easy plane xadddgeometry

8/9/2019 Easy Plane XADDDGeometry

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Geometry - Plane Figures Problems and Solutions Plane fgures, solved problems, examples 1,Example: The area o a !ir!le is " !m# greater then the area o the s$uare ins!ribed into the!ir!le% Then, the area o the !ir!le is&Solution: Given '!ir!le ( 's$uare ) "% '!ir!le ( &

#%Example: *nto a $uarter o a !ir!le +ith the radius 1 ) # ins!ribed is a !ir!le, fnd the radiuso the ins!ribed !ir!le%Solution: Given ( 1 ) #% r ( &

.%Example: /hi!h polygon have t+o times more diagonals then sides&Solution: The number o diagonals in a polygon,

+here n is the number o polygon sides%

+heren

is the number o polygonsides%

0%Example: Given is a regular hexagon +ith the side a into +hi!h is ins!ribed a !ir!le, fnd thearea bet+een the hexagon and the !ir!le%Solution: Given a% ' ( &

% Example: * the ratio o the greater angle bet+een diagonals o a re!tangle to the smallerangle is # : 1 and a 2 b then, the ratio a : b o sides o the re!tangle is&Solution: Given 3 : 41567 - 38 ( # : 1 and a 2 b% a : b ( &



"% Example: *nto a !ir!le ins!ribed is deltoid 49ite8% * one o its angles bet+een e$ual sidessubtends "67 then, the relation o the longer to the shorter diagonal is&Solution: Given a ( "67% e : ( &

%Example: ;etermine the least n 4the number o sides o a regular polygon8, or +hi!h theinterior angle o a regular polygon is greater then 1".7%Solution: Given a 2 1".7% n ( &

The interior angle o a regular polygon,

5%Example: ;eltoid +ith sides length o 1 and # is ins!ribed into a !ir!le% The ratio o greater tothe shorter diagonal is&Solution: Given a ( # and b ( 1% e : ( &

<%Example: =ne side o a parallelogram is 0 !m, other " !m, and the diagonal 0 # !m, length o the other diagonal is&Solution: Given a ( " !m, b ( 0 !m and d1 ( 0s$uare root o # !m % d# ( &



16%Example: Find the area o the rhomb 4diamond8 +ith the side o length #6 !m and thediagonal o length .# !m%Solution: Given a ( #6 !m and e ( .# !m% ' rhombus ( &

11%Example: >ength o the side o the right de!agon is 1#% Find the radius o its !ir!um!ir!le%Solution: Given n ( 16 and a ( 1#% r ( &

1#%Example: *n a !ir!le +ith the radius # !m, on its !hord # # !m long ta9en as a diameter,!onstru!ted is another !ir!le% The area o the part o that !ir!le, +hi!h is outside o the greater!ir!le, is&Solution: Given r ( # !m and t ( # # !m% ' ( &



1.%Example: ;istan!e o !enters o !ir!les +ith radii, #6 !m and 16 !m, is 6 !m% Their !ommonexternal tangent tou!hes one !ir!le at a point ;1 and other at ;#% Find the length o the segment;1;#%Solution: Given r1 ( #6 !m, r# ( 16 !m and S1S# ( 6 !m % ;1;# ( &

10%Example: The !hord o a !ir!le is long 1# !m and !uts o the !ir!le segment +hose height is .!m% Find the radius o the !ir!le% ?eight o the segment e$uals the di@eren!e bet+een the radiuso the !ir!le and the distan!e o the !hord rom the !ir!leAs !enter%Solution: Given t ( 1# and h ( .% r ( &

1 %Example: The perimeter o a !ir!le is divided to the our parts +hose ar!s relate as 1: # : 0 :, +hat are the values o the !orresponding !entral angles&Solution: Given ar!s relationship, 1: # : 0 : % the !orresponding !entral angles &



