Circles — Chord, Tangent & Angle Properties explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
Step 1
Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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Circle theorems describe special relationships in circles.
They help solve angle, chord, and tangent problems without measuring.
These properties are important in geometry proofs and construction.
A chord is a line segment joining two points on a circle.
A tangent touches a circle at exactly one point.
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Equal chords are the same distance from the centre.
The angle in a semicircle is a right angle.
Meaning: A chord joins two points on a circle.
Definition: A chord is a line segment with both endpoints on the circle.
A diameter is the longest chord.
Meaning: A tangent touches a circle at one point.
Definition: A tangent is a line that meets a circle at exactly one point.
The radius drawn to the point of contact is perpendicular to the tangent.
Meaning: Two tangents from the same outside point are equal.
Definition: Tangents drawn from an external point to a circle have equal lengths.
If PA and PB are tangents from P, then PB.
Meaning: An angle standing on a diameter is 90 degrees.
Definition: The angle in a semicircle is a right angle.
If AB is a diameter and C is on the circle, angle AC degrees.
Easy
Question: A radius meets a tangent at P. What is the angle between them?
Medium
Question: Tangents PA and PB are drawn from P. If P cm, find PB.
Exam-level
Question: AB is a diameter and C lies on the circle. Find angle ACB.
Calling every line through a circle a tangent
Why it happens: Touching and cutting are confused.
Correct approach: A tangent touches at one point only.
Using angle in semicircle without a diameter
Why it happens: The theorem is applied by memory.
Correct approach: Check that the subtended line is a diameter.
Forgetting geometry reasons
Why it happens: Only the calculation is written.
Correct approach: State the theorem used in proof questions.
Is a diameter a chord?
Yes. It is the longest chord.
Can a tangent cross the circle?
No. A tangent touches the circle at exactly one point.
Why is the tangent perpendicular to the radius?
At the point of contact, the radius meets the tangent at a right angle by theorem.
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