Right Angle Quarter Circle

Formula for arc length: $\displaystyle s = r\theta$ where $\theta$ is the central angle in radians, $r$ is the radius of the sector and $s$ is the arc length.

We can form the equation $\displaystyle r \cdot \underbrace{\frac{\pi}{2}}_{\theta} = \underbrace{14}_{\large s}$ which can be solved by multiplying by $\displaystyle \frac{2}{\pi}$ :

$\green{\boxed{r = \frac{28}{\pi} = 8.9126768 \dots \approx 9 \text{ (nearest whole number)}}}$

Imagine 3 other figures connected by the straght sides all togeter to form a circle (my apologies if the quarter-circle isn't perfect: I drew it myself freehanded). When you imagine this, you will see that the arc connecting the two striaght lines (obviously not where the to lines meet to form the right angle) is $\frac{1}{4}$ of the total circumference of the circle.

$C = 14 * 4 = 56 cm$

But this isn't what the question is asking. We need the diameter of the circle, and then divide it by 2 to find the value of $s$ ( $s$ = radius of the circle). We can find the diameter from a circumference by rearranging the formula C = πd:

$C = πd$

When we divide each side by π, we get:

$\frac{C}{π}$ $= d$

Now lets plug in the numbers:

$\frac{56}{3.1415}$ $= d$

Now divide by 2:

$\frac{17.82587936}{2}$ $= 8.912939678$

Finally, round to the nearest integer:

$8.912939678 ≈ 9$