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In the diagram, AB is a diameter of the circle, with centre F - NSC Mathematics - Question 9 - 2024 - Paper 2
Question 9
In the diagram, AB is a diameter of the circle, with centre F. AB and CD intersect at G. FD and FC are drawn. BA bisects CAD and \( \hat{D_1} = 37^\circ \). 9.1 Det... show full transcript
Worked Solution & Example Answer:In the diagram, AB is a diameter of the circle, with centre F - NSC Mathematics - Question 9 - 2024 - Paper 2
Step 1
9.1 Determine, giving reasons, any three other angles equal to \( \hat{D_1} \).
Answer
Since AB is the diameter of the circle, by the Inscribed Angle Theorem, an angle subtended by a diameter at the circumference is a right angle. Therefore:
- ( \hat{D} = \hat{BAG} = 90^\circ ) (Angle in a semicircle)
- Since ( BA ) bisects ( CAD ), it follows that ( \hat{D} = \hat{CAD} )
- Therefore, ( \hat{D_1} = \hat{D_2} = \hat{D_3} = 37^\circ ) (angles subtended by the same arc are equal).
Step 2
9.2 Show that \( DG = GC \).
Answer
In triangle ( ADG ), we have:
- ( \hat{A} = 37^\circ ) (as established earlier)
- The angle at the center is twice the angle at the circumference: ( \hat{F_2} = 2 \cdot \hat{D_1} = 74^\circ ).
- Using the triangle angle sum property: ( \hat{D} + \hat{A} + \hat{G} = 180^\circ ) gives ( \hat{G} = 180 - 74 - 90 = 16^\circ ).
- By the properties of isosceles triangles, since ( \hat{D} = \hat{G} ), then ( DG = GC ).
Step 3
9.3 If it is further given that the radius of the circle is 20 units, calculate the length of BG.
Answer
Using the sine rule in triangle ( FBG ):
-
( FG = 20 ) units (radius)
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To find ( BG ), we can use trigonometric ratios. Since ( \hat{F_2} = 74^\circ ), we can find:
[ BG = FG \cdot \sin(37^\circ) = 20 \cdot \sin(37^\circ) ]
Using calculator, ( BG \approx 14.49 ) units. -
Hence, ( BG \approx 14.49 ) units.