A triangle with vertices `(4, 0), (-1,-1), (3,5)`, is

Options

A. isosceles and right angled
B. isoscles but not right angled
C. right angled but not isosceles
D. neither right angeld nor isosceles

Answer

To determine the properties of the triangle with vertices \( A(4, 0) \), \( B(-1, -1) \), and \( C(3, 5) \), we will calculate the lengths of the sides using the distance formula. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

### Step 1: Calculate the length of side AB

Using the coordinates of points A and B:

\[ AB = \sqrt{((-1) - 4)^2 + ((-1) - 0)^2} \]

Calculating this:

\[ AB = \sqrt{(-5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

### Step 2: Calculate the length of side BC

Using the coordinates of points B and C:

\[ BC = \sqrt{(3 - (-1))^2 + (5 - (-1))^2} \]

Calculating this:

\[ BC = \sqrt{(3 + 1)^2 + (5 + 1)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} \]

### Step 3: Calculate the length of side AC

Using the coordinates of points A and C:

\[ AC = \sqrt{(3 - 4)^2 + (5 - 0)^2} \]

Calculating this:

\[ AC = \sqrt{(-1)^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26} \]

### Step 4: Compare the lengths of the sides

Now we have:

- \( AB = \sqrt{26} \)
- \( BC = \sqrt{52} \)
- \( AC = \sqrt{26} \)

From this, we can see that:

- \( AB = AC \) (which indicates that the triangle is isosceles)

### Step 5: Check for right angle using Pythagorean theorem

To check if the triangle is right-angled, we can use the Pythagorean theorem. For triangle ABC to be right-angled, the square of the length of the longest side should equal the sum of the squares of the other two sides.

Here, \( BC \) is the longest side:

\[ BC^2 = AB^2 + AC^2 \]

Calculating:

\[ BC^2 = 52, \quad AB^2 = 26, \quad AC^2 = 26 \]

Now, check if:

\[ 52 = 26 + 26 \]

This holds true, confirming that triangle ABC is also right-angled.

### Conclusion

The triangle with vertices \( A(4, 0) \), \( B(-1, -1) \), and \( C(3, 5) \) is both isosceles and right-angled.