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The answer is NO. The reason is pretty clear and simple as written by Eugen Covaci and Benjamin in the comments section.
A point $3 cm$ away from the centre of a circle of radius $5 cm$ will lie inside the area of a circle itself. And by the definition of a tangent, the reason why the answer is NO becomes even clearer-
A tangent is a line which touches the circle at exactly one point.
So, even if you try to construct a tangent from a point inside the circle, you will not be able to. However, you will be able to construct a chord or a secant, but not a tangent. Hence, you cannot construct a tangent from a point inside the circle. The minimum distance that the point from where the tangent has to be constructed to a circle of a given radius is the radius itself. This is because the tangent is perpendicular to the radius at the point of contact. So, by drawing any radius, you will reach a point on the circle. From that point, you can construct a line perpendicular to the radius you have already drawn. That line perpendicular to the radius will be your tangent.
Another important point -
Always trust yourself if you know you're right. Be confident and give the right arguments in favour of your opinion.
For any point $P$ in the interior of your circle, every line through $P$ meets the circle in two points, hence cannot be a tangent to the circle.
As an example, suppose that the circle is given by $x^2 + y^2 = 25$, and the point is $(3, 0)$. Then any line through the point must be $$ A(x-3) = By $$ for some $A$ and $B$, not both zero.
Substituting to find intersections, we get $$ x^2 + y^2 = 25 \\ x^2 + \left( \frac{A(x-3)}{B} \right)^2 = 25\\ x^2 + \left( \frac{A}{B}\right)^2 (x-3)^2 - 25 = 0. $$ That's a quadratic in $x$ with positive discriminant (you have to do some algebra to show this), hence has two solutions.