Show that the points (4, 0), (2, 1), and (-1,-5) are vertices of a right triangle. Find its area.

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.

Learn how to verify if points (4, 0), (2, 1), and (-1, -5) form a right triangle using the Pythagorean theorem. Calculate the area of the triangle using the distance formula and basic geometric principles. Suitable for high school students studying geometry and trigonometry.

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