Compassvale Sec Circle Properties Revision

1 COMPASSVALE SECONDARY SCHOOL MATHEMATICS DEPARTMENT 2022 SECONDARY 4E5N Practice Circles – Singapore Schools Exam Papers Name: _______________________________( ) Class: __________ Marks: Date: _____________ Question 1 The perpendicular bisector of a chord, RS, cuts it at T and the circumference of the circle at P. If RS = 20 cm and TP = 8 cm, find the radius of the circle. Question 2 In the diagram below, A, C, D and F lie on a circle with centre O. Line OB is perpendicular to chord AC and E is the mid-point of chord DF. Given that AC = 6 cm, DF = 10 cm, OE = 5 cm and OB = y cm, find the value of y correct to 3 significant figures. A C D E F O B 6 cm 10 cm 5 cm y cm 50 20 cm R S O T P 8 cm

2 Question 3 In the diagram, A, B, C and D lie on the circumference of a circle. The straight lines AC and BD intersect at X. AB = 4 cm, BX = 2 cm and CD = 9 cm. (a) Prove that triangles ABX and CDX are similar. (b) Calculate the length of CX. (c) Calculate the ratio of area of triangle ABX : area of triangle DCX. Question 4 The diagram shows a circle with centre O. AC is a tangent to the circle. AOF, FDB and ODC are straight lines. Given that  46 and 24 HAFACO . (a) Write down why  90FAB . A B C D X 2 4 9 A B C D E F G H O 46 24

3 (b) Calculate (i) HEF . (ii) AFB . (iii) HGF . (iv) FED . Question 5 In the diagram, AC is a diameter of the circle with center O. The chords AE and BC meet at D when produced. Given that  22CDE and  39BAC . Find (a) ACB , (b) AEB , (c) CBE . Question 6 The points A, B, C, D and E lie on a circle, centre O. AD is a diameter of the circle. BQ is a tangent to the circle,  24AEB and QAD is a straight line. Calculate, stating your reasons clearly, (a) (i) BAD . (ii) BED . (iii) BQA . (iv) reflex BOD . (v) ABQ . (b) If  xDAE , show that  )114( xQBE . Q B C D E A O 24 D A B O E C 22 39

4 (c) Given that  43ABE , determine whether the lines BO and AE are parallel. Give a reason for your answer. Question 7 In the diagram, AC is a diameter of the circle ABCDE. AC and ED are extended to meet at F. Given that CE = CF,  xAFD and  yBED . (a) Explain with reasons why  xDBC . (b) Find, in terms of x and/or y, (i) ACD . (ii) ACE . (iii) BCD . Question 8 In the figure below, PQRST is a circle with O as the centre. PX and RX are tangents to the circle at points P and R respectively. Given that  40 and 78 SPQPXR , calculate (a) PRX . (b) POR . (c) OST . (d) QRX . T P O S R Q X 40 78

5 Question 9 A, B, C, D and E lie on a circle. AC is a diameter of the circle and AE is parallel to BD. F is the point of intersection of AC and BD.  58ABD . Find, stating reasons, (a) ABC . (b) DBC . (c) CAD . (d) AED . Question 10 In the diagram below, the line AB meets the diameter CD at E. A, B, C and D lie on a circle and . Find (a) , (b) , (c) given that O is the centre of the circle. Give your answer in radians. (d) Hence, calculate the area of the shaded segment shown, given that the radius of the circle is 7 cm.  60ABD CBD CDA ,DOA 60 B C E A D 58 A B C D E F

6 Question 11 In the diagram, A, B, C and D lie on a circle, centre O. The tangents at A and B meet at E and  76AOB . Find, stating your reasons clearly, (a) OBA . (b) ACB . (c) BCD . (d) AEB . Question 12 In the diagram, AC is a diameter of the circle ABCD. The straight lines EBA and ECD cut the circle at A, B, C and D. AC and BD cut at F.  33 and 40 DACCDB . E A B C F D 40 33 E A B A C D O 76

7 Giving your reasons in your working steps, find (a) DCA . (b) BCD . (c) AFB . (d) AED . Question 13 In the diagram, O is the centre of circle ABCDE where DE = DC. Reflex angle  240EOC and angle  100EAB . (You must not assume CT is a tangent to the circle at C.) (a) Find, giving reasons for each answer, (i) angle DCE, (ii) angle CBE, (iii) angle CEB. (b) Given that  20CTE and EBT is a straight line, show that CT is a tangent to the circle at C. Question 14 ABCDEF is a hexagon inscribed in a circle. E A F D C B 116 148

8 Given that  148AFE ,  116CDE and FA = FE, find (a) ACF , (b) ABC . Question 15 In the diagram, O is the centre of circle ABPC.  70MCNBAC and  62POC . The larger circle BMNC passes through O. ACN, BPN and CPM are straight lines. Give reasons and find the value of (a) reflex BOC . (b) ABP . (c) MBN . (d) BMP . (e) ACO . Question 16 A, B, C, D are four points on the circle with centre O. AT is a tangent at A. Form two simultaneous equations in x and y and hence find the value of x and y. O N A C M B P

9 Question 17 O is the centre of the circle through P, Q and S. PR and QR are tangents to the circle. (a) Find POQ . (b) Find PSQ . (c) Give a reason for your answer. C y8 x5 y5 x2 D A T B O

10 Question 18 The diagram shows a circle ABCDE, centre O. CT and ET are tangents to the circle. BE is a diameter and angle BOC = 58 . (a) Explain why angle EAB is a right angle. (b) State two other angles which are right angle. (c) Find, giving reasons for each answer, (i) angle EBC, (ii) angle BEC, (iii) angle EDC, (iv) angle CET, (d) Given that OE is 3 cm, find the area of quadrilateral OETC.