Question

find the exact length x of the diagonal of the rectangle. x = the value of x is between which two whole numbers? and

Explanation:

Step1: Apply Pythagorean theorem

In a rectangle, the diagonal $x$ and the two - sides of the rectangle form a right - triangle. According to the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 8$, $b = 4$, and $c=x$. So $x^{2}=8^{2}+4^{2}$.
$x^{2}=64 + 16$

Step2: Calculate $x^{2}$ and $x$

$x^{2}=80$. Then $x=\sqrt{80}$. Simplifying $\sqrt{80}=\sqrt{16\times5}=4\sqrt{5}$.

Step3: Estimate the value of $x$

We know that $\sqrt{64}=8$ and $\sqrt{81}=9$. Since $64<80<81$, then $\sqrt{64}<\sqrt{80}<\sqrt{81}$, so $8 < \sqrt{80}<9$.

Answer:

$x = 4\sqrt{5}$
The value of $x$ is between $8$ and $9$