To determine which set of numbers represents the lengths of the sides of a right triangle, we can use the Pythagorean theorem, which 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 lengths of the other two sides. Let's go through each set of numbers and apply this theorem:
A. (2,3,6)
We can check if this set of numbers forms a right triangle by applying the Pythagorean theorem:
\(2^2 + 3^2 = 4 + 9 = 13\)
The sum of the squares of the two shorter sides is 13, which is not equal to the square of the longest side (6^2 = 36). Therefore, this set of numbers does not represent the lengths of the sides of a right triangle.
B. {5,5,10}
Applying the Pythagorean theorem:
\(5^2 + 5^2 = 25 + 25 = 50\)
The sum of the squares of the two shorter sides is 50, which is not equal to the square of the longest side (10^2 = 100). Thus, this set of numbers does not represent the lengths of the sides of a right triangle.
C. [4,5,6)
Using the Pythagorean theorem:
\(4^2 + 5^2 = 16 + 25 = 41\)
The sum of the squares of the two shorter sides is 41, which is not equal to the square of the longest side (6^2 = 36). Therefore, this set of numbers does not represent the lengths of the sides of a right triangle.
D. {5,12,13}
Applying the Pythagorean theorem:
\(5^2 + 12^2 = 25 + 144 = 169\)
The sum of the squares of the two shorter sides is 169, which is equal to the square of the longest side (13^2 = 169). Hence, this set of numbers represents the lengths of the sides of a right triangle.
Therefore, the correct answer is D.
Moving on to the second question:
If two legs of a right triangle are 9 and 11, we can find the length of the hypotenuse using the Pythagorean theorem:
\(9^2 + 11^2 = 81 + 121 = 202\)
So, the length of the hypotenuse is \(\sqrt{202}\).
Therefore, the correct answer is A.
For the third question:
If the lengths of the legs of a right triangle are 5 and 12, we can find the length of the hypotenuse using the Pythagorean theorem:
\(5^2 + 12^2 = 25 + 144 = 169\)
So, the length of the hypotenuse is \(\sqrt{169}\), which is equal to 13.
Therefore, the correct answer is D.
Finally, for the fourth question:
If the lengths of the legs of a right triangle are 12 and 16, we can find the length of the hypotenuse using the Pythagorean theorem:
\(12^2 + 16^2 = 144 + 256 = 400\)
So, the length of the hypotenuse is \(\sqrt{400}\), which is equal to 20.
Therefore, the correct answer is A.
It's important to double-check the calculations and verify the answers by applying the Pythagorean theorem and ensuring the sum of the squares of the two shorter sides is equal to the square of the longest side.
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