Tangent Constructions (without Centre, Named)

Tangent Constructions (without Centre, Named)

What is it?

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.

In simple words

• 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

Important terms

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.

What you should learn

• 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.

Examples

• OP radius
• Perp at P
• Unique tangent at P

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.

Full school-level explanation

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.

With centre vs without:

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.

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.

Remember

• perp to radius at P

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.

Remember

• named arc methods

Examples

• Midpoint of arc
• Endpoint of arc
• Not any secant

Remember

• perp at contact if radius known
• named no-centre methods
• perp to OP at P
• one meeting point
• perp to radius at P
• named arc methods
• perp at contact if radius known
• named no-centre methods

How the ideas connect

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

Deeper understanding

Go beyond memorising Tangent Constructions (without Centre, Named). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

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 mistakes

Exam focus

Core Idea

CENTRE KNOWN

CENTRE NOT GIVEN

Worked Method

Exam focus

Quick revision

• perp at contact if radius known
• named no-centre methods
• perp to OP at P
• one meeting point
• TANGENT CONSTRUCTIONS (WITHOUT CENTRE, NAMED): 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. Construct a 90 degree line through P to OP. The c

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.

For long answers on Tangent Constructions (without Centre, Named): define clearly → explain the main parts → give one example → state one exam caution.