Here is the proof by contradiction to prove tangent to the circle and radius are Perpendicular at the point of Contact. Let me know whether the proof is Complete. Consider the circle with center $O$ and tangent drawn to it at the point $H$. By the definition of a tangent, if we join the center $O$ and any other point apart from $H$ on the tangent line say $T$, then it is evident that $$OT>OH=R---(1)$$.
No let us assume that $OH$ is not perpendicular to the tangent. So $\exists$ $OT$ $\perp$ to tangent with $T$ on the tangent line. Now $\Delta OHT$ is a Right angled triangle with the hypotenuse $OH$, so $OH>OT$ contradicting $(1)$. Hence $OH \perp$ to the tangent.