Why Tangents Matter in Everyday Life
Ever watched a bike wheel roll and noticed the point where the tire just kisses the ground? That tiny kiss is a perfect example of a tangent – a line that touches a circle at exactly one point.
💡 In Simple Words: A tangent is a straight line that meets a circle at just one spot, and at that spot it stands straight up like a fence against the circle’s radius.
First Tangent Theorem – Perpendicular at the Point of Contact
What it says: The radius (the line from the centre of the circle to any point on it) meets the tangent at a right angle (90°) where they touch.
Why it’s true: Imagine the circle as a round pond and the radius as a rope tied to the centre. If you pull the rope tight and let it touch a straight board (the tangent), the rope will push against the board straight on – that’s a 90° push.
Proof (step‑by‑step)
- Let O be the centre, P the point of contact, and PT the tangent line.
- Assume another line through P makes an angle less than 90° with OP.
- That line would cut the circle at another point, contradicting the definition of a tangent.
- Hence the only possible angle is exactly 90°.
Second Tangent Theorem – Equal Lengths from an External Point
When you stand outside a circular fence and stretch two ropes to touch the fence at different spots, both ropes end up the same length. That’s the theorem.
Statement: From a point outside the circle, the two tangents drawn to the circle are equal in length.
Proof Sketch
- Take external point A, points of contact B and C.
- Join A to the centre O, and draw radii OB and OC.
- Triangles △AOB and △AOC share side AO and have OB = OC (radii) and ∠OBA = ∠OCA = 90° (first theorem).
- By the Hypotenuse‑Leg (HL) congruence rule, the triangles are identical, so AB = AC.
Third Tangent Theorem – Angle Between Tangent and Chord
If you draw a chord (a line joining two points on the circle) and a tangent at one end of the chord, the angle you get equals the angle made by the chord with the opposite part of the circle.
In plain words: Angle between a tangent and a chord = angle in the alternate segment (the part of the circle opposite the chord).
Proof Idea
- Let the chord be AB and the tangent at A be AT.
- Draw the diameter through A and call its other end D.
- Because AD is a straight line, ∠BAT + ∠BAD = 180°.
- ∠BAD is an inscribed angle that subtends the same arc as ∠BCD (the angle in the opposite segment).
- Rearranging gives ∠BAT = ∠BCD.
Worked Example
Find the length of the tangent from point P(8, 0) to the circle x² + y² = 25.
- Centre O is (0, 0), radius r = 5.
- Distance OP = √(8² + 0²) = 8.
- Using the right‑triangle formed by OP, the radius, and the tangent, we have PT² = OP² – r² = 8² – 5² = 64 – 25 = 39.
- Thus PT = √39 ≈ 6.24 units.
Quick Reference Table
| Theorem | Statement | Key Idea in Proof |
|---|---|---|
| Perpendicular Radius | Radius meets tangent at 90° | Any other angle would intersect the circle again. |
| Equal Tangents | Two tangents from an external point are equal | Congruent right triangles using HL. |
| Angle Between Tangent & Chord | ∠(tangent, chord) = angle in opposite segment | Use inscribed angles and a diameter. |
📝 Likely Exam Questions
- State and prove the theorem that a radius is perpendicular to a tangent at the point of contact.
Answer: The radius drawn to the point of contact makes a right angle with the tangent. Proof follows by assuming a smaller angle would intersect the circle again, which contradicts the definition of a tangent. - From a point outside a circle, two tangents are drawn. Prove that the tangents are equal in length.
Answer: Connect the external point to the centre and draw radii to the points of contact. The two resulting right triangles share the hypotenuse and have equal legs, so by HL they are congruent, giving equal tangent lengths. - Find the length of the tangent from (13, 0) to the circle x² + y² = 144.
Answer: OP = 13, radius r = 12, so PT = √(13²‑12²)=√25=5 units. - Explain why the angle between a tangent and a chord equals the angle in the alternate segment.
Answer: By drawing a diameter through the point of contact and using the property of inscribed angles, the exterior angle formed equals the interior angle subtended by the same arc. - Given a circle with centre O(0,0) and radius 7, a tangent touches the circle at T(7,0). Find the equation of the tangent.
Answer: The tangent at (7,0) is vertical, so its equation is x = 7.