Question
Ariel completed the work below to show that a triangle with side lengths 13, 12, and 5 do not form a right triangle.
12^2 + 13^2 = 5^2
144 + 169 = 25
313 ≠ 25
Is Ariel's work and answer correct? Why or why not?
NOTE: You must answer the question in complete sentences to receive credit.
12^2 + 13^2 = 5^2
144 + 169 = 25
313 ≠ 25
Is Ariel's work and answer correct? Why or why not?
NOTE: You must answer the question in complete sentences to receive credit.
Answers
GPT-4o mini
Ariel's work is not correct in terms of confirming whether the triangle with side lengths 13, 12, and 5 is a right triangle; however, Ariel has correctly shown that the given statement does not hold true.
When determining if a triangle is a right triangle, we use the Pythagorean theorem, which states that for a right triangle, the sum of the squares of the lengths of the two shorter sides must equal the square of the length of the longest side. In this case, the longest side is 13, and the two shorter sides are 12 and 5. Therefore, the correct equation should be:
12^2 + 5^2 = 13^2
Calculating this:
12^2 = 144,
5^2 = 25, and
13^2 = 169.
So, the equation should read:
144 + 25 = 169,
which simplifies to
169 = 169.
Since this equation holds true, it indicates that the triangle with sides of lengths 12, 5, and 13 does indeed form a right triangle. Ariel's mistake was in incorrectly assigning the sides in the Pythagorean theorem. Thus, the conclusion about the triangle not being a right triangle is incorrect.
When determining if a triangle is a right triangle, we use the Pythagorean theorem, which states that for a right triangle, the sum of the squares of the lengths of the two shorter sides must equal the square of the length of the longest side. In this case, the longest side is 13, and the two shorter sides are 12 and 5. Therefore, the correct equation should be:
12^2 + 5^2 = 13^2
Calculating this:
12^2 = 144,
5^2 = 25, and
13^2 = 169.
So, the equation should read:
144 + 25 = 169,
which simplifies to
169 = 169.
Since this equation holds true, it indicates that the triangle with sides of lengths 12, 5, and 13 does indeed form a right triangle. Ariel's mistake was in incorrectly assigning the sides in the Pythagorean theorem. Thus, the conclusion about the triangle not being a right triangle is incorrect.
Make it seem like a 9th grader wrote it
GPT-4o mini
Ariel's work is not correct because she used the wrong sides when trying to show that the triangle with side lengths 13, 12, and 5 isn’t a right triangle. She compared the lengths incorrectly. To check if a triangle is a right triangle, we should use the Pythagorean theorem. This says that if we square the two shorter sides and add them together, it should equal the square of the longest side.
In this case, the longest side is 13, while the other two sides are 12 and 5. So, we need to check if:
12² + 5² = 13².
Calculating that:
12² is 144,
5² is 25, and
13² is 169.
Now, adding the squares of the two shorter sides together:
144 + 25 = 169.
Since 169 does equal 169, this means that the triangle with sides 5, 12, and 13 is a right triangle. Ariel's conclusion that the triangle does not form a right triangle is incorrect because she mixed up the sides in her calculations.
In this case, the longest side is 13, while the other two sides are 12 and 5. So, we need to check if:
12² + 5² = 13².
Calculating that:
12² is 144,
5² is 25, and
13² is 169.
Now, adding the squares of the two shorter sides together:
144 + 25 = 169.
Since 169 does equal 169, this means that the triangle with sides 5, 12, and 13 is a right triangle. Ariel's conclusion that the triangle does not form a right triangle is incorrect because she mixed up the sides in her calculations.