selected topics in math

8/20/2019 selected topics in math

1/13

Mathematics

(algebra)

Special Products and Factoring

1.

2.

3.

4.

5.

6.

7.

Factoring Polynomials

1.

2.

3.

4.

5.

6.

7.

8.

Problem 10Given that , whereand are nonzero real number,!nd

the value o" .

Solution

#rom the given$ and

%hu,

answer


2/13

Direct Variation / DirectlyProportional

& i dire'tl& roortional to , & * $

+ 'ontant o" roortionalit&& varie dire'tl& a i anothertatement e-uivalent to the abovetatement.

Inverse Variation / DirectlyProportional

& i inverel& roortional to , & *1$

+ 'ontant o" roortionalit&& varie inverel& with hold theame meaning a the enten'eabove.

Joint Variation / JointlyProportional

& i dire'tl& roortional to and z$

& i dire'tl& roortional to andinverel& roortional to z$

+ 'ontant o" roortionalit&

Variation to nth poer o! " andmth poer o! #

& i dire'tl& roortional to the -uareo" and varie inverel& to the 'ube o" z$

+ 'ontant o" roortionalit&

(%rigonometr&)

Spherical $riangle/n& e'tion made b& a 'utting lanethat ae through a here i 'ir'le./ great 'ir'le i "ormed when the'utting lane ae through the'enter o" the here. 0heri'altriangle i a triangle bounded b& ar' o" great 'ir'le o" a here.

ote that "or heri'al triangle, idea, b, and ' are uuall& in angular unit./nd li+e lane triangle, angle /, ,and are alo in angular unit.

Sum o! interior angles o! sphericaltriangle %he um o" the interior angle o" aheri'al triangle i greater than 18and le than 54.


3/13

%rea o! spherical triangle %he area o" a heri'al triangle onthe ur"a'e o" the here o" radiu i given b& the "ormula

here i the heri'al e'e indegree.

Spherical e"cess

or

here

Spherical de!ect

ote$9n heri'al trigonometr&, earth i

aumed to be a er"e't here. :neminute ( 1;) o" ar' "rom the 'enter o" the earth ha a ditan'e e-uivalent toone (1) nauti'al mile (68 "eet) onthe ar' o" great 'ir'le on the ur"a'e o" the earth.

1 minute o" ar' 1 nauti'al mile1 nauti'al mile 68 "eet1 tatute mile 528 "eet1 +not 1 nauti'al mile er hour

Solution o! right spherical triangleith an& two -uantitie given (three-uantitie i" the right angle i'ounted), an& right heri'al triangle'an be olved b& "ollowing theaier


4/13

0heri'al triangle 'an have one or twoor three A interior angle. 0heri'altriangle i aid to be right i" onl& oneo" it in'luded angle i e-ual to A. %riangle with more than one Aangle are obli-ue.

De)nition o! obli*ue sphericaltriangle0heri'al triangle are aid to beobli-ue i" none o" it in'luded angle iA or two or three o" it in'ludedangle are A. 0heri'al triangle withonl& one in'luded angle e-ual to A ia right triangle.

Sine la

+osine la !or sides

+osine la !or angles

&apier,s analogies

Summary o! $rigonometric

Identities

-asic Identities

1.

2.

3.

4.

Pythagorean Identities

1.2.3.

Sum and Di.erence o! $o %ngles

1.

2.

3.

4.

5.

6.

Double %ngle Formulas

1.2.

3.

http://www.mathalino.com/reviewer/spherical-trigonometry/oblique-spherical-trianglehttp://www.mathalino.com/reviewer/spherical-trigonometry/right-spherical-trianglehttp://www.mathalino.com/reviewer/spherical-trigonometry/right-spherical-trianglehttp://www.mathalino.com/reviewer/spherical-trigonometry/oblique-spherical-triangle


5/13

al! %ngle Formulas

1.

2.3.

Problem 109n a triangle /, i"

,!nd the value o" angle .

Solution

Bideli'+ here to how or hide theolution

Given$

#rom oine law$

0ubtitute to the given e-uation$

C -uation (1)

-uation (1) ati!e the >&thagoreantheorem "or a right triangle whoeerendi'ular ide are and andh&otenue . %hu, angle .answer

eometry2

De)nition o! a $riangle

%riangle i a 'loed !gure bounded b&three traight line 'alled ide. 9t 'analo be de!ned a ol&gon o" threeide.

