Final answer:
To determine whether a triangle with vertices A=(1,1), B=(4,5), and C=(12,1) is a right triangle, the distances between each pair of points are calculated and the Pythagorean theorem is used. The theorem isn't satisfied, indicating that triangle ABC is not a right triangle.
Step-by-step explanation:
To show that the triangle with vertices A=(1,1), B=(4,5), and C=(12,1) is a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
First, we calculate the distances between each pair of points:
AB = √((4-1)² + (5-1)²)
= √(3² + 4²)
= √(9 + 16)
= √25
= 5
BC = √((12-4)² + (1-5)²)
= √(8² + (-4)²)
= √(64 + 16)
= √80
= 4√5
AC = √((12-1)² + (1-1)²)
= √(11² + 0²)
= √121
= 11
Now, we verify the Pythagorean theorem:
AB² + BC² = 5² + (4√5)²
= 25 + 16· 5
= 25 + 80
= 105
AC² = 11²
= 121
Since AB² + BC² is not equal to AC², triangle ABC cannot be a right triangle. Thus, we have shown that triangle ABC is not a right triangle.