MathematicsChapter 10: Tangents and Angles of a Circle
Two Tangents From An External Point
From a point outside, exactly two tangents.
Learning objectives
- Explain two tangents from an external point using the correct definitions, conditions, and notation.
- Apply two tangents, equal lengths correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
From a point outside, exactly two tangents. They are equal in length. The line from the centre to the external point bisects the angle between the tangents (congruent right triangles). If PB=9 cm then PC=9 cm.
Mathematical form
PB=9 cm then PC
BAC = 40 deg
BOC = 360 90 + 90 + 40 + angle BOC
BOC = 360 angle BOC
Where / symbols
- TWO TANGENTS FROM AN EXTERNAL POINT
- From a point outside, exactly two tangents. They are equal in length. The line from the centre to the external point bisects the angle between the tangents (congruent right triangles). If PB=9 cm then PC=9 cm.
- EQUAL SEGMENTS
- Tangents from P touch at B and C: PB=PC. That is the usual 'find the other tangent' one-liner.
- ANGLES AT THE CENTRE
- Quadrilateral OBPC: two right angles at B and C. Angle BOC + angle BPC = 180 deg in that kite/quad story: 40 deg between tangents -> 140 deg at the centre.
Overview
Congruent right triangles:
Equal Segments
Tangents from P touch at B and C: PB=PC. That is the usual 'find the other tangent' one-liner.
Examples 9 cm and 9 cm · Not half unless a different length is given · Two from outside, one from on the circle
Angles At the Centre
Quadrilateral OBPC: two right angles at B and C. Angle BOC + angle BPC = 180 deg in that kite/quad story: 40 deg between tangents -> 140 deg at the centre.
Examples Angle BOC = 140 if BPC=40 · If BOC=120 then BPC=60 · OT still perp
WHAT — TWO TANGENTS FROM AN EXTERNAL POINT
From a point outside, exactly two tangents. They are equal in length. The line from the centre to the external point bisects the angle between the tangents (congruent right triangles). If PB=9 cm then PC=9 cm. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Two Tangents From An External Point matters because tangent and circle-angle theorems explain the geometry at a point of contact. They support proofs, numerical angle questions, and practical circle constructions.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through EQUAL SEGMENTS, then ANGLES AT THE CENTRE. Worked: tangents from A, angle BAC = 40 deg, O centre. Find angle BOC. angles at B and C in OB and OC to tangents are 90 deg angle OBA + angle OCA + angle BAC + angle BOC = 360 90 + 90 + 40 + angle BOC = 360 220 + angle BOC = 360 angle BOC = 140 deg Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- tangents from A, angle BAC = 40 deg, O centre. Find angle BOC.
- angles at B and C in OB and OC to tangents are 90 deg
- angle OBA + angle OCA + angle BAC + angle BOC = 360
- 90 + 90 + 40 + angle BOC = 360
- 220 + angle BOC = 360
- angle BOC = 140 deg
In simple words
Equal tangents does not mean the tangents are parallel. They meet at P.
Key points
- two tangents, equal lengths
- PB = PC
- 90+90+angle+angle=360
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Two Tangents From An External Point
Sticky Notes · Quick Revision
Core Idea
- From a point outside, exactly two tangents. They are equal in length. The line from the centre to the external point bisects the angle between the tangents (congruent right triangles). If PB=9 cm then PC=9 cm.
EQUAL SEGMENTS
- Tangents from P touch at B and C: PB=PC. That is the usual 'find the other tangent' one-liner.
ANGLES AT THE CENTRE
- Quadrilateral OBPC: two right angles at B and C. Angle BOC + angle BPC = 180 deg in that kite/quad story: 40 deg between tangents -> 140 deg at the centre.
Worked Method
- Worked: tangents from A, angle BAC = 40 deg, O centre. Find angle BOC.
- angles at B and C in OB and OC to tangents are 90 deg
- angle OBA + angle OCA + angle BAC + angle BOC = 360
- 90 + 90 + 40 + angle BOC = 360
- 220 + angle BOC = 360
- angle BOC = 140 deg
Common Misconception
- Equal tangents does not mean the tangents are parallel. They meet at P.
FBISE Exam Check
- Equal tangents does not mean the tangents are parallel. They meet at P.
60-second revision
- From a point outside, exactly two tangents.
- EQUAL SEGMENTS: Tangents from P touch at B and C: PB=PC.
- ANGLES AT THE CENTRE: Quadrilateral OBPC: two right angles at B and C.
- Worked-method reminder: angle BOC = 140 deg
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