This is true!

### Conclusion:
The points (-2, -1), (8, 3), and (1, 6) form the vertices of a right triangle because they satisfy the Pythagorean theorem.

---

Do you want further details or have any questions?

Here are 5 related questions to expand this concept:
1. How do you find the area of a triangle given its vertices?
2. Can a triangle with the same side lengths form a different type of triangle?
3. What are the properties of an equilateral triangle compared to a right triangle?
4. How can we determine the slope of a line between two points?
5. How do you find the equation of a circle passing through the vertices of a triangle?

**Tip**: The Pythagorean theorem is only applicable in Euclidean geometry, and it defines right triangles.

Show that the points (-2,-1), (8,3), and (1,6) are vertices of a right triangle.

Learn how to prove that the points (-2,-1), (8,3), and (1,6) form the vertices of a right triangle using the Pythagorean theorem and distance formula. This solution provides a detailed step-by-step guide, ideal for students in Grades 9-10.

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