- [UNDEFINED Terms] [11 Point; notation Point A is labeled wi th a capi tal letter, A in thi s case [2J Line; notation Line KM is labeled cither KM or MK or li ne I [3J Plane; notation Plane N is labeled eithe r plane 11 or pl a ne AB C if points A , B, and C are o n plane /I 1'.' [TRANSVERSAL LINE Angles] [1J The set of collinea r points going in o ne d irection from onc poin t (the endpoi n t of the y) on a li ne ; notati on: AS w here A is the endpoint; not ice lIB BA because they have differe nt endpoin ts a nd co nta in d ifferent points on the li ne [2J Oppo It r y are collinear, share o nly a common e ndpoin t an d go in o ppos ite direc tions * [DEFINED Terms] [GENERAL Terms] [1J [21 [3J [4J [5J rlCJI Jl" ) Shapes are the same shape an d s ize (II 11"1' Shapes arc the same shape, but can be different sizes i.q Jdl Sets of po ints or numerica l measurements arc exactly t he same JIllor! Describes the result when all of the points are put together (II rs C 101) Describes the points where ind icated sha pes touch [6J p c The set of all points i( C 111"'4' [1J Collinear points are on the sam e line [2J Non colill c I points are not on the same line [3J Inter II lines have one and on ly one point in common [4J PIp I dlcul, r lines intersect and form 90° angles at the intersection; [5J k w lines are not in the same plane , never touch, and go In different directions [61 Tlan versa I lines intersect two or more co-pl a nar lines at diffe re nt points [7J P r II I lines are co-planar (in the same plane) , share no points in common, not intersect , go in the same direction and never touch; II [ANGLES] [1J Th e union of two rays that share o ne and only on e point, the endpoin t of the rays a T he Sid of the angl e are the rays a nd the v r x is the endpoint of the rays b The Interrol is all t he points be tween the two sides of the angle c LABe where B is the vertex or simply IB if there is onl y one angl e w ith vertex B [2J Ov Ilapping angl s share some comm on inte rio r points [3J An ac Jte angle mea sures less than 90° [4J An obtuse angle measures more than 90° [5J A right angle measures exactly 90°; it is indicated on diagrams by drawing a square in the corner by the vertex of the angle [6] A straight angle measures exactly 180° [7J Complem ntary ngles are two a ngles whose m easures total 90° [8J Supplementary angles a re two an g le s whose m easures tota l 180 [9J Vertical angl s are two ang les that share o nly a eommo n vertex a nd whose s ide s form lines [10JAdjacent angles are two angles t hat sha re e xaetly one vertex and one side, b ut no commo n inte rior points ; i.e., they not overlap [11J An ngle bisector is a ray or a line that contain s the vertex of the angle , is in the interior, a nd separates the a ngle into two adjacent angles with equal measures [LINE Segments] [1J T he set of any points on a line and al l of the collin ear poin ts betw een th e m ; AB where A and B are the en d poi nts of the Iine segment [2J The I CJth is the distance between the endpoints; it is a numerical va lu e; AB means the length of A B [3] T he n idpoint is a point exactl y in the midd le of the two endpoints [4] The bisector intersects a line segment at its m idpoint [5J T he perpendicular bls tor intersects a line segmen t at its midpoint and forms 90° angl es at the intersection [1J In erlor ilngll are form ed with thc rays from the li nes and the tra nsversal, such that the interior region s of the angles are located between the lines [2J Alt rn e II trior mgl c a rc inte r ior a ng le s w it h di fferent vertexes a nd interior regions on opposite sides of t he tran sversa l [3J ,1m lel II t nor ngle are inter ior a ngl e s wi th different vertexes an d inte rior reg ions on th e sa m e s ide o f the t ransversal [4J Ex I lor, r gl are formed with rays from the lines and the transversa l, such t hat thc in terior regio ns of the a ngle s are not bet w een t he lines [51 Altern t xt riar lnql a re exterio r a ngles with d ifferent vertexes and in terior regions on opposite sides of the tra nsversa l [6] Carr pondlng angl have differe nt vertexes ; the ir intcrior reg ions are on thc same side of