1) AB is a line segment and M is its mid-point .Three semi circles are drawn with AM, MB and AB as diameters on the same side of the line AB. A circle with radius r units is drawn so that it touches all the three semi circles .Show that AB=6 times r.

2) ABC is a right triangle with angle B=90 degree. A circle with BC as diameter meets hypotenuse AC at point D .Prove that (i) AC*AD=AB*AB. (ii)BD*BD=AD*DC.

3)In a square ABCD, its diagonals AC and BD intersect each other at point O. The bisectors of angle DAO meets BD at point M and the bisector of angle ABD meets AC at N and AM at L. Show that:(i)angle ONL+ angle  OML =180 degree. (ii) angle BAM =angle BMA. (iii)ALOB is a cyclic quadrilateral.

 4)In the given figure, PAT is a tangent to the circle with centre O, at point A on its circumference and is parallel to the chord BC. If CDQ is a line segment, show that :(i) angle BAP =angle ADQ.(ii) angle AOB = angle 2* angle ADQ.(iii) angle ADQ =angle ADB.

 

Asked by ashutoshtripathy2000 | 27th Sep, 2015, 09:17: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
L e t space t h e space l e n g t h space o f space t h e space s e g m e n t space A B equals 2 R L e t space O space b e space t h e space c e n t r e space a n d space r space b e space t h e space r a d i u s space o f space t h e space c i r c l e space w h i c h space t o u c h e s space a l l space t h e space o t h e r space s e m i space c i r c l e s space i n space P comma Q space a n d space R. S i n c e space M space i s space t h e space m i d p o i n t space o f space A B comma space r a d i u s space o f space t h e space b i g g e r space s e m i c i r c l e space i s space R space a n d space r a d i u s space o f space s m a l l e r space s e m i c i r c l e s equals R over 2. T h u s comma space A M equals M B equals R space a n d space A L equals L M equals M N equals N B equals R over 2 A l s o comma space P L equals Q N equals R over 2 N o w space j o i n space O L space a n d space O N. W E space k n o w space t h a t space i f space t w o space c i r c l e s space t o u c h space e a c h space o t h e r space i n t e r n a l l y space o r space e x t e r n a l l y space t h e space p o i n t space o f space c o n t a c t space l i e s space o n space t h e l i n e space j o i n i n g space t h e i r space c e n t r e s. T h u s comma space t h e space p o i n t s space O P L comma space O Q N space a n d space R O M space a r e space c o l l i n e a r. N o w space c o n s i d e r comma space O L equals O P plus P L equals r plus R over 2 a n d space O N equals O Q plus Q N equals r plus R over 2 T h e r e f o r e comma space O L equals O N equals space r plus R over 2 T h u s comma space t h e space t r i a n g l e space triangle O L N space i s space a n space i s o s c e l e s space t r i a n g l e. A n d space M space i s space t h e space m i d p o i n t space o f space A B space a n d space h e n c e space M space i s space t h e space m i d p o i n t space o f space L N. rightwards double arrow O M perpendicular L N C o n s i d e r space t h e space r i g h t space t r i a n g l e space M O N. B y space a p p l y i n g space P y t h a g o r a s space T h e o r e m comma space w e space h a v e comma O N squared equals O M squared plus M N squared rightwards double arrow open parentheses r plus R over 2 close parentheses squared equals open parentheses R M minus O R close parentheses squared plus open parentheses R over 2 close parentheses squared rightwards double arrow open parentheses r plus R over 2 close parentheses squared equals open parentheses R minus r close parentheses squared plus R squared over 4 rightwards double arrow r squared plus R squared over 4 plus fraction numerator 2 R r over denominator 2 end fraction equals R squared plus r squared minus 2 R r plus R squared over 4 rightwards double arrow R r equals R squared minus 2 R r rightwards double arrow R squared equals 3 R r rightwards double arrow R equals 3 r rightwards double arrow 2 R equals 6 r rightwards double arrow A B equals 6 r rightwards double arrow r equals fraction numerator A B over denominator 6 end fraction space u n i t s

Answered by Vimala Ramamurthy | 28th Sep, 2015, 09:41: AM

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