8.1 In the diagram below, tangent KT to the circle at K is parallel to the chord NM - NSC Mathematics - Question 8 - 2016 - Paper 2
Question 8
8.1 In the diagram below, tangent KT to the circle at K is parallel to the chord NM. NT cuts the circle at L. \( \triangle KML \) is drawn. \( M_2 = 40^\circ \) an... show full transcript
Worked Solution & Example Answer:8.1 In the diagram below, tangent KT to the circle at K is parallel to the chord NM - NSC Mathematics - Question 8 - 2016 - Paper 2
Step 1
8.1.1 \( \hat{K}_2 \)
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Answer
( \hat{K}_2 = M_2 = 40^\circ )
Reason: Tan chord theorem states that the angle between the tangent and the chord through the point of contact is equal to the angle in the alternate segment.
Step 2
8.1.2 \( \hat{N}_i \)
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Answer
( \hat{N}_i = \hat{K}_1 )
To find ( \hat{N}_i ), using the property that angles in the same segment are equal, we have:
( \hat{K}_1 = 84^\circ - 40^\circ = 44^\circ )
Step 3
8.1.3 \( \hat{T} \)
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Answer
( \hat{T} = \hat{N}_i = 44^\circ )
Reason: Alternate segment theorem applies, which states that ( \hat{T} ) is equal to the angle subtended by the same chord at point ( N ).
Step 4
8.1.4 \( \hat{L}_2 \)
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Answer
( \hat{L}_2 = \hat{K}_2 + \hat{T} = 40^\circ + 44^\circ = 84^\circ )
Reason: Using the exterior angle theorem, which states that the exterior angle is equal to the sum of the opposite interior angles.
Step 5
8.1.5 \( \hat{L}_i \)
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Answer
( \hat{L}_i = 180^\circ - (44^\circ + 40^\circ + \hat{L}_2) = 180^\circ - 128^\circ = 12^\circ )
Reason: This is done by applying the sum of angles in a triangle.
Step 6
8.2 Calculate, giving reasons, the value of x.
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Answer
( \hat{C} = 108^\circ )
From parallelogram properties, we know that:
[ 2x + 40^\circ + 108^\circ = 180^\circ ]
[ 2x + 148^\circ = 180^\circ ]
[ 2x = 32^\circ ]
[ x = 16 ] \circ
Therefore, the value of x is 16.
