How do you determine if the length 9, 40, 41 form a right triangle?

To determine if the lengths 9, 40, and 41 can form a right triangle, you can use the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Steps to Check for a Right Triangle

• Identify the longest side: In this case, 41 is the longest side.
• Apply the Pythagorean theorem: Check if \( 41^2 = 9^2 + 40^2 \).

Calculating the Squares

Now, let's calculate:

• 41 squared is \( 41 \times 41 = 1681 \).
• 9 squared is \( 9 \times 9 = 81 \).
• 40 squared is \( 40 \times 40 = 1600 \).

Adding the Squares

Next, add the squares of the two shorter sides:

\( 81 + 1600 = 1681 \)

Final Check

Since \( 41^2 = 9^2 + 40^2 \) (or \( 1681 = 1681 \)), the lengths 9, 40, and 41 do indeed form a right triangle.