Cyclic Quadrilateral
Cyclic Quadrilateral
CYCLIC QUADRILATERAL
DEFINITION A quadrilateral is called a cyclic quadrilateral if its all vertices lie on a circle.
A cyclic quadrilateral has some special properties which other quadrilaterals, in general, need not have . We shall state and prove these properties as theorems.
Theorem 1 The sum of either pair of opposite angles of a cyclic quadrilaterals is
OR
The opposite angles of a cyclic quadrilateral are supplementary.
Given A cyclic quadrilateral ABCD. TO PROVE CONSTRUCTION Join AC and BD. PROOF Consider side AB of quadrilateral ABCD as the chord of the circle. Clearly,
Now, consider the side BC of quadrilateral ABCD as the chord of the circle. We find that
Adding equations (i) and (ii) , we get
Hence, The converse of this theorem is also true as given below. |  |
Converse Theorem: If the sum of any pair of opposite angles of a quadriletral is | |
Given: A quadrilateral ABCD in which Tp Prove: ABCD is a cyclic quadrilateral Construction: Draw a circle passing through A, B and C. Proof: Suppose the circle meets CD or CD produced at E. Join AE Now ABCE is a cyclic Quadrilateral. As Opposite angles of a cyclic quadrilateral are supplementary
But, From Equation (1) and (2) we get But this is not possible And we know that the exterior angle of a triangle is equal to sum of interior opposite angles That is exterior angle is always greater than interior opposite angle Thus our assumption is wrong Hence ABCD is a cyclic quadrilateral. |
Case (1)
Case (2) |
| Illustration: In the figure ACDF is a cyclic quadrilateral. A circle passing through A and F meets AC and DF in the points B and E respectively. Prove that BE || CD. | |
Solution: ACDF is a cyclic quadrilateral and In a cyclic quadrilateral opposite angles are supplementary
Also ABEF is a cyclic quadrilateral and In a cyclic quadrilateral opposite angles are supplementary
From Equation 1 and 3 we get But they are corresponding angles when BE and CD are two straight lines and BC is the transversal As they are equal, So BE || CD |  |
If the sum of a pair of opposite angles of a quadrilateral is | |||
| Right Option : B | |||
| View Explanation | |||
If minor arc AB subtends | |||
| Right Option : A | |||
| View Explanation | |||
| |||||
| Right Option : A | |||||
| View Explanation | |||||
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