Complete the following theorem statement:
Angles subtended by a chord of the circle, on the same side of the chord .. - NSC Technical Mathematics - Question 7 - 2021 - Paper 2
Question 7
Complete the following theorem statement:
Angles subtended by a chord of the circle, on the same side of the chord ...
In the diagram below, circle PTRS, with cent... show full transcript
Worked Solution & Example Answer:Complete the following theorem statement:
Angles subtended by a chord of the circle, on the same side of the chord .. - NSC Technical Mathematics - Question 7 - 2021 - Paper 2
Step 1
Determine, stating reasons: (a) Three other angles each equal to $56°$
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Answer
From the circle, we know that angles subtended by the same chord in the same segment are equal. Therefore, in triangle PTRS:
∠PTS=∠R1=56°
This means:
rianglePTS=∠R1=56° (angles in the same segment)
riangleOSR=56° (opposite sides of the same segment)
rianglePTS=∠PTS=56° (angles subtended by the same chord)
Step 2
Determine, stating reasons: (b) The size of $ riangle P_1$
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Answer
Since PSR is in a semicircle, we know:
∠PSR=90°
Using the angles in a triangle sum:
∠P1+∠PSR+∠PTS=180°
Substituting the known angle:
∠P1+90°+56°=180°
This gives:
∠P1=180°−90°−56°=34°
Step 3
Determine, stating reasons: (c) The size of $ riangle S_3$
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Answer
Using the angles in a triangle sum:
∠S1+∠S2+∠S3=90°
From the semicircle property:
∠S1+∠S2=90°
So,
∠S3=180°−90°−34°=56°
Step 4
Prove, stating reasons, that OT is NOT parallel to SR.
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Answer
To prove that OT is not parallel to SR, we can compare the angles:
Given that:
∠O1+∠O=112°/2=56°
Then:
∠SO+∠R1=44°+68°−56°=180°
Thus, since the angles do not equal 180° in corresponding angles, therefore:
OT is not parallel to SR (as the alternate angles are not equal)
