The sets of numbers 7, 24, 25 and 9, 40, 41 are Pythagorean triples. Use what you know about the Pythagorean Theorem and explain or show why they are Pythagorean triples. Be sure to show your work for each set of triples! (5 points)
To show that a set of numbers is a Pythagorean triple, we need to verify that the Pythagorean theorem holds true for that set.
For the set of numbers 7, 24, and 25:
According to the Pythagorean theorem, for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, 7 and 24 are the two sides, and 25 is the hypotenuse.
Let's square each of the numbers:
7^2 = 49
24^2 = 576
25^2 = 625
Now, let's check if the Pythagorean theorem holds true:
49 + 576 = 625
Since the sum of the squares of the two sides (49 + 576) equals the square of the hypotenuse (625), the set of numbers (7, 24, 25) is a Pythagorean triple.
For the set of numbers 9, 40, and 41:
Using the same process, let's square each of the numbers:
9^2 = 81
40^2 = 1600
41^2 = 1681
Now, let's check if the Pythagorean theorem holds true:
81 + 1600 = 1681
Since the sum of the squares of the two sides (81 + 1600) equals the square of the hypotenuse (1681), the set of numbers (9, 40, 41) is also a Pythagorean triple.
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To show that a set of numbers is a Pythagorean triple, we need to verify that the Pythagorean theorem holds true for that set.
For the set of numbers 7, 24, and 25:
According to the Pythagorean theorem, the sum of the squares of the two legs (the shorter sides) is equal to the square of the hypotenuse (the longest side). Let's check if this is true:
7^2 + 24^2 = 49 + 576 = 625 = 25^2
Since the sum of the squares of the two legs (7^2 + 24^2) equals the square of the hypotenuse (25^2), the set of numbers (7, 24, 25) is a Pythagorean triple.
For the set of numbers 9, 40, and 41:
Using the same process, let's check if the Pythagorean theorem holds true:
9^2 + 40^2 = 81 + 1600 = 1681 = 41^2
Again, the sum of the squares of the two legs (9^2 + 40^2) equals the square of the hypotenuse (41^2), so the set of numbers (9, 40, 41) is a Pythagorean triple.
Cade and Tyce both tried to find the missing side of the right triangle.
A right triangle is shown. The legs are labeled 6 and 9. A right angle is labeled in between them.
Cade's Work Tyce's Work
a2 + b2 = c2 a2 + b2 = c2
62 + b2 = 92 62 + 92 = c2
36 + b2 = 81 36 + 81 = c2
b2 = 45 117 = c2
Square root b squared equals square root 45. Square root 117 equals c squared.
b ≈ 6.71 10.82 ≈ c
Is either of them correct? Explain your reasoning. (5 points)
Cade's work is correct, while Tyce's work is incorrect.
Cade correctly applies the Pythagorean theorem by using the formula a^2 + b^2 = c^2. He substitutes the given values 6 for a and 9 for b. Then, he solves for c^2 by simplifying the equation and getting 36 + b^2 = 81. Finally, he solves for b by taking the square root of both sides, getting b ≈ 6.71.
On the other hand, Tyce's work is incorrect. He miscalculates the sum of 6^2 + 9^2 as 36 + 81 = 117. This is an error, as 6^2 + 9^2 is actually equal to 36 + 81 = 117. Furthermore, Tyce incorrectly writes that the square root of 117 is equal to c^2. The correct calculation would be taking the square root of both sides, resulting in √117 ≈ 10.82 ≈ c.
Therefore, Cade's answer is correct with b ≈ 6.71. Tyce's calculation contains errors in both the sum and the calculation of the square root, so his answer is incorrect.