In the diagram, PQRS is a cyclic quadrilateral such that PQ = PR - NSC Mathematics - Question 10 - 2023 - Paper 2
Question 10
In the diagram, PQRS is a cyclic quadrilateral such that PQ = PR. The tangents to the circle through P and R meet QS produced at A. RS is produced to meet tangent AP... show full transcript
Worked Solution & Example Answer:In the diagram, PQRS is a cyclic quadrilateral such that PQ = PR - NSC Mathematics - Question 10 - 2023 - Paper 2
Step 1
10.1 $ar{S}_1 = ar{S}_4$
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Answer
Since PQRS is a cyclic quadrilateral, we have:
Sˉ1=Sˉ4
This follows from the opposite angles of a cyclic quadrilateral being supplementary.
Step 2
10.2 SMRC is a cyclic quadrilateral
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Answer
To prove SMRC is a cyclic quadrilateral, we use the Tangent-Chord Theorem:
Rˉ1+Rˉ2=PQRˉ
This establishes that the angles at points S and A are supplementary to the angle at R. Thus, SMRC indeed is a cyclic quadrilateral.
Step 3
10.3 RP is a tangent to the circle passing through P, S and A at P
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Answer
According to the properties of tangents and chords, we have:
The angle ar{S} = \bar{R}_2 + \bar{P}_2 (using the angle at intersection).
By the Tangent-Chord Theorem, we know that the tangent at point P (RP) is perpendicular to the radius at that point.
Therefore, RP is a tangent to the circle passing through P, S, and A at P.
