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.

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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.