Final answer:
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using the differences in x and y coordinates as the lengths of the triangle's legs, the distance between the two points (12, 0) and (0, 7) can be calculated as the square root of the sum of the squares of the differences, confirming the 14 units distance.
Step-by-step explanation:
Understanding the Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry that describes the relationship between the sides of a right-angled triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This relationship can be expressed as a² + b² = c². When you know the lengths of any two sides, you can use this formula to find the length of the third side.
To apply the Pythagorean theorem to the given points, we can consider the difference in the x-coordinates (12 - 0 = 12) and the y-coordinates (7 - 0 = 7) as the lengths of legs a and b of a right triangle, where the distance between the points is the hypotenuse (c). The calculation would be: distance = √(12² + 7²).
By plugging in the given values, you will find that the calculated distance using the Pythagorean theorem is in fact 14 units, consistent with the information provided in the question. The student should understand that this method provides the straight-line distance between two points, which is the most direct path between them.