Cyclic Quardilateral

PROJECT TOPIC:::::: TOPIC: CYCLIC QUADRILATERAL SUBMITTED BY:::: LAVNEET KAUR (MATH MISTRESS) (G.G.S.S.SCHOOL, (G.G.S.S.SCHOOL SAMRALA)

TOPIC === CYCLIC QUADRILATERAL

DEFINITION

A QUADRILATERAL WHOSE VERTICES LIE ON A CIRCLE IS CALLED A CYCLIC QUADRILATERAL

THEOREM

SUM OF TWO OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL IS 180.

PROOF

∠ABC=1/2∠AOC=1/2 X (ANGLE MADE BY AN ARC OF A CIRCLE AT -----⊕

THE CENTRE OF THE CIRCLE IS TWICE THE ANGLE MADE BY IT ON THE CIRCUMFERENCE)

∠ADC=1/2∠AOC=1/2 Y (

“

-------⊗ ADDING EQN’S ⊗ AND ⊕ , WE GET ∠ABC +∠ADC =1/2 X+1/2 Y =1/2 (X+Y)) =1/2 (360) =180

“

“

“

B

x A

o

OO ∩∩ OO

y D

SIMILARLY, ∠BAD+∠BCD=180 HENCE PROVED.

)

C

CONVERSE OF PREVIOUS THEOREM IS TRUE: CONVERSE:: IF THE SUM OF OPPOSITE ANGLES OF A QUADRILATERAL IS 180, THEN IT IS CYCLIC PROOF:

SUPPOSE ∠ABC+∠ADC=180 THEN

∠BAD+∠BCD= 180 (SUM OF FOUR ANGLES OF ------⊗

QUADRILATERAL IS 360)

SUPPOSE, THE CIRCLE WITH CENTRE ‘C ‘PASSING THROUGH A, B, C DOES NOT PASS THROUGH ‘D ‘. BUT, IF WE EXTEND CD, IT CUTS IT AT POINT ‘E’. A THEN ∠B+∠E=180 (BECAUSE OF PREVIOUS THEOREM) ALSO ∠B+∠D =180 (BY EQN” ⊗ ) THIS IMPLIES ∠D=∠E B WHICH A CONTRADICTION AS IN A TRIANGLE, EXTERIOR ANGLE IS NOT EQUAL TO INTERIOR OPPOSITE ANGLE BUT SUM OF INTERIOR OPPOSITE ANGLE.

∴, OUR SUPPOSITION IS WRONG. HENCE, CIRCLE WITH CENTRE ‘O’ PASSES THROUGH A, B,C& D. HENCE, ABCD IS A CYCLIC QUADRILATERAL

E D .O

D C C

EXAMPLES

1.

IN THE GIVEN FIGURE, FIND ∠ ADC.

D C

A

SOLUTION:

GIVEN THAT ∠ABC=70 SINCE WE KNOW THAT IN A CYCLIC QUADRILATERAL, SUM OF TWO OPPOSITE ANGLE IS 180.

∴BY USING THIS RESULT, WE GET THAT

∠ABC+∠ADC=180 ⇒

70 +∠ADC=180

⇒

∠ADC=180-70

⇒

∠ADC=110 HENCE ∠ADC=110

70

B

2.

PROVE THAT ∠BCE=∠BAD. A

B

SOLUTION: IN THE FIGURE,

∠BAD + ∠BCD=180 (BY THEOREM 1) D ∠BCD +∠BCE =180 (BY LINEAR PAIR PROPERTY)

C

E

∴∠BAD +∠BCD = ∠BCD +∠BCE ⇒ ∠BAD =∠BCE HENCE PROVED.

∗THIS RESULT IS SOMETIME USED AS A PROPERTY.

3 . S H O W T H A T A L L A N G L E S IN T H E G IV E N M EASU RE 90 EAC H .

Q U A D R IL

S O L . G IV E N T H A T A N G L E D A B = 90 N O W A N G L E A B C = 9 0 ( A N G L E IN A S E M I C I R C L E IS R I G H T A N G L E ) S IM IL A R L Y , A N G L E A D C = 90 (

,,

)

N O W B Y U S IN G P R E V I O U S T H E O R E M , A N G L E B C D = 9 0 . ( S U M O F A N G L E S IN A C YC L IC Q U A D R IL A T E R A L IS 3 60 ) H E N C E , E A C H A N G L E IN T H E G IV E N Q U A D R IL A T E R A L IS O F 9 0 .