Given three positive numbers, how can you determine if they form a Pythagorean triple?

• A. Check if all three numbers are consecutive integers.
• B. Check if the product of the two smaller numbers equals the largest number.
• C. Check if the sum of the two smaller numbers equals the largest number.
• D. Check if the sum of the squares of the two smaller numbers equals the square of the largest number, that is, if $a^2+b^2=c^2$.

Answer: (D)

Verified step by step guidance

1. Identify the three positive numbers and label them as \(a\), \(b\), and \(c\), where \(c\) is the largest number.
2. Recall the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides.
3. To check if the numbers form a Pythagorean triple, calculate the squares of the two smaller numbers: \(a^2\) and \(b^2\).
4. Calculate the square of the largest number: \(c^2\).
5. Compare the sum of the squares of the two smaller numbers to the square of the largest number by verifying if \(a^2 + b^2 = c^2\). If this equality holds true, the numbers form a Pythagorean triple.