What Are The Properties Of Circles
- Two circles are congruent, if and only if they have equal radii.
- Two arcs of a circle are congruent if the angles subtended by them at the centre are equal.
- Two arcs subtend equal angles at the centre, if the arcs are congruent.
- If two arcs of a circle are congruent, their corresponding chords are equal.
- If two chords of a circle are equal, their corresponding arcs are equal.
- The angle in a semi-circle is a right angle.
- The arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semicircle.
Read More:
- Parts of a Circle
- Perimeter of A Circle
- Common Chord of Two Intersecting Circles
- Construction of a Circle
- The Area of A Circle
- Sector of A Circle
- The Area of A Segment of A Circle
- The Area of A Sector of A Circle
Properties Of Circles Example Problems With Solutions
In figure ABCD is a cyclic quadrilateral; O is the centre of the circle. If ∠BOD = 160º, find the measure of ∠BPD.
In figure ∆ABC is an isosceles triangle with AB = AC and m ∠ABC = 50º. Find m ∠BDC and m ∠BEC
Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
Suppose you are given a circle. Give a construction to find its centre.
Example 1: O is the centre of the circle. If ∠BOA = 90° and ∠COA = 110°, find ∠BAC.
Solution: Given: A circle with centre O and ∠BOA = 90°, ∠COA = 110°.
Example 2: O is the centre of the circle. If ∠BAC = 50°, find ∠OBC.
Solution:
Example 3: Find the value of x from the given figure, in which O is the centre of the circle.
Solution:
Example 4: P is the centre of the circle . Prove that ∠XPZ = 2 (∠XZY + ∠YXZ).
Solution: Given: A circle with centre P, XY and YZ are two chords.
Example 5: O is the centre of the circle. ∠OAB = 20°, ∠OCB = 55°. Find ∠BOC and ∠AOC.
Solution:
Example 6: If a side of a cyclic quadrilateral is produced, then prove that the exterior angle is equal to the interior opposite angle.
Solution: Given: A cyclic quadrilateral ABCD. Side AB is produced to E.
Example 7: Prove that the right bisector of a chord of a circle, bisects the corresponding arc of the circle.
Solution: Let AB be a chord of a circle having its centre at O. Let PQ be the right bisector of the chord AB, intersecting AB at L and the circle at Q. Since the right bisector of a chord always passes through the centre, so PQ must pass through the centre O. Join OA and OB. In triangles OAL and OBL we have
Example 8: In figure AB = CB and O is the centre of the circle. Prove that BO bisects ∠ABC.
Solution: Join OB and OC. Since the angle subtended by an arc of a circle at its centre is twice the angle subtended by the same arc at a point on the circumference.
Example 9: In fig. ABC is a triangle in which ∠BAC = 30º. Show that BC is the radius of the circumcircle of ∆ABC, whose centre is O.
Solution: Join OB and OC. Since the angle subtended by an arc of a circle at its centre is twice the angle subtended by the same arc at a point on the circumference.
Example 10: Consider the arc BCD of the circle. This arc makes angle ∠BOD = 160º at the centre of the circle and ∠BAD at a point A on the circumference.
Solution: Consider the arc BCD of the circle. This arc makes angle ∠BOD = 160º at the centre of the circle and ∠BAD at a point A on the circumference.
Example 11: In figure ∆ABC is an isosceles triangle with AB = AC and m ∠ABC = 50º. Find m ∠BDC and m ∠BEC
Solution:
Example 12: Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
Solution:
Example 13: Suppose you are given a circle. Give a construction to find its centre.
Solution: (i) Take three points A, B, C on given circle.
(ii) Join B to A & C.
(iii) Draw ⊥ bisectors of BA & BC.
(iv) The intersection point of ⊥ bisecteros is centre.