In the diagram below, O is the centre of the circle. OM = 3 cm and bisects AB. A, B and C are points on the circle such that AB = 10 cm. MN || BC. M is the midpoint of AB. (a) Write down, with a reason, the size of \(M_i\). (b) Determine the length of the radius of the circle. If \(MN = 5,12\text{ cm}\), write down, with a reason, the length of \(BC\).

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In the diagram below, O is the centre of the circle - NSC Technical Mathematics - Question 9 - 2024 - Paper 2

Question 9

In the diagram below, O is the centre of the circle. OM = 3 cm and bisects AB. A, B and C are points on the circle such that AB = 10 cm. MN || BC. M is the midpoint ... show full transcript

Worked Solution & Example Answer:In the diagram below, O is the centre of the circle - NSC Technical Mathematics - Question 9 - 2024 - Paper 2

Step 1

Write down, with a reason, the size of \(M_i\).

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Answer

Since OM is a radius and bisects the chord AB at its midpoint M, by the property of circles, the angle (M_i) is equal to (90^\circ). Therefore, (M_i = 90^\circ\ \text{(Angle from the center to the midpoint of the chord)}).

Step 2

Determine the length of the radius of the circle.

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Answer

Using the Pythagorean theorem in triangle OMB, where OM is the perpendicular bisector, we have:

[ OB^2 = OM^2 + MB^2 ] [ OB^2 = 3^2 + 5^2 ] [ OB^2 = 9 + 25 = 34 ] [ OB = \sqrt{34} \approx 5.83\text{ cm} ]

Thus, the radius of the circle is approximately 5.83 cm.

Step 3

If \(MN = 5,12\text{ cm}\), write down, with a reason, the length of \(BC\).

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Answer

Since MN is parallel to BC and M is the midpoint of AB, by the Midpoint Theorem:

[ BC = 2 \cdot MN = 2 \cdot 5.12 \text{ cm} = 10.24 \text{ cm} ]

Therefore, the length of (BC) is 10.24 cm.

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