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 o... 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 B∩A
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Answer
To find angle B∩A, we can use the tangent-chord theorem which states that the angle formed between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. Thus, we have:
B∩A=46°
(reference: angle A)
Step 2
8.1.2 ∠A3
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Answer
Angle ∠A3 is inscribed in the circle and subtended by the same chord CD. Since CD is parallel to TN, we can state that:
∠A3=∠C1=20°
Step 3
8.1.3 ∠A1
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Answer
Angle ∠A1 is an alternate angle to angle ∠A3, as CD is parallel to TN and is intercepted by the transversal OA. Hence:
∠A1=66°
Step 4
8.1.4 ∠O1
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Answer
Angle ∠O1 is at the center and is double the inscribed angle subtended by the same arc, which is angle ∠B∩A. Therefore:
∠O1=2imes∠B∩A=2imes46°=132°
Step 5
8.2.1 ∠E
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Answer
Angle ∠E is equal to angle ∠D1, being in the same segment as D, thus:
∠E=48°
Step 6
8.2.2 ∠D2
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Answer
Angle ∠D2 is formed by two equal chords HG and FG, thus it is equal to:
∠D2=48°
Step 7
8.2.3 ∠G4
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Answer
Angle ∠G4 is inscribed in a cyclic quadrilateral formed by points E, F, G, and K, and can be calculated using:
