Final answer:
To prove the points A, B, and C form a right triangle, we calculated the distances between the points and confirmed that the Pythagorean theorem holds for these lengths, thus establishing that they are vertices of a right triangle.
Step-by-step explanation:
To prove that the points A(-1,-2), B(2,3), and C(-11,4) are vertices of a right triangle, we can show that the squares of the lengths of two sides add up to the square of the length of the third side, as per the Pythagorean theorem. The distances between the points are calculated using the distance formula, which is √((x2-x1)² + (y2-y1)²).
- Distance AB = √((2 - (-1))² + (3 - (-2))²) = √(3² + 5²) = √34
- Distance BC = √((-11 - 2)² + (4 - 3)²) = √((-13)² + 1²) = √170
- Distance AC = √((-11 - (-1))² + (4 - (-2))²) = √((-10)² + 6²) = √136
Next, we check if the Pythagorean theorem holds for these distances:
(Distance AB)² + (Distance AC)² = 34 + 136 = 170 = (Distance BC)²
Since the sum of the squares of distances AB and AC equals the square of distance BC, the points A, B, and C form a right triangle.