Why Tangents Matter in Circle Problems
Ever noticed how a straight line just kisses a circle at one point? That line is called a tangent, and it shows up a lot in CBSE exams.
💡 In Simple Words: A tangent touches a circle at exactly one point. From any outside point, you can draw two such lines, and they’re always the same length.
What Is a Tangent? (First Time Definition)
A tangent is a line that meets a circle at one and only one point, called the point of contact. Think of it like a pencil just barely touching a round ball without pressing through.
Key Theorems You Must Remember
Here are the three classic theorems that appear in most Class 10 questions.
| Theorem | What It Says | Why It Helps |
|---|---|---|
| Radius‑Tangent Perpendicular Theorem | The radius drawn to the point of contact is perpendicular (forms a 90° angle) to the tangent. | Lets you find right‑angle triangles quickly. |
| Equal Tangents from an External Point | Two tangents drawn from the same outside point have equal lengths. | Useful for calculating unknown sides. |
| Angle Between Tangent and Chord | The angle formed by a tangent and a chord through the point of contact equals the angle in the alternate segment (the part of the circle opposite the chord). | Turns a tricky angle problem into a simple arc‑angle one. |
Proof Sketches (No Heavy Algebra)
Let’s walk through why each theorem is true. No need to memorize long algebra—just understand the picture.
1. Radius‑Tangent Perpendicular Proof
Draw a circle with centre O and a tangent line touching at point T. Suppose the line isn’t perpendicular, so we can drop a perpendicular from O to the line, meeting it at P. Then OP is shorter than OT (the radius), but both OP and OT reach the same line. That contradicts the definition of a radius being the shortest distance from the centre to the circle. Hence OT must be at a right angle to the tangent.
2. Equal Tangents from an External Point Proof
Take an external point P and draw two tangents PT and PS touching the circle at T and S. Connect O to T, O to S, and O to P. You now have two right‑angled triangles ΔOTP and ΔOSP (right angles at T and S because of the first theorem). They share the side OP, and OT = OS (both are radii). By the hypotenuse‑leg (HL) congruence rule, the triangles are identical, so PT = PS.
3. Angle Between Tangent and Chord Proof
Consider chord AB and tangent at A. Draw the radius OA. Extend the chord to meet the circle again at C, forming triangle ABC. The angle between the tangent at A and chord AB equals the angle in the alternate segment, which is ∠ACB. This follows from the fact that the central angle ∠AOB subtends the same arc as ∠ACB, and the exterior angle at A (tangent‑chord) equals half the central angle (by the inscribed angle theorem). So both angles end up equal.
Worked Example
Problem: A circle has centre O and radius 5 cm. From an external point P, two tangents PT and PS are drawn, each touching the circle at T and S. If OP = 13 cm, find the length of PT.
Solution:
- Draw OT and OS (both 5 cm). Connect O, P, T, S forming two right‑angled triangles ΔOTP and ΔOSP.
- Use the Pythagorean theorem in ΔOTP: OP² = OT² + PT².
- Plug in: 13² = 5² + PT² → 169 = 25 + PT² → PT² = 144.
- Take square root: PT = 12 cm.
Notice we used the “equal tangents” theorem to know both PT and PS are the same, so solving one gives the other.
Quick Revision Checklist
- Identify the point of contact.
- Remember radius to that point is ⟂ (perpendicular) to the tangent.
- If you have two tangents from the same outside point, write them as equal.
- When a tangent meets a chord, match the angle with the opposite arc’s angle.
Common Mistakes to Avoid
• Assuming a tangent can intersect the circle at two points – that’s a secant, not a tangent.
• Forgetting to draw the radius to the point of contact before applying the perpendicular theorem.
• Mixing up the “alternate segment” angle with the angle inside the triangle formed by the chord and radius.
📝 Likely Exam Questions
- Question: Prove that the tangents drawn from an external point to a circle are equal.
Answer: Join the centre O to the external point P and to the points of contact T and S. Use the radius‑tangent perpendicular theorem to get two right triangles ΔOTP and ΔOSP. Since OT = OS (radii) and OP is common, the triangles are congruent (HL), giving PT = PS. - Question: In a circle of radius 7 cm, a tangent at point A meets the extension of chord AB at C. If ∠ACB = 50°, find ∠TAB (tangent‑chord angle).
Answer: By the tangent‑chord theorem, ∠TAB = ∠ACB = 50°. - Question: From point P, two tangents PT and PS touch a circle of radius 4 cm. If PT = 9 cm, find OP.
Answer: In right triangle ΔOTP, OP² = OT² + PT² = 4² + 9² = 16 + 81 = 97 → OP = √97 cm. - Question: Show that the angle between a tangent and a chord is half the intercepted arc.
Answer: Let the tangent touch at A and chord AB cut the circle at B. Draw radii OA and OB. Central angle ∠AOB subtends the same arc AB. The inscribed angle theorem says ∠ACB = ½∠AOB. The tangent‑chord angle ∠TAB equals ∠ACB (tangent‑chord theorem), so ∠TAB = ½ (arc AB). - Question: A circle with centre O has a tangent at T. If OT = 6 cm and the tangent meets a line through O at point Q, find the distance OQ.
Answer: Since OT ⟂ tangent, OQ is the hypotenuse of right triangle ΔOTQ with OT = 6 cm and OQ = ?. Without extra data, we use Pythagoras if another side is given; typical exam provides it. (This tests recognition that OT ⟂ tangent.)