Question

find the value of x. leave your answer in simplest radical form.

Explanation:

Step1: Apply Pythagorean theorem in rectangle

In the rectangle, if the length is $x$ and the diagonal is 6. Let the width be $w$. For the right - angled triangle formed by the length, width and diagonal of the rectangle, by the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 6$ (diagonal), $a=x$ (length) and $b = w$. First, consider the right - angled triangle at the top. The two legs of the right - angled triangle at the top are 2 and 2. The width of the rectangle is equal to the hypotenuse of the right - angled triangle at the top.

Step2: Calculate the width of the rectangle

Using the Pythagorean theorem for the right - angled triangle at the top with legs $a = 2$ and $b = 2$. The hypotenuse $w=\sqrt{2^{2}+2^{2}}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}$.

Step3: Calculate the value of $x$

Now, use the Pythagorean theorem for the rectangle's right - angled triangle with diagonal $c = 6$ and width $b=2\sqrt{2}$. We have $x^{2}+(2\sqrt{2})^{2}=6^{2}$. Expand the equation: $x^{2}+8 = 36$. Then, isolate $x^{2}$: $x^{2}=36 - 8=28$. Take the square root of both sides: $x=\sqrt{28}=\sqrt{4\times7}=2\sqrt{7}$.

Answer:

$2\sqrt{7}$