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Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 17 - Circles

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Selina Textbook Solutions Chapter 17 - Circles

Selina Textbook Solutions are a perfect way to ace your examination with high marks. These Textbook Solutions are extremely helpful for solving difficult questions in the ICSE Class 10 Mathematics exam. Our Selina Textbook Solutions are written by our subject experts. Find all the answers to the Selina textbook questions of Chapter 17 - Circles.

All Selina textbook questions of Chapter 17 - Circles solutions are created in accordance with the latest ICSE syllabus. These free Textbook Solutions for ICSE Class 10 Selina Concise Mathematics will give you a deeper insight on the fundamentals in this chapter and will help you to score more marks in the final examination. ICSE Class 10 students can refer to these solutions while doing their homework and while studying and revising for the Mathematics exam. 

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Exercise/Page

Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 17 - Circles Page/Excercise 17(A)

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

 

Given : space Two space chords space AB space and space CD space intersect space each space other space at space straight P space inside space the space circle. space OA comma
OB comma space OC space and space OD space are space joined.
To space prove : space angle AOC plus angle BOD space equals 2 angle APC
Construction : space Join space AD.
Proof : space Arc space AC space subtends space angle AOC space at space the space centre space and space angle ADC space at space the space remaining space
part space of space the space circle.
angle AOC space equals space 2 angle ADC space...... left parenthesis 1 right parenthesis
Similarly comma
angle BOD space equals space 2 angle BAD....... left parenthesis 2 right parenthesis
Adding space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma
angle AOC space plus angle BOD space equals space 2 angle ADC space plus 2 angle BAD
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 left parenthesis angle ADC plus angle BAD right parenthesis....... left parenthesis 3 right parenthesis
But space in space increment PAD comma
Ext. space angle APC equals angle PAD plus angle ADC
space space space space space space space space space space space space space space space space space space space space space equals angle BAD space plus angle ADC space space space space space space space space space space space space space space......... left parenthesis 4 right parenthesis
From space left parenthesis 3 right parenthesis space and space left parenthesis 4 right parenthesis comma
angle AOC space plus angle BOD space equals 2 angle APC

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39



(i) angleAEB = 90 to the power of 0

(Angle in a semicircle is a right angle)

Therefore angle EBA = 90 to the power of 0 - angleEAB = 90 to the power of 0 - 63 to the power of 0= 27 to the power of 0


(ii) AB parallel to ED

Therefore angle DEB = EBA = 27 to the power of 0                        (Alternate angles)

Therefore BCDE is a cyclic quadrilateral

Therefore angle DEB  angleBCD = 180 to the power of 0

[Pair of opposite angles in a cyclic quadrilateral are supplementary]

Therefore angle BCD = 180 to the power of 0 - 27 to the power of 0 = 153 to the power of 0


Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

Solution 50

Solution 51

Solution 52

  1. ABCD is a cyclic quadrilateral

mDAB = 180° - DCB

= 180° - 130°

= 50°

  1. In ∆ADB,

mDAB + mADB + mDBA = 180°

50° + 90° + mDBA = 180°

mDBA = 40°

Solution 53

Solution 54

Solution 55

Solution 56

Solution 57

Solution 58

Solution 59

Solution 60

Solution 61


Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 17 - Circles Page/Excercise 17(B)

Solution 1

 

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Solution 2

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.


Solution 3

 

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Solution 4

 

Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.

Solution 5

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Solution 6

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Solution 7

 

Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.

Solution 8

 

Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.

Solution 9

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Solution 10

 

 

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Selina Concise Mathematics - Part II Solution for Class 10 Mathematics Chapter 17 - Circles Page/Excercise 17(C)

Solution 1

Syntax error from line 1 column 49 to line 1 column 73. Unexpected 'mstyle '.

