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.
    Properties-Of-Circle-1
  2. Two arcs of a circle are congruent if the angles subtended by them at the centre are equal.
    Properties-Of-Circle-2
  3. Two arcs subtend equal angles at the centre, if the arcs are congruent.
    Properties-Of-Circle-3
  4. If two arcs of a circle are congruent, their corresponding chords are equal.
    Properties-Of-Circle-4
  5. If two chords of a circle are equal, their corresponding arcs are equal.
    Properties-Of-Circle-4
  6. The angle in a semi-circle is a right angle.
    Properties-Of-Circle-5
  7. The arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semicircle.
    Properties-Of-Circle-6

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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°.
Properties-Of-Circle-Example-1

Example 2:    O is the centre of the circle. If ∠BAC = 50°, find ∠OBC.
Solution:
Properties-Of-Circle-Example-2
Properties-Of-Circle-Example-2-1

Example 3:    Find the value of x from the given figure, in which O is the centre of the circle.
Solution:
Properties-Of-Circle-Example-3
Properties-Of-Circle-Example-3-1

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.
Properties-Of-Circle-Example-4
Properties-Of-Circle-Example-4-1

Example 5:    O is the centre of the circle. ∠OAB = 20°, ∠OCB = 55°. Find ∠BOC and ∠AOC.
Properties-Of-Circle-Example-5
Solution:
Properties-Of-Circle-Example-5-1
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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.
Properties-Of-Circle-Example-6

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
Properties-Of-Circle-Example-7

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.
Properties-Of-Circle-Example-8

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.
Properties-Of-Circle-Example-9
Properties-Of-Circle-Example-9-1

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.
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