Unit Test: Circles

Time: 1 hour
M.M. 30 
Attempt all questions. 
Question numbers 1 to 5 carry 1 mark each. 
Question numbers 6 to 8 carry 2 marks each. 
Question numbers 9 to 11 carry 3 marks each. 
Question numbers 12 & 13 carry 5 marks each.

Q1. How many circles pass through three non-collinear points? (1 Mark)
(a) one
(b) two
(c) three
(d) four

Q2. What is the region enclosed between a chord and its corresponding arc called? (1 Mark)
(a) Radius
(b) Diameter
(c) Sector
(d) Segment

Q3. What is the sum of opposite angles in a cyclic quadrilateral? (1 Mark)
(a) 90°
(b) 180°
(c) 360°
(d) 120°

​Q4. If a line segment subtends equal angles at two points on the same side of the line, what can be concluded? (1 Mark)
(a) The points are collinear
(b) The points lie on a circle
(c) The points form a square
(d) The points are equidistant

Q5. What is the angle in a semicircle? (1 Mark)
(a) 45°
(b) 90°
(c) 180°
(d) 360°

​Q6. If a chord is perpendicular to the radius at its midpoint, what is the angle between the radius and the chord? (2 Marks)

Q7. State the converse of the theorem: "Angles in the same segment of a circle are equal." (2 Marks)

Q8. In a cyclic quadrilateral ABCD, if ∠A = 95°, find ∠C and justify your answer. (2 Marks)

Q9. On a common hypotenuse AB, two right triangles, ACB and ADB, are situated on opposite sides. Prove that ∠BAC = ∠BDC. (3 Marks)

Q10. ABCD is a parallelogram. The circle through A, B and C intersects (produce if necessary) at E. Prove that AE = AD. (3 Marks)

Q11. The circumcenter of the triangle ABC is O. Prove that ∠OBC + ∠BAC = 90º. (3 Marks)

​Q12. If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side. (5 Marks)

​Q13. ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70° and ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD. (5 Marks)