Answer:
After calculating the distances between points A(1,4), B(4,5), and C(7,0) and applying the Pythagorean theorem, it was determined that these points do not form a right triangle as the sum of the squares of the two shorter sides differs from the square of the longest side.
Step-by-step explanation:
Checking if Points Form a Right Triangle
To determine whether the points A(1,4), B(4,5), and C(7,0) form a right triangle, we will calculate the distances between each pair of points to find the lengths of the sides of the triangle they might form.
Using the distance formula, which is derived from the Pythagorean theorem, we find the lengths of the three sides. For the points to form a right triangle, the square of the longest side must equal the sum of the squares of the other two sides.
- Distance AB: √((4-1)² + (5-4)²) = √(9 + 1) = √10
- Distance BC: √((7-4)² + (0-5)²) = √(9 + 25) = √34
- Distance CA: √((7-1)² + (0-4)²) = √(36 + 16) = √52
Next, we check the Pythagorean relationship: (AB² + BC² = CA²).
(√10)² + (√34)² ≠ (√52)²
10 + 34 ≠ 52
44 ≠ 52
The sum of the squares of AB and BC does not equal the square of CA, therefore, the points A, B, and C do not form a right triangle.