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To determine if the points \( A(-7, -7) \), \( B(2, -3) \), and \( C(1, 1) \) form a right triangle, we can use the distance formula to calculate the lengths of the sides. If the square of one side equals the sum of the squares of the other two sides, then the triangle is a right triangle.

### Step 1: Calculate the distances between points
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Distance \( AB \)
\[
AB = \sqrt{(2 - (-7))^2 + (-3 - (-7))^2} = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97}
\]

#### Distance \( BC \)
\[
BC = \sqrt{(1 - 2)^2 + (1 - (-3))^2} = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17}
\]

#### Distance \( CA \)
\[
CA = \sqrt{(1 - (-7))^2 + (1 - (-7))^2} = \sqrt{8^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2}
\]

### Step 2: Check for the Pythagorean Theorem
For a right triangle, we should have:
\[
\text{(Hypotenuse)}^2 = (\text{Other Side 1})^2 + (\text{Other Side 2})^2
\]
Let's verify if one of the sides is the hypotenuse.

1. \( AB^2 = 97 \)
2. \( BC^2 = 17 \)
3. \( CA^2 = 128 \)

Since \( CA \) has the largest length, it is the potential hypotenuse. Now we check:
\[
CA^2 = AB^2 + BC^2
\]
\[
128 = 97 + 17 = 114 \quad \text{(which is not true)}
\]

Since the distances do not satisfy the Pythagorean theorem, **the triangle is not a right triangle**.

**Answer:** No.

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### Related Questions:
1. How do you determine if three points are collinear?
2. What are other methods to determine if a triangle is right-angled?
3. How can we use vector analysis to determine the type of triangle?
4. What is the significance of the Pythagorean theorem in coordinate geometry?
5. How can you verify if a triangle is an equilateral or isosceles triangle?

**Tip:** In problems involving right triangles, always check the largest distance as the hypotenuse candidate!

Determine whether or not the points A(-7, -7), B(2, -3), and C(1, 1) form a right triangle.

Find out whether the points A(-7, -7), B(2, -3), and C(1, 1) form a right triangle using the distance formula and the Pythagorean theorem. This problem involves concepts from coordinate geometry and is suitable for students in Grades 8-10.

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