Let's analyze Ariel's work and determine whether the triangle with side lengths 13, 12, and 5 forms a right triangle.
In a right triangle, the Pythagorean theorem states that if \(c\) is the length of the longest side (the hypotenuse), and \(a\) and \(b\) are the lengths of the other two sides, then the relationship must hold:
\[ a^2 + b^2 = c^2 \]
For the triangle with side lengths 13, 12, and 5:
- First, identify the longest side, which is 13 (hypotenuse, \(c\)).
- The other two sides are 12 and 5 (\(a\) and \(b\)).
Now we calculate \(a^2 + b^2\) and compare it with \(c^2\):
-
Calculate \(a^2 + b^2\): \[ 12^2 + 5^2 = 144 + 25 = 169 \]
-
Calculate \(c^2\): \[ 13^2 = 169 \]
Now we see if \(a^2 + b^2 = c^2\): \[ 169 = 169 \]
Since both sides are equal, this indicates that the triangle with sides 13, 12, and 5 does form a right triangle.
Ariel's work contains several errors:
- Ariel incorrectly stated \(122 + 132 = 52\) instead of \(12^2 + 5^2 = 144 + 25\).
- Ariel mistakenly wrote \(144 + 169 = 25\) instead of calculating \(13^2\).
- Lastly, his equality \(313 ≠ 25\) does not correspond to the calculations or comparisons that need to be made.
Therefore, Ariel's conclusion is incorrect because, based on the Pythagorean theorem, the triangle with side lengths 13, 12, and 5 does form a right triangle.
1 answer