The figure below is a square. Find the length of side x in simplest radical form with a rational denominator. 12
Power Operation
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Exponents
The exponent or power or index of a variable/number is the number of times that variable/number is multiplied by itself.
The figure below is a square. Find the length of side xx in simplest radical form with a rational denominator.
![**Problem Statement:**
The figure below is a square. Find the length of side \( x \) in simplest radical form with a rational denominator.
[Image Description]:
The image shows a square with one of its diagonal lines present. The square is tilted slightly to the left. One diagonal is marked with a label "12", indicating its length. One segment of a side of the square within the diagonal is labeled with an "x".
**Solution:**
To find the side length \( x \) of the square, which is the side of the square, we'll apply the properties of squares and the Pythagorean theorem.
Since the given shape is a square, all its sides are of equal length. The diagonals of a square are also equal and they intersect at right angles, forming two 45-45-90 right triangles.
The properties of a 45-45-90 triangle dictate that if the legs have length \( x \), the hypotenuse will be \( x\sqrt{2} \). Here, the hypotenuse is given as 12.
Given the the properties of 45-45-90 triangles:
\[ x\sqrt{2} = 12 \]
Solve for \( x \):
\[ x = \frac{12}{\sqrt{2}} \]
Rationalize the denominator:
\[ x = \frac{12}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{12\sqrt{2}}{2} = 6\sqrt{2} \]
Therefore, the length of side \( x \) in simplest radical form with a rational denominator is \( 6\sqrt{2} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c50f00e-8711-4f2c-bf3e-a4378eae7e9d%2F801f4746-77d1-4084-8f4b-1d30fc4ede26%2Flbq1947_processed.png&w=3840&q=75)
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