Question

Which of the following could be the side lengths of a right triangle?

• A. $3,13,$ and $14$
• B. $4,5,$ and $6$
• C. $4,9,$ and $10$
• D. $5,10,$ and $15$
• E. $5,12,$ and $13$

Answer 1

The correct option is E $5,12,$ and $13$

If a triangle is a right triangle, then the lengths of its sides satisfy the Pythagorean Theorem, $a^2+b^2=c^2$. To determine which choice is correct, test each set of values by substituting them into the Pythagorean Theorem. Start with the first set of numbers: 3, 13, and 14.
$3^2+{13}^2={14}^2$
$9+169=196$
$178=196$
Since the result is not a true equality, the first set of values does not represent the side lengths of a right triangle. Test the other four choices. The only values that satisfy the Pythagorean Theorem are 5, 12, and 13.
$5^2+{12}^2={13}^2$
$25+144=169$
$169=169$