To determine if the triangle formed by the points $A(55, 22)$, $B(0, 55)$, and $C(-66, -55)$ is a right triangle, we can use the distance formula to calculate the lengths of the sides and then apply the Pythagorean theorem.

Step 1: 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:

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

1. Length of side $AB$:

$AB = \sqrt{(0 - 55)^2 + (55 - 22)^2} = \sqrt{(-55)^2 + (33)^2} = \sqrt{3025 + 1089} = \sqrt{4114}$

2. Length of side $BC$:

$BC = \sqrt{(-66 - 0)^2 + (-55 - 55)^2} = \sqrt{(-66)^2 + (-110)^2} = \sqrt{4356 + 12100} = \sqrt{16456}$

3. Length of side $AC$:

$AC = \sqrt{(-66 - 55)^2 + (-55 - 22)^2} = \sqrt{(-121)^2 + (-77)^2} = \sqrt{14641 + 5929} = \sqrt{20570}$

Step 2: Check if the Pythagorean theorem holds.

The Pythagorean theorem states that for a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

Compare:

$AB^2 + AC^2 = \text{Is it equal to } BC^2?$

Calculate the squares:

• $AB^2 = 4114$
• $AC^2 = 20570$
• $BC^2 = 16456$

Now check if:

$4114 + 16456 = 20570$

Adding the squares:

$20570 = 20570 \quad \text{(True)}$

Since the Pythagorean theorem holds, the triangle is a right triangle.

Would you like further details or have any other questions? Here are some related questions to deepen your understanding:

1. Can you determine which angle in the triangle is the right angle?
2. What is the area of this right triangle?
3. How would you find the equation of the line forming the hypotenuse?
4. What would be the perimeter of the triangle?
5. How can you verify the triangle is right-angled using slopes of the sides?

Tip: Always compare the squares of the sides when using the Pythagorean theorem, and remember that the longest side must be the hypotenuse for a right triangle.