9.3-Cyclic Quadrilaterals

9.3-Cyclic Quadrilaterals Important Formulae

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Grade 9 → Math → Circles → 9.3-Cyclic Quadrilaterals

After successful completion of this topic, you should be able to:

  • Understand properties of a cyclic quadrilateral.

Theorem 9.7 : The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.:

The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference. If $\angle AOB$ is at the center and $\angle ACB$ is on the circle, then $\angle AOB = 2\angle ACB$.


Theorem 9.8 : Angles in the same segment of a circle are equal.:

In a circle, angles subtended by the same arc at any point on the remaining part of the circle are equal. If points A and B are on the circle, and C and D are points in the same segment, then $\angle ACB = \angle ADB$.


Theorem 9.9 : If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).:

If a line segment joining points A and B subtends equal angles at points C and D on the same side, then points A, B, C, and D are concyclic. Mathematically, if $\angle ACB = \angle ADB$, the four points lie on a circle.


Theorem 9.10 : The sum of either pair of opposite angles of a cyclic quadrilateral is 180$^\circ$.:

In a cyclic quadrilateral, the sum of either pair of opposite angles is $180^\circ$. Specifically, if quadrilateral ABCD is cyclic, then $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.


Theorem 9.11 : If the sum of a pair of opposite angles of a quadrilateral is 180$^\circ$, the quadrilateral is cyclic.:

If the sum of a pair of opposite angles of a quadrilateral is $180^\circ$, then the quadrilateral is cyclic. This means that the vertices of the quadrilateral lie on a single circle. Mathematically, if $\angle A + \angle C = 180^\circ$, the quadrilateral ABCD is cyclic.



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In Fig. 9.23, A,B and C are three points on a circle with centre O such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Solution:
150°, 30°

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In Fig. 9.24,∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.

In Fig.9.25, ∠ABC=69°, ∠ACB=31°, find ∠ BDC.

In Fig.9.26, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠ BEC = 130°and ∠ ECD = 20°. Find ∠ BAC.

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠ DBC = 70°, ∠ BAC is 30°, find ∠ BCD. Further, if AB = BC, find ∠ ECD.

Solution:
∠BCD = 80° and ∠ECD = 50°

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If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

Solution:
Draw perpendiculars AM and BN on CD (AB || CD and AB < CD). Show ∆ AMD ≅ ∆ BNC. This gives ∠ C = ∠ D and, therefore, ∠ A + ∠ C = 180°.

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If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see Fig. 9.27). Prove that ∠ACP=∠QCD.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.

ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD=∠CBD.

Prove that a cyclic parallelogram is a rectangle.