Ever wondered why a line just kisses a circle at one spot? That single touch point holds a lot of math magic.
Think of a circle as a round pond. A tangent is like a straight board that just barely touches the water without sinking. The board meets the pond at exactly one point, and the pond’s radius at that point stands straight up like a pole.
What is a Tangent to a Circle?
A tangent (noun) is a straight line that meets a circle at exactly one point. That point is called the point of contact. If you draw the radius (the line from the centre of the circle to any point on the edge) to the point of contact, it will be at a right angle (90°) to the tangent. This perpendicular relationship is the heart of most tangent theorems.
Key Theorems About Tangents
Theorem 1: Perpendicular Radius
When a tangent touches a circle, the radius drawn to the point of contact is perpendicular to the tangent.
Why it works: Imagine the circle as a flexible hoop. If you push a straight stick against it, the hoop pushes back exactly opposite to the stick. That push direction is the radius, and it meets the stick at a right angle.
Proof (short version): Let O be the centre, T the point of contact, and L the tangent line. Suppose a line through O meets L at P (different from T). Triangle OTP would have two sides OT (radius) and OP (any line). If angle OTP were not 90°, you could draw a shorter line from O to L than OT, contradicting the definition of a radius as the shortest distance from the centre to the line. Hence angle OTP = 90°.
Theorem 2: Two Tangents from an External Point are Equal
If you stand outside a circle and draw two tangents to it, those two tangent segments have the same length.
Everyday picture: Think of holding a rope tied to a pole and pulling it tight in two directions. Both rope pieces will feel the same tension because they’re equally stretched.
Proof: Let P be the external point, and let the tangents touch the circle at A and B. Join OA and OB (radii). Triangles OAP and OBP share side OP and have OA = OB (both radii). Also, angles OAP and OBP are right angles (by Theorem 1). By the RHS (right‑angle‑hypotenuse‑side) congruence rule, the two triangles are identical, so PA = PB.
Theorem 3: Angle Between Tangent and Chord
The angle formed between a tangent and a chord through the point of contact equals the angle in the alternate segment of the circle.
In plain words, if you draw a chord (a line connecting two points on the circle) that starts at the touch point, the angle between that chord and the tangent equals the angle subtended by the chord at any point on the opposite side of the circle.
Proof sketch: Let the tangent touch at T, chord TC, and let a point X be on the opposite arc. Draw radii OT and OC. Triangle OTC is isosceles (OT = OC). The angle between OT and the tangent is 90° (Theorem 1). Using the exterior angle theorem and the fact that angles in the same segment are equal, you arrive at the stated relationship.
Worked Example: Using Theorem 2
Problem: From point P(8, 0) draw tangents to the circle (x‑2)² + (y‑3)² = 25. Find the length of each tangent.
Solution:
- Centre O is (2, 3); radius r = 5.
- Distance OP = √[(8‑2)² + (0‑3)²] = √[36 + 9] = √45 = 3√5.
- Using the right‑triangle formed by OP, the radius, and the tangent, we have PT² = OP² – r² = (3√5)² – 5² = 45 – 25 = 20.
- Thus PT = √20 = 2√5. Both tangents have the same length 2√5 units.
Quick Summary Table
| Theorem | Statement | Key Idea |
|---|---|---|
| Perpendicular Radius | Radius to point of contact ⟂ tangent | Shortest distance from centre to line is a radius |
| Equal Tangents | Two tangents from an external point are equal | Congruent right triangles (RHS) |
| Angle Between Tangent & Chord | ∠(tangent, chord) = angle in alternate segment | Angles subtended by same chord are equal |
Common Mistakes to Avoid
- Assuming a tangent can intersect the circle at two points – it can’t.
- Mixing up the angle between a tangent and a chord with the angle between two chords.
- Forgetting that the equal‑tangent theorem only works when both lines truly touch the circle.
📝 Likely Exam Questions
- State and prove the theorem that the radius drawn to the point of contact of a tangent is perpendicular to the tangent.
Answer: (Provide statement and short proof as in Theorem 1.) - From a point P outside a circle, two tangents touch the circle at A and B. Show that PA = PB.
Answer: Use RHS congruence of triangles OAP and OBP. - In a circle, a tangent at point T meets chord TC. Prove that ∠(tangent, chord) equals the angle in the alternate segment.
Answer: Refer to Theorem 3 proof sketch. - Find the length of the tangent from (12, 0) to the circle x² + y² = 49.
Answer: Centre O(0,0), r=7, OP=12, PT=√(12²‑7²)=√(144‑49)=√95. - Explain why two different tangents drawn from the same external point cannot have different lengths.
Answer: They form congruent right triangles sharing the same hypotenuse (the line from the external point to the centre).