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Chapter -XIII Cyclic Quadrilateral

Chapter -XIII Cyclic Quadrilateral. Maths - 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

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Chapter -XIII Cyclic Quadrilateral

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  1. Chapter-XIII CyclicQuadrilateral Maths - Grade IX

  2. 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

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

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

  5. 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. • .

  6. Activity!! • Let us do the following activity to find out the relation between its angles. • Draw a circle with centreO. • Inscribe a quadrilateral KLMN in it. • Measure its angles and find out K + Mand L+N. • Repeat the process by drawing two more cyclic quadrilaterals and record your results in the table given: • .

  7. 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…

  8. Theorem5 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°

  9. 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 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.

  10. Converse of Theorem 5 If the opposite angles of a quadrilateral are supplementary, then it is cyclic.I

  11. Know This! Brahmagupta (628 AD.) an Jndian astronomer and mathematician in his masterpiece ‘BrahmasphutaSiddhanta” 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?

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

  13. 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.

  14. Example 1 ABCD is a cyclic quadrilateral. If A = 85°, B = 70°, Find the measures of Cand D ?

  15. Construction of a CyclicQuadrilateral 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

  16. Construction of a CyclicQuadrilateral Step 1: Draw the rough figure and mark the measurements

  17. Construction of a CyclicQuadrilateral (continued) Step 2: Construct triangle PQS.

  18. Construction of a CyclicQuadrilateral (continued) Step 3: . Draw the perpendicular bisector of any two sìdes of the triangle and draw its circumcircle

  19. Construction of a CyclicQuadrilateral (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

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

  21. 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

  22. 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°

  23. 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.

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