MathematicsChapter 11: Practical Geometry of Circles
Tangent Constructions (without Centre, Named)
A tangent can be constructed without first marking the centre by using the alternate segment theorem.
Learning objectives
- Explain tangent constructions (without centre, named) using the correct definitions, conditions, and notation.
- Apply perp at contact if radius known, named no-centre methods correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
A tangent can be constructed without first marking the centre by using the alternate segment theorem. At endpoint A of chord AB, copy at A the angle subtended by chord AB at another point C on the arc, but construct it outside the circle. The new line through A is tangent. If the centre is known, the simpler method is to draw the line through A perpendicular to radius OA.
Where / symbols
- TANGENT CONSTRUCTIONS (WITHOUT CENTRE, NAMED)
- A tangent can be constructed without first marking the centre by using the alternate segment theorem. At endpoint A of chord AB, copy at A the angle subtended by chord AB at another point C on the arc, but construct it outside the circle. The new line through A is tangent. If the centre is known, the simpler method is to draw the line through A perpendicular to radius OA.
- CENTRE KNOWN
- When O is known, join O to the point of contact P. Construct a 90 degree line through P to OP. The converse of the radius-tangent theorem proves that this line is tangent.
- CENTRE NOT GIVEN
- When O is not known, choose a chord from the required point and use an angle in the opposite arc. Copy that angle outside the circle at the chord endpoint. By the alternate segment theorem, the copied line is the tangent.
Overview
With centre vs without:
Centre Known
When O is known, join O to the point of contact P. Construct a 90 degree line through P to OP. The converse of the radius-tangent theorem proves that this line is tangent.
Examples OP radius · Perp at P · Unique tangent at P
Centre Not Given
When O is not known, choose a chord from the required point and use an angle in the opposite arc. Copy that angle outside the circle at the chord endpoint. By the alternate segment theorem, the copied line is the tangent.
Examples Midpoint of arc · Endpoint of arc · Not any secant
WHAT — TANGENT CONSTRUCTIONS (WITHOUT CENTRE, NAMED)
A tangent can be constructed without first marking the centre by using the alternate segment theorem. At endpoint A of chord AB, copy at A the angle subtended by chord AB at another point C on the arc, but construct it outside the circle. The new line through A is tangent. If the centre is known, the simpler method is to draw the line through A perpendicular to radius OA. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Tangent Constructions (Without Centre, 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 CENTRE KNOWN, then CENTRE NOT GIVEN. Worked: tangent at P when O is known. draw radius OP construct the line through P perpendicular to OP that line is the tangent at P Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- tangent at P when O is known.
- draw radius OP
- construct the line through P perpendicular to OP
- that line is the tangent at P
In simple words
A line perpendicular to an arbitrary chord through the point is not necessarily a tangent. Use the radius-perpendicular method or the alternate-segment construction.
Key points
- perp at contact if radius known
- named no-centre methods
- perp to radius at P
- named arc methods
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Tangent Constructions (without Centre, Named)
Sticky Notes · Quick Revision
Core Idea
- A tangent can be constructed without first marking the centre by using the alternate segment theorem. At endpoint A of chord AB, copy at A the angle subtended by chord AB at another point C on the arc, but construct it outside the circle. The new line through A is tangent. If the centre is known, the simpler method is to draw the line through A perpendicular to radius OA.
CENTRE KNOWN
- When O is known, join O to the point of contact P. Construct a 90 degree line through P to OP. The converse of the radius-tangent theorem proves that this line is tangent.
CENTRE NOT GIVEN
- When O is not known, choose a chord from the required point and use an angle in the opposite arc. Copy that angle outside the circle at the chord endpoint. By the alternate segment theorem, the copied line is the tangent.
Worked Method
- Worked: tangent at P when O is known.
- draw radius OP
- construct the line through P perpendicular to OP
- that line is the tangent at P
Common Misconception
- A line perpendicular to an arbitrary chord through the point is not necessarily a tangent. Use the radius-perpendicular method or the alternate-segment construction.
FBISE Exam Check
- A line perpendicular to an arbitrary chord through the point is not necessarily a tangent. Use the radius-perpendicular method or the alternate-segment construction.
60-second revision
- A tangent can be constructed without first marking the centre by using the alternate segment theorem.
- CENTRE KNOWN: When O is known, join O to the point of contact P.
- CENTRE NOT GIVEN: When O is not known, choose a chord from the required point and use an angle in the opposite arc.
- Worked-method reminder: that line is the tangent at P
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