**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.

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