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. We'll also find the area of the triangle.

### Step 1: Calculate the distances between the points
We use the distance formula to find the lengths of the sides of the triangle:

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

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
The Pythagorean theorem states that for a right triangle:

\[
\text{Hypotenuse}^2 = \text{Side}_1^2 + \text{Side}_2^2
\]

Here, we compare the squares of the distances:

- \(\text{Hypotenuse} = d_3 = 5\sqrt{2}\)
- \(\text{Side}_1 = d_2 = 3\sqrt{5}\)
- \(\text{Side}_2 = d_1 = \sqrt{5}\)

Now, check if:

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

Calculate each side:

- \( (5\sqrt{2})^2 = 50 \)
- \( (3\sqrt{5})^2 = 45 \)
- \( (\sqrt{5})^2 = 5 \)

Since:

\[
50 = 45 + 5
\]

The equation holds true, confirming that 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:

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

Here, we can take \(d_1 = \

Learn how to verify if the points (4, 0), (2, 1), and (-1, -5) form a right triangle using the Pythagorean theorem. Calculate distances between points and confirm the triangle properties. Find the area of the right triangle with detailed step-by-step calculations.

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