Solution 2

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Solution 3




J o i n space A P space a n d space p r o d u c e space A B space t o space m e e t space t h e space b i g g e r space c i r c l e space a t space C.
A B equals A C minus B C equals 5 c m minus 3 c m equals 2 c m.
B u t comma space M space i s space t h e space m i d minus p o i n t space o f space A B.
therefore A M equals 2 over 2 equals 1 c m
N o w space i n space r i g h t space increment A P M comma
A P squared equals M P squared plus A M to the power of 2 space end exponent space left square bracket P y t h a g o r a s space T h e o r e m right square bracket
rightwards double arrow left parenthesis 5 right parenthesis squared equals M P squared plus 1 squared
rightwards double arrow 25 equals M P squared 1
rightwards double arrow M P squared equals 25 minus 1
rightwards double arrow M P squared equals 24
rightwards double arrow M P equals square root of 24 equals square root of 4 cross times 6 end root equals 2 square root of 6 c m
therefore P Q equals 2 M P equals 2 cross times 2 square root of 6 equals 4 square root of 6 c m
rightwards double arrow P Q equals 4 cross times 2.45 equals 9.8 c m


Solution 4

G i v e n space minus space I n space t h e space f i g u r e space A B C space i s space a space t r i a n g l e space i n space w h i c h space angle A equals 30 degree.
T o space p r o v e minus B C space i s space t h e space r a d i u s space o f space c i r c u m c i r c l e space o f space increment A B C space w h o s e space c e n t r e
i s space O.
C o n s t r u c t i o n minus J o i n space O B space a n d space O C.
P r o o f :
angle B O C equals 2 angle B A C equals 2 cross times 30 degree equals 60 degree
N o w space i n space increment O B C comma
O B equals O C space space space space space space space space space space space space space left square bracket R a d i i space o f space t h e space s a m e space c i r c l e right square bracket
angle O B C equals angle O C B
B u t comma space i n space increment B O C comma
angle O B C plus angle O C B plus angle B O C equals 180 degree space space left square bracket A n g l e s space o f space a space t r i a n g l e right square bracket
rightwards double arrow angle O B C plus angle O B C plus 60 degree equals 180 degree
rightwards double arrow 2 angle O B C plus 60 degree equals 180 degree
rightwards double arrow 2 angle O B C equals 180 degree minus 60 degree
rightwards double arrow 2 angle O B C equals 120 degree
rightwards double arrow angle O B C equals fraction numerator 120 degree over denominator 2 end fraction equals 60 degree
rightwards double arrow angle O B C equals angle O C B equals angle B O C equals 60 degree
rightwards double arrow increment B O C space i s space a n space e q u i l a t e r a l space t r i a n g l e.
rightwards double arrow B C equals O B equals O C
B u t comma space O B space a n d space O C space a r e space t h e space r a d i i space o f space t h e space c i r c u m minus c i r c l e.
therefore B C space i s space a l s o space t h e space r a d i u s space o f space t h e space c i r c u m minus c i r c l e.

Solution 5




G i v e n minus I n increment A B C comma A B equals A C space a n d space a space c i r c l e space w i t h space A B space a s space d i a m e t e r space i s space d r a w n
w h i c h space i n t e r s e c t s space t h e space s i d e space B C space a n d space D.
T o space p r o v e minus D space i s space t h e space m i d space p o i n t space o f space B C.
C o n s t r u c t i o n minus J o i n space A D.
P r o o f minus angle 1 equals 90 degree space space space left square bracket A n g l e space i n space a space s e m i space c i r c l e right square bracket
B u t space angle 1 plus angle 2 equals 180 degree space space left square bracket L i n e a r space p a i r right square bracket
therefore angle 2 equals 90 degree
N o w space i n space r i g h t space increment A B D space a n d space increment A C D comma
H y p. space A B equals H y p. A C space space left square bracket G i v e n right curly bracket
S i d e space A D equals A D space space space space space space space space space space space left square bracket C o m m o n right square bracket
therefore B y space t h e space R i g h t space A n g l e minus H y p o t e n u s e minus S i d e space c r i t e r i o n space o f space c o n g r u e n c e comma space w e space h a v e
increment A B D approximately equal to increment A C D space space space space space space space space space left square bracket R H S space c r i t e r i o n space o f space c o n g r u e n c e right square bracket
T h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g r u e n t.
therefore B D equals D C space space space space space space space space space space space space space left square bracket c. p. c. t right square bracket
H e n c e space D space i s space t h e space m i d space p o i n t space o f space B C.


