Question

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Hint:

Remember: Perpendicularity is equivalent to the shortest distance property. This theorem is the foundation for solving most circle-tangent problems.

Answer 1

Step 1: Understanding the Concept:
We need to prove that if $XY$ is a tangent to a circle with center $O$ at point $P$, then $OP \perp XY$.
Step 2: Key Formula or Approach:
We use the principle that the shortest distance from a point to a line is the perpendicular distance.
Step 3: Detailed Explanation:
1. Let $XY$ be a tangent to a circle with center $O$ at point $P$.
2. Take any point $Q$ on $XY$ other than $P$ and join $OQ$.
3. The point $Q$ must lie outside the circle. (If $Q$ lies inside the circle, $XY$ would be a secant and not a tangent).
4. Since $Q$ lies outside the circle, $OQ$ must be longer than the radius $OP$.
5. Thus, $OQ>OP$.
6. Since this is true for every point on the line $XY$ except point $P$, $OP$ is the shortest distance from the center $O$ to the line $XY$.
7. Therefore, $OP \perp XY$.
Step 4: Final Answer:
The tangent is perpendicular to the radius at the point of contact.