MathematicsChapter 11: Practical Geometry of Circles
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 per
Learning objectives
- 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.
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 tangent. Two intersections give the two tangents. This uses Chapter 10, not a new theorem.
Where / symbols
- 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.
Overview
Semicircle trick:
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.
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.
Examples Radius perp tangent · Two tangents equal · P outside, not on the circle
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.
Worked example
- 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
In simple words
If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radius.
Key points
- diameter OP
- angle in semicircle 90
- auxiliary diameter OP
- OT perp PT
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Tangent From An External Point (named)
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- 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 Misconception
- If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radius.
FBISE Exam Check
- If P is on the circle, this auxiliary-diameter method is the wrong picture: you want the single tangent perp to the radius.
60-second revision
- 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
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.