%rea o! triangle %he area o" the triangle i given b& the

"ollowing "ormula$Given the bae and the altitude

Given two ide and in'luded angle

http://www.mathalino.com/reviewer/plane-trigonometry/10-solving-angle-triangle-abchttp://www.mathalino.com/reviewer/plane-trigonometry/10-solving-angle-triangle-abchttp://www.mathalino.com/reviewer/plane-trigonometry/10-solving-angle-triangle-abchttp://www.mathalino.com/reviewer/plane-trigonometry/10-solving-angle-triangle-abc


6/13

Given three ide (ee the derivationo" Bero; "ormula)

where, 'alled theemi=erimeter.

Given one ide and three angle (a&angle /, , and , and ide b aregiven)

Center of a triangle

This page will define the following:

incenter, circumcenter, orthocenter,centroid, and Euler line.

Incenter

Incenter is the center of the inscribed

circle (incircle) of the triangle, it is the

point of intersection of the angle

bisectors of the triangle.

The radius of incircle is given by the

formula

where t ! area of the triangle and s !

" (a # b # c). $ee the derivation of

formula for radius of incircle.

Circumcenter

Circumcenter is the point of intersection

of perpendicular bisectors of the

triangle. It is also the center of the

circumscribing circle (circumcircle).

s you can see in the figure above,

circumcenter can be inside or outside

the triangle. In the case of the right

triangle, circumcenter is at the midpoint

of the hypotenuse. %iven the area of the

triangle t, the radius of the

circumscribing circle is given by the

formula

&ou may want to ta'e a loo' for

the derivation of formula for radius of

circumcircle.

Orthocenter

rthocenter of the triangle is the point

of intersection of the altitudes. i'e

circumcenter, it can be inside or outside

the triangle as shown in the figure

below.

http://www.mathalino.com/reviewer/derivation-formulas/derivation-heron-s-hero-s-formula-area-trianglehttp://www.mathalino.com/reviewer/derivation-formulas/derivation-heron-s-hero-s-formula-area-trianglehttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#angle-bisectorhttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#angle-bisectorhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-incirclehttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-incirclehttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#perpendicular-bisectorhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-circumcirclehttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-circumcirclehttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#altitudehttp://www.mathalino.com/reviewer/derivation-formulas/derivation-heron-s-hero-s-formula-area-trianglehttp://www.mathalino.com/reviewer/derivation-formulas/derivation-heron-s-hero-s-formula-area-trianglehttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#angle-bisectorhttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#angle-bisectorhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-incirclehttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-incirclehttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#perpendicular-bisectorhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-circumcirclehttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-formula-for-radius-of-circumcirclehttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#altitude


7/13

Centroid

The point of intersection of

the medians is the centroid of the

triangle. Centroid is the geometric

center of a plane figure.

Euler Line

The line that would pass through the

orthocenter, circumcenter, and centroid

of the triangle is called the Euler line.

Side0ide o" a triangle i a line egmentthat 'onne't two verti'e. %riangleha three ide, it i denoted b& a, b,and ' in the !gure below.

Verte"Derte i the oint o" intere'tion o"two ide o" triangle. %he threeverti'e o" the triangle are denoted b&/, , and in the !gure below. oti'ethat the ooite o" verte / i ide a,

ooite to verte i ide , andooite to verte i ide '.

Included %ngle or Verte" %ngle9n'luded angle i the angle ubtended

b& two ide at the verte o" thetriangle. 9t i alo 'alled verte angle.#or 'onvenien'e, ea'h in'luded angleha the ame notation to that o" theverte, ie. angle / i the in'ludedangle at verte /, and o on. %he umo" the in'luded angle o" the trianglei alwa& e-ual to 18.

%ltitude3 h/ltitude i a line "rom verteerendi'ular to the ooite ide.

%he altitude o" the triangle willintere't at a 'ommon oint'alled ortho'enter.

9" ide a, b, and ' are +nown, olve

one o" the angle uing oineEaw then olve the altitude o" thetriangle b& "un'tion o" a righttriangle. 9" the area o" the triangle /t i+nown, the "ollowing "ormula areue"ul in olving "or the altitude.

http://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#medianhttp://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#orthocenterhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/plane-trigonometry/functions-of-a-right-trianglehttp://www.mathalino.com/reviewer/plane-trigonometry/functions-of-a-right-trianglehttp://www.mathalino.com/reviewer/plane-geometry/properties-of-a-triangle#medianhttp://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#orthocenterhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/plane-trigonometry/functions-of-a-right-trianglehttp://www.mathalino.com/reviewer/plane-trigonometry/functions-of-a-right-triangle


8/13

-ase %he bae o" the triangle i relative towhi'h altitude i being 'onidered.#igure below how the bae o" the

triangle and it 'orreonding altitude.

