Chapter-XIII Cyclic Quadrilateral Maths - Grade IX

Module Objectives Define Cyclic Quadrilateral . 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 ? .

We call KLMN a Cyclic Quadrilateral. 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 5 The 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 Reason BAD = ½ BOD Inscribed angle is half central angle BCD = ½ reflex BOD Inscribed angle is half central angle BAD + BCD= ½ [BOD+½ reflex BOD) adding(1)&(2) i.e BAD + BCD = [BOD + reflex BOD)) Taking ½ common i.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 BCE Measure 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 cm Step 5: Join RS to get the required cyclic quadrilateral PQRS

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

Example 2 Construct 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.