In the diagram, PQRS is a cyclic quadrilateral - NSC Mathematics - Question 10 - 2022 - Paper 2
Question 10
In the diagram, PQRS is a cyclic quadrilateral. KP is a tangent to the circle at P. C and D are points on chords PQ and PS respectively and CD produced meets RS prod... show full transcript
Worked Solution & Example Answer:In the diagram, PQRS is a cyclic quadrilateral - NSC Mathematics - Question 10 - 2022 - Paper 2
Step 1
10.1 $ar{S_1} = ar{T_2}$
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Answer
To prove that ar{S_1} = ar{T_2}, we can invoke 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. Thus, since KP is tangent to the circle at P, we have:
ar{S_1} = ar{Q_2}\ \text{(by the tangent-chord theorem)}
Step 2
10.2 $\frac{AD}{AS} = \frac{AR}{AC}$
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Answer
In triangle ACD, we can establish that:
ar{A} = \bar{A} (common angle),
ar{S_1} = \bar{C_2} (proved above),
ar{T_1} = \bar{C_2} (alternate angles).
Therefore, by the criteria for similar triangles, we find:
ASAD=ACAR(corresponding sides in proportion)
Step 3
10.3 $AC \times SD = AR \times TC$
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Answer
Applying the properties of similar triangles, specifically in triangles ACD and ACR, we relate the sides:
