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Tangent from an External Point (named)
If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT). Then PT is tangent. Two intersections give the two tangents. This uses Chapter 10, not a new theorem.
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Tangent from an External Point (named)
What is it?
If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT). Then PT is tangent. Two intersections give the two tangents. This uses Chapter 10, not a new theorem.
In simple words
- If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT).
- THE STEPS: Segment OP as diameter of an auxiliary circle.
- WHY 90 DEG: Angle in a semicircle at T is 90, so OT perp PT.
- Worked-method reminder: those are the two tangents from P
Important terms
TANGENT FROM AN EXTERNAL POINT (NAMED): If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT). Then PT is tangent. Two intersections give the two tangents. This uses Chapter 10, not a new theorem.
THE STEPS: Segment OP as diameter of an auxiliary circle. Intersections T, T' with the original circle. Draw PT and PT'. Those are the two tangents.
WHY 90 DEG: Angle in a semicircle at T is 90, so OT perp PT. But OT is a radius, so PT is tangent. Same for T'. Lengths PT=PT' as a check.
What you should learn
- Explain tangent from an external point (named) using the correct definitions, conditions, and notation.
- Apply diameter OP, angle in semicircle 90 correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- Circle on OP
- Meet at T and T'
- PT and PT' tangents
THE STEPS
Segment OP as diameter of an auxiliary circle. Intersections T, T' with the original circle. Draw PT and PT'. Those are the two tangents.
WHY 90 DEG
Angle in a semicircle at T is 90, so OT perp PT. But OT is a radius, so PT is tangent. Same for T'. Lengths PT=PT' as a check.
Recommended for Class 10
Full school-level explanation
If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT). Then PT is tangent. Two intersections give the two tangents. This uses Chapter 10, not a new theorem.
Semicircle trick:
WHAT — TANGENT FROM AN EXTERNAL POINT (NAMED)
If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT). Then PT is tangent. Two intersections give the two tangents. This uses Chapter 10, not a new theorem. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Tangent From An External Point (Named) matters because accurate compass-and-straightedge constructions apply circle theorems in a visible way. Each construction works because a locus or perpendicular condition fixes the required centre, circle, or tangent.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE STEPS, then WHY 90 DEG. Worked: named steps, tangent from P (centre O known). join O to P draw the circle with diameter OP let it meet the given circle at T and T' draw PT and PT' those are the two tangents from P Finish by checking signs, restrictions, units, and whether the answer matches the question.
THE STEPS
Segment OP as diameter of an auxiliary circle. Intersections T, T' with the original circle. Draw PT and PT'. Those are the two tangents.
Remember
- auxiliary diameter OP
Examples
- Circle on OP
- Meet at T and T'
- PT and PT' tangents
WHY 90 DEG
Angle in a semicircle at T is 90, so OT perp PT. But OT is a radius, so PT is tangent. Same for T'. Lengths PT=PT' as a check.
Remember
- OT perp PT
Examples
- Radius perp tangent
- Two tangents equal
- P outside, not on the circle
Remember
- diameter OP
- angle in semicircle 90
- circle on diameter OP
- P must be outside
- auxiliary diameter OP
- OT perp PT
- diameter OP
- angle in semicircle 90
How the ideas connect
Worked: named steps, tangent from P (centre O known). join O to P draw the circle with diameter OP let it meet the given circle at T and T' draw PT and PT' those are the two tangents from P
Deeper understanding
Go beyond memorising Tangent from an External Point (named). Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: named steps, tangent from P (centre O known). join O to P draw the circle with diameter OP let it meet the given circle at T and T' draw PT and PT' those are the two tangents from P
Common mistakes
Exam focus
Core Idea
THE STEPS
WHY 90 DEG
Worked Method
Exam focus
Quick revision
- diameter OP
- angle in semicircle 90
- circle on diameter OP
- P must be outside
- TANGENT FROM AN EXTERNAL POINT (NAMED): If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T
- THE STEPS: Segment OP as diameter of an auxiliary circle. Intersections T, T' with the original circle. Draw PT
If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radius.
For long answers on Tangent from an External Point (named): define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- diameter OP
- angle in semicircle 90
- circle on diameter OP
- P must be outside
Important formulas
- diameter OP
- angle in semicircle 90
- circle on diameter OP
Common exam traps
- If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radius.
- Do not treat THE STEPS and WHY 90 DEG as the same step; each has a different purpose.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radius.
- Mistake
Do not treat THE STEPS and WHY 90 DEG as the same step; each has a different purpose.
- Mistake
Do not jump straight to the answer; omitted signs, brackets, restrictions, or construction reasons can change the result.
- Mistake
If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radius.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
diameter OP
angle in semicircle 90
circle on diameter OP
P must be outside
If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT). Then PT is ta
- Definition
- If the centre is known: join OP, draw the circle on OP as diameter, meet the given circle again at T (angle in a semicircle is 90, so OT perp PT). Then PT is ta
- Example
- Circle on OP
- Critical warning
- If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radi
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Exam Practice
Define
Q1. Define Tangent from an External Point (named).
Explain
Q2. Explain: THE STEPS.
Example
Q3. Give one clear example of Tangent from an External Point (named) and say why it fits.
Mistake
Q4. State one common mistake students make about Tangent from an External Point (named).
Exam note
Q5. Write a short exam note on Tangent from an External Point (named): definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.