In the diagram below, O is the centre of the circle - NSC Technical Mathematics - Question 8 - 2024 - Paper 2
Question 8
In the diagram below, O is the centre of the circle.
TN is a tangent to the circle at A.
∠A = 46°
CD || TN and ∠C1 = 20°
Determine, with reasons, the size of each ... show full transcript
Worked Solution & Example Answer:In the diagram below, O is the centre of the circle - NSC Technical Mathematics - Question 8 - 2024 - Paper 2
Step 1
8.1.1 BˆCÅ
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Answer
The angle BˆCÅ is equal to 46°. This follows from the tangent-chord theorem which states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
Step 2
8.1.2 Aˆ3
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Answer
The angle Aˆ3 is equal to 44°, as it is vertically opposite to angle A (∠A = 46°) and angles opposite in equal intersecting lines are equal.
Step 3
8.1.3 Aˆ1
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The angle Aˆ1 is equal to 66°. This is derived from the property that alternate angles created by parallel lines (CD || TN) are equal, where ∠C1 = 20° and ∠CˆD = 2 ∙ 20°.
Step 4
8.1.4 Oˆ1
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Answer
The angle Oˆ1 is equal to 132°, calculated as 180° - ∠A = 180° - 48°. Given that angles at center are twice the angles at circumference, we have ∠O1 = 2 ∙ 66°.
Step 5
8.2.1 Eˆ
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The angle Eˆ is equal to 48°, because angles in the same segment are equal.
Step 6
8.2.2 Dˆ2
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The angle Dˆ2 is equal to 48°, since Dˆ2 and Dˆ1 are equal as they subtend equal arcs (equal chords).
Step 7
8.2.3 Gˆ4
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The angle Gˆ4 is equal to 96°, derived from the cyclic quadrilateral property that states opposite angles sum to 180°. Therefore, Gˆ4 = 180° - 48°.
