Grade 11 Tangents: Short Notes

11 ශ්‍රේණිය · 3 වාරය · 22 පරිච්ඡේදය · ස්පර්ශක

38%

of Grade 11 term tests include it (23 of 61)

312

marks in past papers (36 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

Use: the tangent is perpendicular to the radius at the point of contact, and its converse
Use: two tangents from an external point are equal and subtend equal angles at the centre
Use the alternate segment theorem
Combine tangent facts with Pythagoras and other circle theorems to find angles and lengths

The rope at the well

At a village well, the rope runs over a pulley wheel and drops straight down to the bucket. It leaves the wheel touching it at just one point, and there it meets the spoke at a perfect right angle. That straight line is a tangent, and tangents follow a few neat rules.

What is a tangent?

A tangent is a straight line that meets a circle at exactly one point. That point is the point of contact.A line that cuts the circle at two points is not a tangent; the part inside the circle is a chord.From a point outside a circle you can draw exactly two tangents to it.

Tangent ⟂ radius

Theorem: the tangent at A is perpendicular to the radius OA, so ∠OAT = 90°.Converse: a line drawn through the end A of a radius, perpendicular to OA, is a tangent.

✏️ Spot the right angle

A circle has centre O and radius 5 cm. P is 13 cm from O, and PA touches the circle at A. Find the length of the tangent PA.

1OA is a radius and PA a tangent, so ∠OAP = 90°.
2Pythagoras in △OAP: PA² = OP² − OA².
3PA² = 169 − 25 = 144, so PA = 12 cm.

✅ PA = 12 cm

Two tangents from one point

PA and PB are tangents from P. Triangles OAP and OBP are congruent (RHS): right angles at A and B, OA = OB (radii), OP common.

Three facts from that

For two tangents PA and PB from an external point P:• PA = PB: the tangents are equal.• ∠APO = ∠BPO: OP bisects the angle between the tangents.• ∠AOP = ∠BOP: the tangents subtend equal angles at the centre.Also, OAPB has two right angles, so ∠AOB + ∠APB = 180°.

✏️ Angles around two tangents

PA and PB are tangents from P to a circle with centre O, and ∠APB = 50°. Find (i) ∠AOB (ii) ∠OPA (iii) ∠OAB.

1(i) ∠OAP = ∠OBP = 90°, so ∠AOB = 360 − 90 − 90 − 50 = 130°.
2(ii) OP bisects ∠APB, so ∠OPA = 25°.
3(iii) OA = OB, so △OAB is isosceles: ∠OAB = (180 − 130) ÷ 2 = 25°.

✅ (i) 130° (ii) 25° (iii) 25°

⚠️ Forgetting equal tangents

O/L examiners note that candidates miss the right angle between radius and tangent, and forget that tangents from one point are equal. Whenever you see a tangent, mark 90° at the point of contact; whenever two tangents come from one point, mark them equal. Most of the answer follows from those marks.

The alternate segment theorem

ST is the tangent at A and AB is a chord. The angle between them, ∠BAT, equals the angle in the alternate segment, ∠ACB. C can be anywhere on the arc on the other side of AB.

Which segment is 'alternate'?

The chord AB splits the circle into two segments. ∠BAT opens towards one of them. The alternate segment is the other one, on the far side of the chord.On the other side of A, ∠BAS matches angles in the smaller segment. Since ∠BAT + ∠BAS = 180°, angles in the two segments add to 180°, as in a cyclic quadrilateral.

⚠️ Choosing the wrong angle

A frequent error is matching the tangent-chord angle with an angle in the same segment. Trace it: the tangent-chord angle sits on one side of the chord, so look across the chord. Examiners also note candidates forget other circle facts: the angle in a semicircle is 90° and opposite angles of a cyclic quadrilateral add to 180°.

📌 Remember

• Tangent ⟂ radius at the point of contact (and conversely).• Two tangents from one point are equal, and OP bisects the angle between them; ∠AOB + ∠APB = 180°.• Tangent-chord angle = angle in the alternate segment.• Mark 90° at every point of contact and equal tangents first; then use triangle, semicircle and cyclic-quadrilateral facts.

🧾 Quick revision sheet

Tangent	A line that meets a circle at exactly one point (the point of contact).
Angle between a tangent and the radius at the point of contact	90°
Two tangents PA, PB from P: lengths?	PA = PB
What does OP do to ∠APB?	Bisects it: ∠APO = ∠BPO.
∠AOB + ∠APB = ?	180°
Alternate segment theorem	Angle between tangent and chord = angle in the alternate segment.
Angle in a semicircle	90°