In the diagram, O is the centre of the circle - NSC Mathematics - Question 9 - 2019 - Paper 2
Question 9
In the diagram, O is the centre of the circle. ST is a tangent to the circle at T. M and P are points on the circle such that TM = MP, OT, OP and TP are drawn. Let O... show full transcript
Worked Solution & Example Answer:In the diagram, O is the centre of the circle - NSC Mathematics - Question 9 - 2019 - Paper 2
Step 1
\( O_1 = 360^\circ - x \)
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Answer
The angle at point O1 is given as the exterior angle of the triangle formed by the points S, T, and M. Therefore, we have:
∠SOT=360∘−∠O1
Step 2
\( M = 180^\circ - 2 \angle T \text{ at circumference} \)
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Answer
The angle M at point M can be expressed as:
M=180∘−2⋅2∠T
Where ( \angle T ) is the angle subtended by chord TP at the circumference.
Step 3
\( T_2 + P_1 = 180 - M \)
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Answer
Using the properties of angles in a triangle:
T2+P1=180∘−M
Step 4
\( T_2 = P_1 = \frac{180 - M}{2} \)
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Answer
Since ( T_2 ) and ( P_1 ) are opposite angles in an isosceles triangle, they are equal:
T2=P1=2180∘−M
Step 5
\( STM = \frac{1}{4} x \)
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Answer
Using the property of opposite equal sides in triangle STM and summing the angles, we conclude:
STM=41(360∘−O1)
By substituting the earlier derived relationships, we can show that ( STM = \frac{1}{4} x ).
