Write the given wave in standard form

For a plane electromagnetic wave, $E = E_0 \cos(\omega t - kx)$

Comparing with the given expression $E = 20\cos(2 \times 10^{10} t - 200x)\ \text{V/m}$

we get: $\omega = 2 \times 10^{10}\ \text{rad/s}, \qquad k = 200\ \text{rad/m}$

Find the wave speed in the medium

The phase velocity is $v = \frac{\omega}{k}$

So, $v = \frac{2 \times 10^{10}}{200} = 10^8\ \text{m/s}$

Use the relation between speed and dielectric constant

In a medium, $v = \frac{1}{\sqrt{\mu \varepsilon}}$

Since the medium is non-magnetic, $\mu_r = 1$

Also, $v = \frac{c}{\sqrt{\mu_r \varepsilon_r}} = \frac{c}{\sqrt{\varepsilon_r}}$

Therefore, $\sqrt{\varepsilon_r} = \frac{c}{v}$

Taking $c = 3 \times 10^8\ \text{m/s}$

we get $\sqrt{\varepsilon_r} = \frac{3 \times 10^8}{10^8} = 3$

Hence, $\varepsilon_r = 9$

Match with the options

The dielectric constant of the medium is: $\boxed{9}$

So the correct option is A.