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In the diagram, chords KM, MN and KN are drawn in the circle O with centre O - NSC Mathematics - Question 11 - 2017 - Paper 2

Question 11

In the diagram, chords KM, MN and KN are drawn in the circle O with centre O. PNQ is the tangent to the circle at N. Prove the theorem which states that MNQ = K.

Worked Solution & Example Answer:In the diagram, chords KM, MN and KN are drawn in the circle O with centre O - NSC Mathematics - Question 11 - 2017 - Paper 2

Step 1

Prove that angle MNR = 90°

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Answer

Since PNQ is a tangent to the circle at point N, by the property of tangents, we have:

$\angle MNR = 90°$

This is because the tangent line at any point of a circle is perpendicular to the radius drawn to that point.

Step 2

Establish the relationship between angles in a semi-circle

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Answer

Using the property of angles subtended by a chord in a semi-circle:

$\angle MNQ = \angle MKN$

Since both angles subtend the same chord MN, we can conclude that:

$\angle MNQ + \angle MNR = 180°$

Step 3

Final conclusion

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Answer

Now substituting the value of ( \angle MNR ):

$\angle MNQ + 90° = 180°$

Thus,

$\angle MNQ = 90°$

Since MNQ and K are both equivalent angles, we conclude that:

$MNQ = K$

This proves the desired theorem.

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