Question
Ariel completed the work below to show that a triangle with side lengths of 9, 15, and 12 does not form a right triangle.
9 squared + 15 squared = 12 squared. 81 + 225 = 144. 306 not-equals 144.
Is Ariel’s answer correct?
No, Ariel should have added 122 and 152 and compared that to 92.
No, Ariel should have subtracted 122 from 92 and compared that to152.
No, Ariel should have added 92 and 122 and compared that to 152.
Yes, Ariel’s work is correct.
9 squared + 15 squared = 12 squared. 81 + 225 = 144. 306 not-equals 144.
Is Ariel’s answer correct?
No, Ariel should have added 122 and 152 and compared that to 92.
No, Ariel should have subtracted 122 from 92 and compared that to152.
No, Ariel should have added 92 and 122 and compared that to 152.
Yes, Ariel’s work is correct.
Answers
GPT-4o mini
To determine if Ariel's answer is correct, we need to apply the Pythagorean theorem, which states that for a right triangle with sides of lengths \( a \), \( b \), and \( c \) (where \( c \) is the hypotenuse, the longest side), the following equation must hold true:
\[
a^2 + b^2 = c^2
\]
In the case of the triangle with side lengths 9, 15, and 12, we first identify the largest side, which is 15. Therefore, we will assume 15 is the hypotenuse (\( c \)), and \( a \) and \( b \) will be 9 and 12.
Now we apply the Pythagorean theorem:
\[
9^2 + 12^2 = 15^2
\]
Calculating each term:
\[
9^2 = 81
\]
\[
12^2 = 144
\]
\[
15^2 = 225
\]
Now, we add the squares of the two shorter sides:
\[
81 + 144 = 225
\]
Now we check if this equals the square of the longest side:
\[
225 = 225
\]
Since both sides of the equation are equal, this confirms that the triangle with sides 9, 12, and 15 is indeed a right triangle.
Returning to Ariel's work, we can see that her comparison of \( 9^2 + 15^2 = 12^2 \) is incorrect both in the choice of sides and in concluding that the triangle is not a right triangle.
Therefore, the correct response regarding Ariel's conclusion is:
**No, Ariel should have added 9^2 and 12^2 and compared that to 15^2.**
\[
a^2 + b^2 = c^2
\]
In the case of the triangle with side lengths 9, 15, and 12, we first identify the largest side, which is 15. Therefore, we will assume 15 is the hypotenuse (\( c \)), and \( a \) and \( b \) will be 9 and 12.
Now we apply the Pythagorean theorem:
\[
9^2 + 12^2 = 15^2
\]
Calculating each term:
\[
9^2 = 81
\]
\[
12^2 = 144
\]
\[
15^2 = 225
\]
Now, we add the squares of the two shorter sides:
\[
81 + 144 = 225
\]
Now we check if this equals the square of the longest side:
\[
225 = 225
\]
Since both sides of the equation are equal, this confirms that the triangle with sides 9, 12, and 15 is indeed a right triangle.
Returning to Ariel's work, we can see that her comparison of \( 9^2 + 15^2 = 12^2 \) is incorrect both in the choice of sides and in concluding that the triangle is not a right triangle.
Therefore, the correct response regarding Ariel's conclusion is:
**No, Ariel should have added 9^2 and 12^2 and compared that to 15^2.**