Chapter 10 · Question 2
State and prove Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Q2
State and prove Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Answer Revealed
Direct Answer:
Statement: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Proof: Let be the centre, be the tangent at point , and be the radius. Take any point on other than . Join . Since lies outside the circle, (radius). As this holds for every point on except , is the shortest distance from to the line . The shortest distance from a point to a line is the perpendicular distance. Hence, .
Simple Explanation
The radius drawn to the point where a tangent touches the circle is always at a right angle () to the tangent line. This is because the radius is the shortest possible line from the centre to any point on the tangent.
Exam-Ready Structure
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Given: A circle with centre , a tangent touching the circle at point . To Prove: . Construction: Take a point on other than . Join . Proof: Since is a tangent, the point must lie outside the circle (if were inside the circle, would intersect the circle at two points, making it a secant, not a tangent). Therefore, is longer than the radius : . This is true for every point on line except the point . Thus, is the shortest distance from centre to the line . It is known that the shortest distance from a point to a line is the perpendicular distance. Hence, is perpendicular to , i.e., . Remark: The line containing the radius through the point of contact is called the 'normal' to the circle at that point.
Key Points
- Statement: Radius through point of contact is perpendicular to the tangent
- Take any point Q on tangent other than P
- Q lies outside the circle, so OQ > OP (radius)
- OP is the shortest distance from centre to the tangent line
- Shortest distance = perpendicular distance, so OP ⟂ XY