To verify that the points $(4, 0)$, $(2, 1)$, and $(-1, -5)$ form the vertices of a right triangle, we can follow these steps:

Step 1: Calculate the distances between each pair of points.

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Calculate the three sides:

1. Distance between $(4, 0)$ and $(2, 1)$:

$d_1 = \sqrt{(2 - 4)^2 + (1 - 0)^2} = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$

2. Distance between $(2, 1)$ and $(-1, -5)$:

$d_2 = \sqrt{(-1 - 2)^2 + (-5 - 1)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$

3. Distance between $(4, 0)$ and $(-1, -5)$:

$d_3 = \sqrt{(-1 - 4)^2 + (-5 - 0)^2} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$

Step 2: Check if the triangle formed is a right triangle.

A triangle is a right triangle if the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides.

Let’s compare $d_3^2$ with $d_1^2 + d_2^2$:

1. $d_1^2 = (\sqrt{5})^2 = 5$
2. $d_2^2 = (3\sqrt{5})^2 = 9 \times 5 = 45$
3. $d_3^2 = (5\sqrt{2})^2 = 25 \times 2 = 50$

Now, add $d_1^2$ and $d_2^2$:

$d_1^2 + d_2^2 = 5 + 45 = 50$

Since $d_1^2 + d_2^2 = d_3^2$, the triangle formed by the points $(4, 0)$, $(2, 1)$, and $(-1, -5)$ is indeed a right triangle.

Conclusion

The points $(4, 0)$, $(2, 1)$, and $(-1, -5)$ form the vertices of a right triangle.

Would you like any further details or have any questions? Here are some related questions:

1. How do you find the area of the triangle formed by these points?
2. Can these points form any other type of polygon?
3. How can you find the equation of the lines formed by these points?
4. What are the angles of the triangle at each vertex?
5. How can you find the centroid of the triangle formed by these points?

Tip: When verifying a right triangle, always remember to compare the squares of the sides to identify the hypotenuse.