To use the Pythagorean Theorem, we first need to determine if the triangle described is a right triangle. The Pythagorean Theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), the relationship is given by:
\[ c^2 = a^2 + b^2 \]
In your case, you provide three sides: 39 cm, 80 cm, and \(x\). We need to check if \(x\) can be the hypotenuse or one of the legs of a right triangle.
Let's first calculate \(x\) assuming \(x\) is the hypotenuse. If \(x\) is the hypotenuse, then:
\[ x^2 = 39^2 + 80^2 \]
Calculating the squares:
\[ 39^2 = 1521 \] \[ 80^2 = 6400 \]
Now substituting these values back into the equation:
\[ x^2 = 1521 + 6400 \] \[ x^2 = 7921 \]
Taking the square root of both sides gives:
\[ x = \sqrt{7921} \] \[ x = 89 \]
Now, let's check if \(x\) could be one of the legs instead. We can assume 80 cm is the hypotenuse:
\[ 80^2 = 39^2 + x^2 \]
Calculating the squares again:
\[ 6400 = 1521 + x^2 \]
Now, rearranging the equation:
\[ x^2 = 6400 - 1521 \] \[ x^2 = 4879 \]
Taking the square root of both sides gives:
\[ x = \sqrt{4879} \approx 69.9 \]
So, we have two possible scenarios for \(x\):
- If \(x\) is the hypotenuse: \(x = 89\) cm
- If \(x\) is one of the legs: \(x \approx 69.9\) cm
In conclusion, \(x\) can be either \(89\) cm (if it is the hypotenuse) or approximately \(69.9\) cm (if it is one of the legs).
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