We investigate and prove the key circle theorems: relationships between angles, chords, and tangents. We use these theorems to solve riders (proof-based problems).
Week 5
8.1 Circle Theorems
Prove and apply: angle at centre = twice angle at circumference
Prove and apply: angles in the same segment are equal
Prove and apply: angle in semicircle = 90°
Prove and apply: tangent-radius theorem and tangent-chord angle
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Real-World Connection
The circle theorems were used by ancient astronomers to predict planetary positions. Today, engineers designing circular gears and satellite dishes use exactly these relationships to calculate forces and alignments with millimetre precision.
Property / Rule
Theorem 1: Angle at Centre
The angle subtended by an arc at the centre of a circle is TWICE the angle subtended by the same arc at the circumference.
∠AOB=2∠ACB(same arc AB)
Property / Rule
Theorem 2: Angles in Same Segment
Angles subtended by the same chord (arc) at the circumference, on the same side, are equal.
∠ACB=∠ADB(same arc AB)
Property / Rule
Theorem 3: Angle in Semicircle
An angle inscribed in a semicircle (subtended by a diameter) is always 90°.
If AB is diameter, then ∠ACB=90°
Property / Rule
Theorem 4: Tangent-Radius
A tangent to a circle is perpendicular to the radius at the point of tangency.
OT⊥tangent at T
Property / Rule
Theorem 5: Tan-Chord Angle
The angle between a tangent and a chord equals the inscribed angle on the opposite side (alternate segment theorem).
∠(tan-chord)=∠(inscribed angle in alt. segment)
Worked Examples
Worked Example
Apply circle theorems
Problem
O is the centre of a circle. ∠BOC=100°. Find ∠BAC (where A, B, C are on the circle).
Activity — 8 Questions
CAPS Cognitive Level Distribution
L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
In a circle, the angle at the centre subtended by arc PQ is 140°. Find the angle at the circumference subtending the same arc.
2
L1 · Knowledge1 mark
State the angle in a semicircle theorem.
3
L2 · Routine Procedures2 marks
Two inscribed angles ∠ADB=35° and ∠ACB subtend the same chord AB from the same side. Find ∠ACB.
4
L2 · Routine Procedures1 mark
A tangent at T meets radius OT with OT=8. The tangent-radius angle is ___°.
5
L3 · Complex Procedures4 marks
In a circle with centre O: A, B, C on circle; ∠BAC=42°; find ∠BOC and the reflex ∠BOC.
6
L3 · Complex Procedures3 marks
PT is a tangent from external point P to circle with centre O. OT=5, OP=13. Find the tangent length PT.
7
L4 · Problem Solving5 marks
Prove that opposite angles of a cyclic quadrilateral are supplementary.
8
L4 · Problem Solving5 marks
In circle with centre O, chord AB and tangent at B. Show that the angle between tangent and chord AB equals 21 arc AB.