Which of the following numbers completes the Pythagorean triple: 12, 16, ____?
18
20
22
24
20
To find the missing number in the Pythagorean triple, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the given sides are 12 and 16. Let's calculate the squares of these numbers:
12^2 = 144
16^2 = 256
Now, let's find the square root of the sum of these squares:
√(144 + 256) = √400 = 20
So, the missing number in the Pythagorean triple is 20. Therefore, the correct answer is 20.
To complete a Pythagorean triple, we need to find the missing number that satisfies the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the lengths of the two legs of the right triangle, and c is the length of the hypotenuse.
In this case, we are given the lengths of the two legs: 12 and 16. Let's find the missing number that completes the triple.
To determine which number satisfies the Pythagorean theorem, we need to apply the theorem to each option and check if it holds true.
1. For 18:
a^2 + b^2 = c^2
12^2 + 16^2 = 144 + 256 = 400
The square of 18 is not equal to 400, so it does not satisfy the Pythagorean theorem.
2. For 20:
a^2 + b^2 = c^2
12^2 + 16^2 = 144 + 256 = 400
The square of 20 is equal to 400, so it satisfies the Pythagorean theorem.
3. For 22:
a^2 + b^2 = c^2
12^2 + 16^2 = 144 + 256 = 400
The square of 22 is not equal to 400, so it does not satisfy the Pythagorean theorem.
4. For 24:
a^2 + b^2 = c^2
12^2 + 16^2 = 144 + 256 = 400
The square of 24 is not equal to 400, so it does not satisfy the Pythagorean theorem.
Therefore, the number that completes the Pythagorean triple for the lengths given is 20.