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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.
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Two Tangents from an External Point
What is it?
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.
In simple words
- 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
Important terms
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.
What you should learn
- 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.
Examples
- 9 cm and 9 cm
- Not half unless a different length is given
- Two from outside, one from on the circle
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.
Recommended for Class 10
Full school-level explanation
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.
Congruent right triangles:
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.
EQUAL SEGMENTS
Tangents from P touch at B and C: PB=PC. That is the usual 'find the other tangent' one-liner.
Remember
- PB = PC
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.
Remember
- 90+90+angle+angle=360
Examples
- Angle BOC = 140 if BPC=40
- If BOC=120 then BPC=60
- OT still perp
Remember
- two tangents, equal lengths
- 140 deg
- they meet at P
- PB = PC
- 90+90+angle+angle=360
- two tangents, equal lengths
- two tangents, equal lengths
- mathematical reasoning
How the ideas connect
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
Deeper understanding
Go beyond memorising Two Tangents from an External Point. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
EQUAL SEGMENTS
ANGLES AT THE CENTRE
Worked Method
Exam focus
Quick revision
- two tangents, equal lengths
- 140 deg
- they meet at P
- PB = PC
- TWO TANGENTS FROM AN EXTERNAL POINT: From a point outside, exactly two tangents. They are equal in length. The line from the centre to th
- EQUAL SEGMENTS: Tangents from P touch at B and C: PB=PC. That is the usual 'find the other tangent' one-liner.
Equal tangents does not mean the tangents are parallel. They meet at P.
For long answers on Two Tangents from an External Point: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- two tangents, equal lengths
- 140 deg
- they meet at P
- PB = PC
Important formulas
- two tangents, equal lengths
- 140 deg
- they meet at P
Common exam traps
- Equal tangents does not mean the tangents are parallel. They meet at P.
- Not half unless a different length is given
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Equal tangents does not mean the tangents are parallel. They meet at P.
- Mistake
Not half unless a different length is given
- Mistake
Do not treat EQUAL SEGMENTS and ANGLES AT THE CENTRE as the same step; each has a different purpose.
- Mistake
Equal tangents does not mean the tangents are parallel. They meet at P.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
two tangents, equal lengths
140 deg
they meet at P
PB = PC
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 (con
PB = PC
- 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 (con
- Formula
- PB = PC
- Example
- 9 cm and 9 cm
- Critical warning
- Equal tangents does not mean the tangents are parallel. They meet at P.
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Exam Practice
Define
Q1. Define Two Tangents from an External Point.
Explain
Q2. Explain: EQUAL SEGMENTS.
Example
Q3. Give one clear example of Two Tangents from an External Point and say why it fits.
Mistake
Q4. State one common mistake students make about Two Tangents from an External Point.
Exam note
Q5. Write a short exam note on Two Tangents from an External Point: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.