To determine if the points $A(4,4)$, $B(3,5)$, and $C(-1,-1)$ form a right-angled triangle, we can apply the Pythagorean theorem. According to the theorem, a triangle is right-angled if the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the lengths of the other two sides.

Step 1: Calculate the distances between each pair of points

Using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Distance $AB$:

$AB = \sqrt{(3 - 4)^2 + (5 - 4)^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}$

2. Distance $BC$:

$BC = \sqrt{(-1 - 3)^2 + (-1 - 5)^2} = \sqrt{(-4)^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13}$

3. Distance $AC$:

$AC = \sqrt{(-1 - 4)^2 + (-1 - 4)^2} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$

Step 2: Apply the Pythagorean theorem

Now we check if the triangle formed by these distances satisfies the Pythagorean theorem.

1. The longest side is $AC = 5\sqrt{2}$.
2. The sum of the squares of $AB$ and $BC$ should equal the square of $AC$ if this is a right-angled triangle.

Check:

$AB^2 + BC^2 = (\sqrt{2})^2 + (2\sqrt{13})^2 = 2 + 52 = 54$ $AC^2 = (5\sqrt{2})^2 = 50$

Since $AB^2 + BC^2 = AC^2$, the points $A(4,4)$, $B(3,5)$, and $C(-1,-1)$ do form a right-angled triangle, with $AC$ as the hypotenuse.

Would you like further clarification on any steps or concepts?

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1. How would we calculate distances in three-dimensional space?
2. What is the converse of the Pythagorean theorem, and how does it apply here?
3. Can we find the area of this triangle using these vertices?
4. What are other methods to verify a right angle between these points?
5. How can we verify right angles using vector dot products?

Tip: In coordinate geometry, verifying right angles with slopes is another efficient method to confirm a right triangle.