The points `(4,4)`, `(3,5)` and `(-1,-1)` are the vertices of

Options

A. a right angled triangle.
B. equilateral triangle.
C. isosceles triangle
D. None

Answer

To determine if the points (4,4), (3,5), and (-1,-1) are the vertices of a right-angled triangle, we can use the distance formula to calculate the lengths of the sides of the triangle formed by these points. Then we will check if the Pythagorean theorem holds true.

### Step 1: Assign the points
Let:
- A = (4, 4)
- B = (3, 5)
- C = (-1, -1)

### Step 2: Calculate the lengths of the sides
We will use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

#### Length of AB
Using points A and B:
\[ AB = \sqrt{(3 - 4)^2 + (5 - 4)^2} \]
\[ = \sqrt{(-1)^2 + (1)^2} \]
\[ = \sqrt{1 + 1} \]
\[ = \sqrt{2} \]

#### Length of BC
Using points B and C:
\[ BC = \sqrt{(-1 - 3)^2 + (-1 - 5)^2} \]
\[ = \sqrt{(-4)^2 + (-6)^2} \]
\[ = \sqrt{16 + 36} \]
\[ = \sqrt{52} \]
\[ = 2\sqrt{13} \]

#### Length of AC
Using points A and C:
\[ AC = \sqrt{(-1 - 4)^2 + (-1 - 4)^2} \]
\[ = \sqrt{(-5)^2 + (-5)^2} \]
\[ = \sqrt{25 + 25} \]
\[ = \sqrt{50} \]
\[ = 5\sqrt{2} \]

### Step 3: Check the Pythagorean theorem
For triangle ABC to be a right triangle, the sum of the squares of the lengths of the two shorter sides must equal the square of the length of the longest side.

1. Calculate \( AB^2 \):
\[ AB^2 = (\sqrt{2})^2 = 2 \]

2. Calculate \( AC^2 \):
\[ AC^2 = (\sqrt{50})^2 = 50 \]

3. Calculate \( BC^2 \):
\[ BC^2 = (2\sqrt{13})^2 = 4 \times 13 = 52 \]

Now, check if \( AB^2 + AC^2 = BC^2 \):
\[ 2 + 50 = 52 \]

Since this is true, we conclude that triangle ABC satisfies the Pythagorean theorem.

### Conclusion
Thus, the points (4,4), (3,5), and (-1,-1) are the vertices of a right-angled triangle.