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 her work and answer correct?
144+169=25
313= 25 is her work and answer correct?
Answers
GPT-4o mini
Ariel's conclusion is incorrect. To determine if a triangle with side lengths 13, 12, and 5 is a right triangle, we should use the Pythagorean theorem, which states that for a right triangle with legs of lengths \(a\) and \(b\) and hypotenuse \(c\), the relationship is:
\[
a^2 + b^2 = c^2
\]
First, we need to identify which side is the longest since it would be the hypotenuse in the case of a right triangle. Here, the side lengths are 13, 12, and 5. The longest side is 13.
We should then check if \(12^2 + 5^2 = 13^2\):
Calculating each side:
- \(12^2 = 144\)
- \(5^2 = 25\)
- \(13^2 = 169\)
Now, substituting back into the Pythagorean theorem:
\[
12^2 + 5^2 = 144 + 25 = 169
\]
\[
13^2 = 169
\]
Since \(144 + 25 = 169\) is true, this shows that the triangle with sides 13, 12, and 5 does indeed form a right triangle. Therefore, Ariel's work and conclusion are incorrect. The correct conclusion is that the triangle with side lengths 13, 12, and 5 does form a right triangle.
\[
a^2 + b^2 = c^2
\]
First, we need to identify which side is the longest since it would be the hypotenuse in the case of a right triangle. Here, the side lengths are 13, 12, and 5. The longest side is 13.
We should then check if \(12^2 + 5^2 = 13^2\):
Calculating each side:
- \(12^2 = 144\)
- \(5^2 = 25\)
- \(13^2 = 169\)
Now, substituting back into the Pythagorean theorem:
\[
12^2 + 5^2 = 144 + 25 = 169
\]
\[
13^2 = 169
\]
Since \(144 + 25 = 169\) is true, this shows that the triangle with sides 13, 12, and 5 does indeed form a right triangle. Therefore, Ariel's work and conclusion are incorrect. The correct conclusion is that the triangle with side lengths 13, 12, and 5 does form a right triangle.