• 9" h/ i ta+en a altitude thenide a i the bae

• 9" h i ta+en a altitude thenide b i the bae

• 9" h i ta+en a altitude thenide ' i the bae

Median3 mFedian o" the triangle i a line "romverte to the midoint o" the ooiteide. / triangle ha three median,and thee three will intere't atthe 'entroid. %he !gure below howthe median through / denoted b& m/.

Given three ide o" the triangle, themedian 'an be olved b& two te.

1. 0olve "or one in'luded angle,a& angle , uing oine Eaw.#rom the !gure above, olve "or in triangle /.

2. ing triangle /?, determinethe median through / b& oineEaw.

%he "ormula below, though notre'ommended, 'an be ued to olve

"or the length o" the median.

here m/, m, and m are medianthrough /, , and , ree'tivel&.

%ngle -isector

/ngle bie'tor o" a triangle i a linethat divide one in'luded angle intotwo e-ual angle. 9t i drawn "romverte to the ooite ide o" thetriangle. 0in'e there are threein'luded angle o" the triangle, thereare alo three angle bie'tor, andthee three will intere't atthe in'enter. %he !gure hown belowi the bie'tor o" angle /, it length"rom verte / to ide a i denoted ab/.

%he length o" angle bie'tor i givenb& the "ollowing "ormula$

http://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#centroidhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#incenterhttp://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#centroidhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#incenter


9/13

where 'alled the emi=erimeter and b/, b, and b arebie'tor o" angle /, , and ,ree'tivel&. %he given "ormula arenot worth memorizing "or i" &ou aregiven three ide, &ou 'an eail& olvethe length o" angle bie'tor b& uingthe oine and 0ine Eaw.

Perpendicular -isector>erendi'ular bie'tor o" the trianglei a erendi'ular line that 'roethrough midoint o" the ide o" thetriangle. %he three erendi'ularbie'tor are worth noting "or itintere't at the 'enter o" the'ir'um'ribing 'ir'le o" the triangle. %he oint o" intere'tion i 'alledthe 'ir'um'enter. %he !gure belowhow the erendi'ular bie'torthrough ide b.

4uadrilateral

4uadrilateral is a polygon o! !oursides and !our vertices5 It is alsocalled tetragon and *uadrangle5In the triangle3 the sum o! theinterior angles is 16078 !or*uadrilaterals the sum o! the

interior angles is alays e*ual to9:07

+lassi)cations o! 4uadrilaterals$here are to broadclassi)cations o!*uadrilaterals8 simple and comple

x 5 $he sides o! simple*uadrilaterals do not cross eachother hile to sides o! comple"*uadrilaterals cross each other5

Simple *uadrilaterals are !urtherclassi)ed intoto; convex and concave5 +onve"i! none o! the sides pass throughthe *uadrilateral hen prolongedhile concave i! the prolongationo! any one side ill pass insidethe *uadrilateral5

$he !olloing !ormulas areapplicable only to conve"*uadrilaterals5

eneral 4uadrilateral

%ny conve" *uadrilateral can usethe !olloing !ormulas;

Perimeter3 P applicable to all*uadrilaterals3 simple andcomple"2

http://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-sine-lawhttp://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#circumcenterhttp://www.mathalino.com/reviewer/plane-geometry/trianglehttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-cosine-lawhttp://www.mathalino.com/reviewer/derivation-of-formulas/derivation-of-sine-lawhttp://www.mathalino.com/reviewer/plane-geometry/centers-of-a-triangle#circumcenterhttp://www.mathalino.com/reviewer/plane-geometry/triangle


10/13

%rea3 %

heres < semi perimeter < =P> < = % ? +2 or > < = - ? D2

$he area can also be e"pressed interms o! diagonals d1 and d@

Aength o! one side !or ma"imumarea o! trape#oid solution byeometry2

Problem-+ o! trape#oid %-+D is tangentat any point on circular arc DBhose center is C5 Find the lengtho! -+ so that the area o! %-+D isma"imum5