Solution 6


 

J o i n space O E.
A r c E C space s u b t e n d s angle E O C space a t space t h e space c e n t r e space a n d angle E B C space a t space t h e space r e m a i n i n g
p a r t space o f space t h e space c i r c l e.
angle E O C equals 2 angle E B C equals 2 cross times 65 degree equals 130 degree.
N o w space i n space increment O E C comma O E equals O C space space space space space space space space space space left square bracket R a d i i space o f space t h e space s a m e space c i r c l e right square bracket
therefore angle O E C equals angle O C E
B u t comma space i n space increment E O C comma
angle O E C plus angle O C E plus angle E O C equals 180 degree space space left square bracket A n g l e s space o f space a space t r i a n g l e right square bracket
rightwards double arrow angle O C E plus angle O C E plus angle E O C equals 180 degree
rightwards double arrow 2 angle O C E plus 130 degree equals 180 degree
rightwards double arrow 2 angle O C E equals 180 degree minus 130 degree
rightwards double arrow 2 angle O C E equals 50 degree
rightwards double arrow angle O C E equals fraction numerator 50 degree over denominator 2 end fraction equals 25 degree
therefore A C parallel to E D space space space space space space space space space space space left square bracket g i v e n right square bracket
therefore angle D E C equals angle O C E space space space space left square bracket A l t e r n a t e space a n g l e s right square bracket
rightwards double arrow angle D E C equals 25 degree


Solution 7




G i v e n minus T w o space c h o r d s space A B space a n d thin space C D space i n t e r s e c t
e a c h space o t h e r space a t space P space i n s i d e space t h e space c i r c l e
w i t h space c e n t r e space O space a n d space A P equals C P
T O space p r o v e minus A B equals C D
P r o o d minus T w o space c h o r d s space A B space a n d space C D space i n t e r s e c t space e a c h space o t h e r space i n s i d e space t h e space
c i r c l e space a t space P.
therefore A P cross times P B equals C P cross times P D
rightwards double arrow fraction numerator A P over denominator C P end fraction equals fraction numerator P D over denominator P B end fraction
B u t space A P equals C P space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space space left square bracket g i v e n right square bracket
therefore P D equals P B space o r space P B equals P D space space... left parenthesis 2 right parenthesis
A d d i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis
A P plus P B equals C P plus P D
rightwards double arrow A B equals C D


Solution 8



G i v e n minus A B C D space i s space a space c y c l i c space q u a d r i l a t e r a l space a n d space P Q R S space i s space a
q u a d r i l a t e r a l space f o r m e d space b y space t h e space a n g l e
b i s e c t o r s space o f space a n g l e space angle A comma angle B comma angle C space a n d space angle D.
T o space p r o v e minus space P Q R S space i s space a space c y c l i c space q u a d r i l a t e r a l.
P r o o f minus space I n space increment A P D comma
angle P A D plus angle A D P plus angle A P D equals 180 degree space space space space space space... left parenthesis 1 right parenthesis
S i m i l a r l y comma space I N space increment B Q C comma
angle Q B C plus angle B C Q plus angle B Q C equals 180 degree space space space space space space space... left parenthesis 2 right parenthesis
A d d i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space g e t
angle P A D plus angle A D P plus angle A P D plus angle Q B C plus angle B C Q plus angle B Q C equals 180 degree plus 180 degree
rightwards double arrow angle P A D plus angle A D P plus angle Q B C plus angle B C Q plus angle A P D plus angle B Q C equals 360 degree space space space space.... left parenthesis 3 right parenthesis
B u t space angle P A D plus angle A D P plus angle Q B C plus angle B C Q equals 1 half left square bracket angle A plus angle B plus angle C plus angle D right square bracket
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 half cross times 360 degree equals 180 degree
therefore angle A P D plus angle B Q C equals 360 degree minus 180 degree equals 180 degree space space space space space space left square bracket f r o m space left parenthesis 3 right parenthesis right square bracket
B u t space t h e s e space a r e space t h e space s u m space o f space o p p o s i t e space a n g l e s space o f space q u a d r i l a t e r a l space P R Q S.
therefore Q u a d. space P R Q S space i s space a space c y c l i c space q u a d r i l a t e r a l.
space space space space space









Solution 9

  

