To show that the points $(4, 0)$, $(2, 1)$, and $(-1, -5)$ are vertices of a right triangle, we need to verify that the triangle formed by these points satisfies the Pythagorean theorem:

$\text{For a right triangle: } c^2 = a^2 + b^2$

where $c$ is the hypotenuse (the longest side) and $a$ and $b$ are the other two sides.

Step 1: Calculate the distances between the points

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: Verify the Pythagorean theorem

To confirm the points form a right triangle, check if:

$\text{Largest side}^2 = \text{Sum of the squares of the other two sides}$

We observe that:

$d_3 = 5\sqrt{2}, \quad d_2 = 3\sqrt{5}, \quad d_1 = \sqrt{5}$

Now, check:

$(5\sqrt{2})^2 = (3\sqrt{5})^2 + (\sqrt{5})^2$

$50 = 45 + 5$

Since the equation holds true, the points form a right triangle.

Step 3: Calculate the area of the right triangle

The area $A$ of a right triangle is given by:

$A = \frac{1}{2} \times \text{base} \times \text{height}$

Taking $d_1 = \sqrt{5}$ as the base and $d_2 = 3\sqrt{5}$ as the height:

$A = \frac{1}{2} \times \sqrt{5} \times 3\sqrt{5} = \frac{1}{2} \times 15 = 7.5 \text{ square units}$

Final Answer:

The points $(4, 0)$, $(2, 1)$, and $(-1, -5)$ are the vertices of a right triangle, and the area of this triangle is $7.5$ square units.

Do you have any questions or would you like more details?

Related Questions:

1. What are the other methods to find the area of a triangle given its vertices?
2. How can you determine if three points are collinear?
3. How do you find the centroid of a triangle given its vertices?
4. How can the distance formula be adapted to check for equilateral triangles?
5. What is the formula for the circumradius of a right triangle?

Tip:

When working with triangles, verifying the Pythagorean theorem is a quick way to check if the triangle is a right triangle.