Solution

%s described by %le"ander-ogomolny o! cuttheEnot5org3!or ma"imum area o! trape#oid3the point o! tangency should be at

the midline o! %- and D+3 thus is the midpoint o! -+5

From the )gure;

For the angle theta;

answer

$he +yclic 4uadrilateral

/ -uadrilateral i aid to be '&'li' i" itverti'e all lie on a 'ir'le. 9n '&'li'-uadrilateral, the um o" two ooiteangle i 18 (or H radian)I in otherword, the two ooite angle areulementar&.

http://www.cut-the-knot.org/Optimization/CrookedTrapezoid.shtml#solutionhttp://www.cut-the-knot.org/Optimization/CrookedTrapezoid.shtml#solutionhttp://www.cut-the-knot.org/Optimization/CrookedTrapezoid.shtml#solutionhttp://www.cut-the-knot.org/Optimization/CrookedTrapezoid.shtml#solution


11/13

%he area o" '&'li' -uadrilateral igiven b&

0ee the derivation o" area o" '&'li'-uadrilateral "or ro"ound detail.

Ptolemy,s $heorem !or +yclic4uadrilateral#or an& '&'li' -uadrilateral, therodu't o" the diagonal i e-ual tothe um o" the rodu't o" non=ad@a'ent ide. 9n other word

4uadrilateral +ircumscribing a+ircle

Juadrilateral 'ir'um'ribing a 'ir'le(alo 'alled tangential -uadrilateral) ia -uadrangle whoe ide are tangentto a 'ir'le inide it.

/rea,

here r radiu o" in'ribed 'ir'leand emi=erimeter (a K b K 'K d)2

Derivation !or area

Eet : and r be the 'enter and radiu o" the in'ribed 'ir'le, ree'tivel&.

%otal area

(okay!)

Some Enon properties

1. :oite ide ubtendulementar& angle at the'enter o" in'ribed 'ir'le. #romthe !gure above, L/: KL:? 18 and L/:? KL: 18.

http://www.mathalino.com/reviewer/derivation-formulas/derivation-formula-area-cyclic-quadrilateralhttp://www.mathalino.com/reviewer/derivation-formulas/derivation-formula-area-cyclic-quadrilateralhttp://www.mathalino.com/reviewer/derivation-formulas/derivation-formula-area-cyclic-quadrilateralhttp://www.mathalino.com/reviewer/derivation-formulas/derivation-formula-area-cyclic-quadrilateral


12/13

2. %he area 'an be divided into"our +ite. 0ee !gure below.

3. 9" the ooite angle are e-ual(/ and ?), it i a

rhombu.

$he circle2

$he !olloing areshort descriptions o! the circle shonbelo5

$angent is a line thatould pass through

one point on the circle5Secant is a line that ouldpass through to points onthe circle5+hord is a secant thatould terminate on thecircle itsel!5Diameter3 d is a chord thatpasses through the centero! the circle5(adius3 r is onehal! o! thediameter5

%rea o! the circle

+ircum!erence o! the circle

Sector o! a +ircle

Aength o! arc;

%rea o! the sector;

Segment o! a +ircle


13/13

%rea o! circular segment

ith s

%rea o! circular segmentith s = c;

secant

(elationship -eteen +entral%ngle and Inscribed %ngle

entral angle /ngle ubtended b&an ar' o" the 'ir'le "rom the 'enter o"the 'ir'le.9n'ribed angle /ngle ubtended b&an ar' o" the 'ir'le "rom an& oint onthe 'ir'um"eren'e o" the 'ir'le. /lo

'alled circumferentialangle and peripheral angle.

#igure below how a 'entral angleand in'ribed angle inter'eting theame ar' /. %he relationhi betweenthe two i given b&

i" and onl& i" both angle inter'etedthe ame ar'. 9n the !gurebelow, M and Ninter'eted the amear' /.
http://www.mathalino.com/reviewer/plane-geometry/relationship-between-central-angle-and-inscribed-anglehttp://www.mathalino.com/reviewer/plane-geometry/relationship-between-central-angle-and-inscribed-anglehttp://www.mathalino.com/reviewer/plane-geometry/relationship-between-central-angle-and-inscribed-anglehttp://www.mathalino.com/reviewer/plane-geometry/relationship-between-central-angle-and-inscribed-angle

Documents

selected topics in math