1"% Example: *nto a !ir!le o the radius r ins!ribed is a s$uare, to the s$uare ins!ribed is a !ir!leand into the !ir!le ins!ribed is a s$uare, fnd the length o the side o the latter s$uare%Solution: Given r% a1 ( &

1 %Example: *nto a se!tor +ith ( < and a ( "67 ins!ribed is a !ir!le, fnd the radius o the!ir!le%Solution: Given ( < and a ( "67% r ( &

15%Example: T+o !ir!les +ith radii . !m and 1 !m tou!h externally% The distan!e o the point oits !onta!t rom their !ommon external tangent is&Solution: Given ( . !m and r ( 1 !m% d ( &



1<%Example: The ratio o the area o the !ir!um!ir!le and the area o the ins!ribed regulardode!agon is&Solution: Given n ( 1#% ' !ir!le : '1# ( &

#6% Example: *n a !ir!le the !hord 'B subtends the !entral angle a and the !hord C; subtendsthe !entral angle b, i given are, d4'B8 ( a and angles a and b, then d4C;8 is&Solution: Given d4'B8 ( a, a and b% d4C;8 ( &

#1%Example: The area o the se!tor ' ( "0 and the perimeter P ( .#, fnd the length o its ar!%Solution: Given ' ( "0 and P ( .#% ar! l ( &

##%Example: Bases o an isos!eles trapeDium are, #1 !m and < !m, and its height 5 !m% Theradius o its !ir!um!ir!le is&Solution: Given a ( #1 !m, ! ( < !m and h ( 5 !m% r ( &



#.%Example: Find the area o the interse!tion o t+o !ir!les +ith the same radius r ( 5 +hose!enters lay on ea!h other !ir!um eren!es%Solution: Given r ( 5% ' interse!tion ( &

#0%Example: The area o an annulus e$uals the $uarter o the area o the inner !ir!le% The ratioo the radius o the inner !ir!le to the radius o the outer !ir!le is&Solution: Given 'anulus ( 1 0 'r#% r# : r1 ( &

# %Example: The ratio o the radius o the in!ir!le to the radius o the !ir!um!ir!le o a righttriangle +ith legs, a ( . and b ( 0, is&Solution: Given a ( . and b ( 0% r : ( &

#"%Example: 'n isos!eles trapeDoid +ith the height has perpendi!ular diagonals% Find the areao the trapeDoid%Solution: Given h ( % ' trapeDoid ( &



# %Example: The ratio o the radius o the in!ir!le to the radius o the !ir!um!ir!le o a regularheptagon 4n ( 8 is&Solution: Given n ( % r : ( &

#5%Example: Cir!um eren!e o a se!tor is e$ual to double diameter 0r o the !orresponding!ir!le% The area o the se!tor is&Solution: Given P ( 0r% ' se!tor ( &

#<%Example: From the midpoint o the side o a s$uare o the length # !ir!ums!ribed is the !ir!leo the radius #% The area o the interse!tion o the s$uare and the !ir!le is&Solution: Given a ( r ( #% ' interse!tion ( &



.6%Example: Through the !entroid o a triangle 'BC, parallel to the side 'B, dra+n is a line l+hi!h interse!ts the side 'C at the point '1 and the side BC at B1% * the area o the triangle 'BCis '; ( # !m# fnd the area o the trapeDoid 'BB1'1 %Solution: Given '; ( # !m#% ' trapeDoid 'B'1B1 ( &

.1% From the right triangle 'BC sho+n belo+, 'B (06 !m and BC ( .6 !m% Points E and F are pro3e!tions o point ; rom hypotenuse 'C to the perpendi!ularlegs 'B and BC, respe!tively% ?o+ ar is ; rom 'B so that length EF is minimal&

EF and B; are diagonals o re!tangle EBF;, thus, EF ( B;% >ength o EF is minimal only i B; isperpendi!ular to 'C 4B; as altitude through B o triangle 'BC8% There ore,

easy plane xadddgeometry

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