Solution 10

G i v e n minus I n space increment A B C comma space A B equals A C space a n d space D space a n d space E space a r e space p o i n t s space o n space A B space a n d space A C
s u c h space t h a t space A D equals A E. space D E space i s space j o i n e d.
T o space p r o v e space B comma C comma E comma D space a r e space c o n c y c l i c.
P r o o f minus I n increment A B C comma A B equals A C
therefore angle B equals angle C space left square bracket A n g l e s space o p p o s i t e space t o space e q u e a l space s i d e s right square bracket
S i m i l i a r l y comma space I n space increment A D E comma space A D equals A E space space space left square bracket g i v e n right square bracket
therefore angle A D E equals angle A E D space left square bracket A n g l e s space o p p o s i t e space t o space e q u a l space s i d e s right square bracket
I n space increment A B C comma
because fraction numerator A P over denominator A B end fraction equals fraction numerator A E over denominator A C end fraction
therefore D E parallel to B C
therefore angle A D E equals angle B space space space space space left square bracket c o r r e s p o n o d i n g space a n g l e s right square bracket
B u t space angle B equals angle C space space space space space left square bracket p r o v e d right square bracket
therefore E x t. angle A D E equals i t s space i n t e r i o r space o p p o s i t e space angle C
therefore B C E D space i s space a space c y c l i c space q u a d r i l a t e r a l.
H e n c e space B comma C comma E space a n d space D space a r e space c o n c y c l i c.





Solution 11

I n space c y c l i c space q u a d. space A B C D comma
A F parallel to C B space a n d space D A space i s space p r o d u c e d space t o space E space s u c h space t h a t space angle A D C equals 92 degree space a n d space angle F A E equals 20 degree
N o w space w e space n e e d space t o space f i n d space t h e space m e a s u r e space o f space angle B C D
I n space c y c l i c space q u a d. space A B C D comma
angle B plus angle D equals 180 degree
rightwards double arrow angle B plus 92 degree equals 180 degree
rightwards double arrow angle B equals 180 degree minus 92 degree
rightwards double arrow angle B equals 88 degree
S i n c e space A F parallel to C B comma space angle F A B equals angle B equals 88 degree
B u t comma space angle F A E equals 20 degree space space space space space left square bracket g i v e n right square bracket
E x t. angle B A E equals angle B A F plus angle F A E
equals 88 degree plus 22 degree equals 108 degree
B u t comma space E x t. angle B A E equals angle B C D
therefore angle B C D equals 108 degree



Solution 12



J o i n space D B space a n d space D C comma space I B space a n d space I C comma
angle B A C equals 66 degree comma angle A B C equals 80 degree comma space I space i s space t h e space i n c e n t r e space o f space t h e space increment A B C comma
left parenthesis i right parenthesis space S i n c e space angle D B C space a n d space angle D A C space a r e space i n space t h e space s a m e space s e g m e n t comma
angle D B C equals angle D A C
B u t comma angle D A C equals 1 half angle B A C equals 1 half cross times 66 degree equals 33 degree
therefore angle D B C equals 33 degree
left parenthesis i i right parenthesis space S i n c e space I space i s space t h e space i n c e n t r e space o f space increment A B C comma space I B space b i s e c t s space angle A B C
therefore angle I B C equals 1 half angle A B C equals 1 half cross times 80 degree equals 40 degree
left parenthesis i i i right parenthesis space therefore angle B A C equals 66 degree space a n d space angle A B C equals 80 degree
I n space increment A B C comma angle A C B equals 180 degree minus left parenthesis angle A B C plus angle B A C right parenthesis
rightwards double arrow angle A C B equals 180 degree minus left parenthesis 80 degree plus 66 degree right parenthesis
rightwards double arrow angle A C B equals 180 degree minus left parenthesis 156 degree right parenthesis
rightwards double arrow angle A C B equals 34 degree
S i n c e space I C space b i s e c t s space t h e space angle C comma
therefore angle I C B equals 1 half angle C equals 1 half cross times 34 degree equals 17 degree
N o w space i n space increment I B C comma
angle I B C plus angle I C B plus angle B I C equals 180 degree
rightwards double arrow 40 degree plus 17 degree plus angle B I C equals 180 degree
rightwards double arrow 57 degree plus angle B I C equals 180 degree
rightwards double arrow angle B I C equals 180 degree minus 57 degree
rightwards double arrow angle B I C equals 123 degree



