In the diagram, PQRS is a cyclic quadrilateral - NSC Mathematics - Question 8 - 2019 - Paper 2
Question 8
In the diagram, PQRS is a cyclic quadrilateral. Chord RS is produced to T. K is a point on RS and W is a point on the circle such that QRKW is a parallelogram.
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Worked Solution & Example Answer:In the diagram, PQRS is a cyclic quadrilateral - NSC Mathematics - Question 8 - 2019 - Paper 2
Step 1
8.1.1 Ř
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Answer
We know that ∠R is the co-interior angle to ∠QW || RK.
Thus, we have:
ext∠R=180°−∠QWext∠R=180°−100°=80°
Therefore, Ř = 80°.
Step 2
8.1.2 ŷ
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Answer
The angle ŷ is the opposite angle to ∠R in the cyclic quadrilateral.
By the properties of cyclic quadrilaterals:
ext∠y^=180°−∠Rext∠y^=180°−80°=100°
Thus, ŷ = 100°.
Step 3
8.1.3 PQW
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Answer
To find ∠PQW, we can use the properties of angles in a cyclic quadrilateral:
Since ∠PQR = 136° and ∠QRW = ∠PQS,
We have:
ext∠PQW=∠PQR−∠PQS
Using the external angle property:
ext∠PQW=∠PQR+y^ext∠PQW=136°+100°=236°
And since angles in a triangle sum to 180°:
ext∠PQW=180°−(∠S+angle∠PQS)
Thus, PQW = 36°.
Step 4
8.1.4 Ũ₂
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Answer
To determine Ũ₂, we refer to the angles around point U:
Using alternate angles, we have:
ext∠Uˉ2=∠PUˉW
Since ∠PŪW = ∠QW = 136°,
Thus, Ũ₂ = 136°.
