ABCD is a cyclic quadrilateral - NSC Mathematics - Question 9 - 2016 - Paper 2
Question 9
ABCD is a cyclic quadrilateral. AS is a tangent. CBS is a straight line. AD || SC and AD = BD.
5.1 Name, with reasons, FIVE other angles each equal to x.
5.2 Prove... show full transcript
Worked Solution & Example Answer:ABCD is a cyclic quadrilateral - NSC Mathematics - Question 9 - 2016 - Paper 2
Step 1
Name, with reasons, FIVE other angles each equal to x.
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Answer
Angle ABC = x, since angles subtended by the same arc (AC) are equal.
Angle ADC = x, as angles opposite equal chords AC are equal.
Angle DAB = x, since it's corresponding to angle ABC in cyclic quadrilateral.
Angle DCA = x, because of the alternate segment theorem (tangent AS).
Angle DBS = x, as it also subtends the same arc AB.
Step 2
Prove that ASCD is a parallelogram.
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Answer
To prove ASCD is a parallelogram, we need to show that opposite sides are equal and parallel:
Since AD || SC, angles ACD and DSA are equal (corresponding angles).
Angle DAB = angle ASB (both equal to x).
By the properties of cyclic quadrilaterals, angle ACD + angle DAB = 180° (supplementary).
Therefore, ASCD meets the definition of a parallelogram as opposite sides are parallel and equal.
Step 3
Name a triangle in the figure similar to ΔADB.
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Answer
Triangle BSC is similar to triangle ADB because they share angle ADB and angle ABC equals angle BSC (Angle-Angle similarity).
Step 4
Hence prove that SC.SB = DC^2.
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Answer
Using similar triangles, we can set up the proportion:
