What Are The Properties Of Circles

What Are The Properties Of Circles

1. Two circles are congruent, if and only if they have equal radii.
   
2. Two arcs of a circle are congruent if the angles subtended by them at the centre are equal.
   
3. Two arcs subtend equal angles at the centre, if the arcs are congruent.
   
4. If two arcs of a circle are congruent, their corresponding chords are equal.
   
5. If two chords of a circle are equal, their corresponding arcs are equal.
   
6. The angle in a semi-circle is a right angle.
   
7. 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.