Solution 13

G i v e n minus I n space t h e space f i g u r e comma space A B equals A D equals D C equals P B space a n d space D B C equals X degree
J o i n space A C space a n d space B D.
T o space f i n d space : space t h e space m e a s u r e space o f space angle A B D space a n d space angle A P B.
P r o o f space : space angle D A C equals angle D B C equals X space space space space space space space space space space space space space space space space left square bracket a n g l e s space i n space t h e space s a m e space s e g m e n t right square bracket
B u t space angle D C A equals angle D A C equals X space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because A D equals D C right square bracket
A l s o comma space w e space h a v e comma space angle A B D equals angle D A C space space space space space space space space space space space space space space left square bracket a n g l e s space i n space t h e space s a m e space s e g m e n t right square bracket
I n space increment A B P comma space e x t. angle A B C equals angle B A P plus angle A P B
B u t comma angle B A P equals angle A P B space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because A B equals B P right square bracket
2 cross times X equals angle A P B plus angle A P B equals 2 angle A P B
therefore 2 angle A P B equals 2 X
rightwards double arrow angle A P B equals X
therefore angle A P B equals angle D B C equals X comma
B u t space t h e s e space a r e space c o r r e s p o n d i n g space a n g l e s
therefore A P parallel to D B




Solution 14




G i v e n minus I n space t h e space f i g u r e comma A B C comma A E Q space a n d space C E P space a r e space s t r a i g h t space l i n e
T o space p r o v e minus angle A P E plus angle C Q E equals 180 degree
C o n s t r u c t i o n minus J o i n space E B
P r o o f minus I n space c y c l i c space q u a d. A B E P comma
angle A P E plus angle A B E equals 180 degree space space....... left parenthesis 1 right parenthesis
S i m i l a r l y comma space i n space c y c l i c space q u a d. B C Q E comma
angle C Q E plus angle C B E equals 180 degree space...... left parenthesis 2 right parenthesis
A d d i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma
angle A P E plus angle A B E plus angle C Q E plus angle C B E equals 180 degree plus 180 degree equals 360 degree
rightwards double arrow angle A P E plus angle A B E plus angle C B E equals 360 degree
B u t comma space angle A B E plus angle C B E equals 180 degree space space space space space space space space space space space space space space space space space left square bracket L i n e a r space p a i r right square bracket
therefore angle A P E plus angle C Q E plus 180 degree equals 360 degree
rightwards double arrow angle A P E plus angle C Q E equals 360 degree minus 180 degree equals 180 degree
H e n c e space angle A P E space A N D space angle C Q E space a r e space s u p p l e m e n t a r y.


Solution 15



A r c space A C space s u b t e n d s space angle A O C space a t space t h e space c e n t r e space a n d space angle A D C space a t space t h e space r e m a i n i n g space p a r t
o f space t h e space c i r c l e
therefore angle A O C equals 2 angle A D C
rightwards double arrow angle A O C equals 2 cross times 32 degree equals 64 degree
S i n c e space angle A O C space a n d space angle B O C space a r e space l i n e a r space p a i r comma space w e space h a v e
angle A O C plus angle B O C equals 180 degree
rightwards double arrow 64 degree plus angle B O C equals 180 degree
rightwards double arrow angle B O C equals 180 degree
rightwards double arrow angle B O C equals 180 degree minus 64 degree
rightwards double arrow angle B O C equals 116 degree




