Chapter-XIII Cyclic QuadrilateralMaths - Grade IX

Module Objectives• .

• Define Cyclic Quadrilateral

• Identify and state the property of cyclic quadrilaterals

• Prove the theorem on cyclic quadrilateral logically.

• Construct cyclic quadrilaterals

• Solve problems and ridérs based on the theorem on cyclic quadrilateral

Introduction

Cyclic Quadrilateral:A circle and a quadrilateral can be at different positions in the same plane.

Observe the position of the vertices of the quadrilaterals given below: How is the fourth case different from the others ?

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We notice that in fig 4 all the four vertices of the quadrilateral KLMN lie on the circle.

We call KLMN a Cyclic Quadrilateral.

A quadrilateral, whose vertices lie on the circle is called a cyclic quadrileral. It is an inscribed quadrilateral.

KLMN is a quadrilateral. Hence K + L + M + N = 360°

KLMN is a cyclic quadrilateral. There may also exist some other relations between its angles.

Activity!!• .

Let us do the following activity to find out the relation between its angles.

• Draw a circle with centre O. • Inscribe a quadrilateral KLMN in it. • Measure its angles and find out K + M and L+N.

Repeat the process by drawing two more cyclic quadrilaterals and record your results in the table given:

Cyclic Quadrilateral

K L M N K + M L+N

1

2

3

Activity (contd)

You will observe that in each of the above cases K + M = 180° andl L + N = 180°Hence we can conclude that:

The opposite angles of a cyclic quadrilateral are supplementary.

Now Jet us prove that statement logically…

Theorem 5The opposite angles of a cyclic quadrilatéral are supplementary

Data: ‘O’ is the centre of the circle. ABCD is a cyclic quadrilateral

To Prove: BAD + BCD = 180° ABC + ADC = 180°

Theorem 5 (contd..)

• Construction: Join OB and OD• Proof:

Statement ReasonBAD = ½ BOD Inscribed angle is half central angleBCD = ½ reflex BOD Inscribed angle is half central angleBAD + BCD= ½ [BOD+½ reflex BOD) adding(1)&(2)i.e BAD + BCD = [BOD + reflex BOD)) Taking ½ commoni.e BAD + BCD = ½ x 360° Complete angle at the centre = 360° BAD+ BCD = 180°

Similarly ABC+ ADC=180°

Hence it is proved that the opposite angles of a cyclic quadrilateral are supplementary.

Converse of Theorem 5

If the opposite angles of a quadrilateral are supplementary, then it is cyclic.

I

Know This! Brahmagupta (628 AD.) an Jndian astronomer and mathematician in his masterpiece ‘Brahmasphuta Siddhanta” states that ” the exact area of a cyclic quadrilateral is the square root of the product of four sets of half the sum of the respective sides diminished by the sides.

Think!

Is a square a rectangle and an isosceles trapezium cyclic? why?

Activity

Draw a cyclic quadrilateral ABCD as shown above. Produce DC to E. Measure exterior BCEMeasure interior opposite BAD. Record it

Activity!!

Exterior BCE =Interior BAD =

You will observe that the exterior angle of a cyclic quadrilateral is equal to its interior opposite angle

Verify by producing sides CB, BA and AD and measuring the corresponding exterior and interior opposite angles.

Example 1

ABCD is a cyclic quadrilateral. If A = 85°, B = 70°, Find the measures of C and D ?

Construction of a Cyclic Quadrilateral

Recall that a cyclic quadrilateral is a quadrilateral inscribed in circle.

Let us learn how to construct a cyclic quadrilateral.

Example : Construct a cyclic quadrilateral PQRS, given PQ = 3.6 cru, QR = 5.5 cm. QS = 6.5cm and SP=5.6cm

Construction of a Cyclic Quadrilateral

Step 1: Draw the rough figure and mark the measurements

Construction of a Cyclic Quadrilateral (continued)

Step 2: Construct triangle PQS.

Construction of a Cyclic Quadrilateral (continued)

Step 3: . Draw the perpendicular bisector of any two sìdes of the triangle and draw its circumcircle

Construction of a Cyclic Quadrilateral (continued)

Step 4: To locate ‘R’, draw an arc on Circle from Q with radius 5.5 cmStep 5: Join RS to get the required cyclic quadrilateral PQRS

Think!!To construct a quadrilateral, five elements are needed. But toconstruct a cyclic quadrilateral, only four elements are sufficient. Why?

Example 2Construct a cyclic quadrilateral ABCD, Given: AB=4 cm, BC=3.5cm, CD=4.2cm D = 80°Step 1. Draw the rough figure and mark the measurements on it

Example 2 (contd..)Step 2. To construct ∆ ABC, we know AB=4cm, BC=3.5cm. But what about the third element? We can find the value of ABC

ABC = 180° - ADC (Theorem 5)ABC = 180° - 80°ABC = 100°

Step 3. Construct ∆ ABC, we know AB=4cm, BC=3.5cm, ABC = 100°

Example 2 (contd..)Step 4: Draw the perpendicular bisectors of any two sides of ∆ ABC and draw the circumcircle.Step 5 : To locate point D, draw an arc of radius 4.2 cm from C.Step 6 : Join AD. Then ABCD is the required cyclic quadrilateral.