Photo AI
In the diagram, points A, B, D and C lie on a circle - NSC Mathematics - Question 8 - 2017 - Paper 2
Question 8
In the diagram, points A, B, D and C lie on a circle. CE || AB with E on AD produced. Chords CB and AD intersect at F. \( D_1 = 50° \) and \( C_1 = 15° \). 8.1 Calc... show full transcript
Worked Solution & Example Answer:In the diagram, points A, B, D and C lie on a circle - NSC Mathematics - Question 8 - 2017 - Paper 2
Step 1
8.1.1 Calculate, with reasons, the size of: \( \hat{A} \)
Answer
Given ( E = 50° - 15° = 35° ) (by the exterior angle of triangle AFB) and ( \hat{A} + \hat{E} = 180° ) since they are co-interior angles with ( CE || AB ).
Thus, ( \hat{A} = 180° - (130° + 15°) = 35° ) (by alternate angles, given ( CE || AB )). Therefore, ( \hat{A} = 35° ).
Step 2
Step 3
8.2 Prove, with a reason, that CF is a tangent to the circle passing through points C, D and E.
Answer
To prove that CF is a tangent to the circle at point C, we observe that the angle between CF and the chord CD is equal to angle C1 (the angle in the alternate segment). Hence, since ( C_1 = 15° ) and ( C_2 = 35° ) are equal (angles in the same segment), we conclude that CF is tangent to the circle at point C (by the converse of the tangent-chord theorem).