Solution 16



P Q R S space i s space a space c y c l i c space q u a d r i l a t e r a l space i n space w h i c h space angle P Q R equals 135 degree
S i d e s space S P space a n d space R Q space a r e space p r o d u c e d space t o space m e e t space a t space A space a n d
S i d e s space P Q space a n d space S R space a r e space p r o d u c e d space t o space m e e t space a t space B.
angle A equals angle B equals 2 : 1
L e t space angle A equals 2 x comma t h e n angle B minus x
N o w comma space i n space c y c l i c space q u a d. P Q R S comma
S i n c e comma angle P Q R equals 135 degree comma angle S equals 180 degree minus 135 degree equals 45 degree
open square brackets S i n c e space s u m space o f space o p p o s i t e space a n g l e s space o f space a space c y c l i c space q u a d r i l a t e r a l space a r e space s u p p l e m e n t a r y close square brackets
S i n c e comma space angle P Q R space a n d space angle P Q A space a r e space l i n e a r space p a i r comma
angle P Q R plus angle P Q A equals 180 degree
rightwards double arrow 135 degree plus angle P Q A equals 180 degree
rightwards double arrow angle P Q A equals 180 degree minus 135 degree equals 45 degree
N o w comma space i n space increment P B S comma
angle P equals 180 degree minus left parenthesis 45 degree plus x right parenthesis equals 180 degree minus 45 degree minus x equals 135 degree minus x space space space.... left parenthesis 1 right parenthesis
A g a i n comma space i n space increment P Q A comma
E x t. angle P equals angle P Q A plus angle A equals 45 degree plus 2 x space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 2 right parenthesis
F r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma
45 degree plus 2 x equals 135 degree minus x
rightwards double arrow 2 x plus x equals 135 degree minus 45 degree
rightwards double arrow 3 x equals 90 degree
rightwards double arrow x equals 30 degree
H e n c e comma angle A equals 2 x equals 2 cross times 30 degree equals 60 degree
a n d space angle B equals x equals 30 degree



Solution 17



G i v e n minus I n space t h e space f i g u r e comma A B space i s space t h e space d i a m e t e r space o f space a space c i r c l e space w i t h space c e n t r e space O.
C D space i s space t h e space c h o r d space w i t h space l e n g h space e q u a l space r a d i u s space O A.
A C space a n d space B D space p r o d u c e d space m e e t space a t space p o i n t space P
T o space p r o v e space : space angle A P B equals 60 degree
C o n s t r u c t i o n minus J o i n space O C space a n d space O D
P r o o f minus W e space h a v e space C D equals O C equals O D space space space space space left square bracket g i v e n right square bracket
T h e r e f o r e comma increment O C D space i s space a n space e q u i l a t e r a l space t r i a n g l e
therefore angle O C D equals angle O D C equals angle C O D equals 60 degree
I n space increment A O C comma O A equals O C space space space space space space space space space space space space space space space left square bracket r a d i i space o f space t h e space s a m e space c i r c l e right square bracket
therefore angle A equals angle 1
S i m i l a r l y comma space i n space increment B O D comma O B equals O D space space space space left square bracket r a d i i space o f space t h e space s a m e space c i r c l e right square bracket
therefore angle B equals angle 2
N o w comma space i n space c y c l i c space q u a d. A C D B comma
sin c e comma angle A C D plus angle B equals 180 degree
open square brackets S i n c e space s u m space o f space o p p o s i t e space a n g l e s space o f space a space c y c l i c space q u a d r i l a t e r a l space a r e space s u p p l e m e n t a r y close square brackets
rightwards double arrow 60 degree plus angle 1 plus angle B equals 180 degree
rightwards double arrow angle 1 plus angle B equals 180 degree minus 60 degree
rightwards double arrow angle 1 plus angle B equals 120 degree
B u t comma space angle 1 equals angle A
therefore angle A plus angle B equals 120 degree space space space space space space space space.... left parenthesis 1 right parenthesis
N o w comma space i n space increment A P B comma
angle P plus angle A plus angle B equals 180 degree space space space space space space space space space space space left square bracket S u m space o f space a n g l e s space o f space a space t r i a n g l e s right square bracket
rightwards double arrow angle P plus 120 degree equals 180 degree
rightwards double arrow angle P equals 180 degree minus 120 degree space space space space space space space space space space space space left square bracket f r o m space left parenthesis 1 right parenthesis right square bracket
rightwards double arrow angle P equals 60 degree space o r space angle A P B equals 60 degree
















Solution 18


 

G i v e n minus A B C D space i s space a space c y c l i c space q u a d r i l a t e r a l space i n space w h i c h space A D parallel to B C
B i s e c t o r space o f space angle A space m e e t s space B C space a t space E space a n d space t h e space g i v e n space c i r c l e space a t space F.
D F space a n d space B F space a r e space j o i n e d.
T o space p r o v e minus
left parenthesis i right parenthesis space E F equals F C
left parenthesis i i right parenthesis space B F equals D F
P r o o f minus A B C D space i s space a space c y c l i c space q u a d r i l a t e r a l space a n d space A D parallel to B C
because A F space i s space t h e space b i s e c t o r space o f space angle A comma angle B A F equals angle D A F
A l s o comma space angle D A E equals angle B A E
angle D A E equals angle A E B space left square bracket A l t e r n a t e space a n g l e s right square bracket
left parenthesis i right parenthesis I n space increment space A B E comma space angle A B E equals 180 degree minus 2 angle A E B
angle C E F equals angle A E B space left square bracket V e r t i c a l l y space O p p o s i t e space a n g l e s right square bracket
angle A D C equals 180 degree minus angle A B C equals 180 degree minus left parenthesis 180 degree minus 2 angle A E B right parenthesis
angle A D C equals 2 angle A E B
angle A F C equals 180 degree minus angle A D C
space space space space space space space space space space space space space equals 180 degree minus 2 angle A E B space left square bracket S i n c e space A D C F space i s space a space c y c l i c space q u a d r i l a t e r a l right square bracket
angle E C F equals 180 degree minus left parenthesis angle A F C plus angle C E F right parenthesis
space space space space space space space space space space space space equals 180 degree minus left parenthesis 180 degree minus 2 angle A E B plus angle A E B right parenthesis
space space space space space space space space space space space space equals angle A E B
therefore E C space equals space E F
left parenthesis i i right parenthesis therefore A r c space B F equals A r c space D F space space space space space space space space space space space space space space space space space space left square bracket E q u a l space a r c s space s u b t e n d s space e q u a l space a n g l e s right square bracket
rightwards double arrow B F equals D F space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket E q u a l space a r c s space h a v e space e q u a l space c h o r d s right square bracket

Solution 19




G i v e n minus I n space a space c i r c l e comma space A B C D space i s space a space c y c l i c space q u a d r i l a t e r a l space A B space a n d space D C
a r e space p r o d u c e d space t o space m e e t space a t space E space a n d space B C space a n d space A D space a r e space p r o d u c e d space t o space m e e t space a t space F.
angle D C F : angle F : angle E equals 3 : 5 : 4
L e t space angle D C F equals 3 X comma angle F equals 5 x comma angle E equals 4 x
N o w comma space w e space h a v e space t o space f i n d comma angle A comma angle B comma angle C space A N D space angle D
I n space c y c l i c space q u a d. space A B C D comma space B C space i s space p r o d u c e d.
therefore angle A equals angle D C F equals 3 x
I n space increment C D F comma
E x t. angle C D A equals angle D C F plus angle F equals 3 x plus 5 x equals 8 x
I n space increment B C E comma
E x t. angle A B C equals angle B C E plus angle E space space space space space space space space left square bracket angle B C E equals angle D C F comma v e r t i c a l l y space o p p o s i t e space a n g l e s right square bracket
equals angle D C F plus angle E
equals 3 x plus 4 x equals 7 x
N o w comma space i n space c y c l i c space q u a d. A B C D comma
sin c e comma angle B plus angle D equals 180 degree
space space space space space space space space space space space space space space space space space space space space space left square bracket S i n c e space s u m space o f space o p p o s i t e space o f space a space c y c l i c space q u a d r i l a t e r a l space a r e space s u p p l e m e n t a r y right square bracket
rightwards double arrow 7 x plus 8 x equals 180 degree
rightwards double arrow 15 x equals 180 degree
rightwards double arrow x equals fraction numerator 180 degree over denominator 15 end fraction equals 12 degree
therefore angle A equals 3 x equals 3 cross times 12 degree equals 36 degree
angle B equals 7 x equals 7 cross times 12 degree equals 84 degree
angle C equals 180 degree minus angle A equals 180 degree minus 36 degree equals 144 degree
angle D equals 8 x equals 8 cross times 12 degree equals 96 degree


Solution 20