the transversa l a nd in t he same positions relative to the lines a nd the transversa l; one of th e pair of corresponding angles is a n interi o r ang lc an d th e oth er is an exterior angle iIIr , = " m Z ~ [POLYGONS] [1J Polygons arc plan (flat), closed s hapes that are formed by line seg m ents tha t inter ect only at thcir endpoints a N ot T hcy are na m ed by listing the endpoints of th e line seg m ents in ord er, g oin g either clockwise or co untercl oc kwisc, sta rt in g at anyo ne of the endpoi nts b The sides are li ne egments c T he int riar is all of t he points e nclosed by the s ides d T he xterior is all o f the point · on the plane of the polygo n , but neithe r on t he si des nor in t he interio r e The vertic (or vertexes) are th e iIIr e ndpoi nts of the li ne segm e nts , f Inc ludc a ll the points on the s ides ( line segments) and the ve rtices g T he int rior angl o r a po lygon have " the same vert ices as the verticcs of the polygon , have side s tha t cont ain the sides o r the po lygo n, and have in teri o r reg ion s that co nta in the interior of the ~ po lygo n- every pol ygon has as many interior ang les as it has vertices = m Z r~ ,,= ~~ Polygo ns (continued) h Consecutive interior angles have ve rtices th at are endpoi nts of the same side of the polygon I The exterior angles are formed when the sides o f the po lygon are exte nded; each has a ve rte x and one si de that are als o a vertex a nd co ntain on e side of th e pol ygon; th e se cond si d e of the exterior an g le is the ex te nsio n of th e other polygon side contain ing the angle verte x; the interior of the exte rior angle is part of the exterior region of the polygon; ex teri o r an g les are s upp le ments of their adjacent interior angles j Diagonals of a polygon are line segments wit h endp o ints that are vertices of th e po lygo n , but the di agon als are not sides of the polygo n [2) CONCAVE polygons have at lea st one interio r angle mea suring more th an 180 • [3) CONVEX polyg on s have no inter ior an gles more th an 180 and all interior angles each measure less than 180° [4) REGULAR po lygons have all si de lengths eq ua l and a ll interi or an gl e meas ure s equ a l [5) CLASSIFICATIONS OF POLYGONS a Classifi ed by the number of si de s; eq ua l to the nu mber of vertices b Th~ side len g ths an d a ng le m eas ure s arc not nec essa rily eq ual un les s the word " regular" is also used to na me the po lygon c Categories • • • • • • • • • Triangle s have three sides Q uadrilate ls hav e four sides Pentagons have five sides Hexagons have six si de s Heptagons have se ven sides Oc tagons have e ight side s Nonagons have nine si des Decago ns have ten si de s n- go ns have n s ides [6) SPECIAL POLYGONS a Triangles • Po lygons with sides and vertic e s; the symbol fo r a triangle is ~; tri angle ABC is written ~AB C • An altitude (height) is a line segment with a vertex of the triangle as one endpoint and the point on the line containing the opposite side of t he triangle where the altitude is perp e n d icular to that line; every triangl e has altitudes • A base is a side of the triangle on the line perpendic ular to an altitude; every triangle has bases • Formula for area A =tllb or iI=thb where a=altitude , b=base or where h=heig ht (altitude), b=base • necessarily the side on the bottom of th e tria ngl e c] Th e base angles of a n iso sc eles trian gle have the base contai ned in one of th eir sides ; they are always equal in measu re • Right Triangles a] The hypotenuse is oppo si te the ri gh t a ngle an d is the lo ngest sid e b ]Th e legs arc th e sides that a rc not the hy poten use; the lin e seg ments contained in th e sides o f the right angle b Quadrilaterals • 4-s ided polygons • Ha ve d ia gonal s and vert ices Class~fied in ways, by side length s and by angle measurem ents a] When classified by side lengths: • Scalene have no side lengths=, • Isosceles have at least side lengths equal, • Equilateral have all side lengths equal; note it is also an isoscel es triangle b]When classified by angle measure ments: • Obtuse have ex actly one a ngle measurement more tha n 90 • Right have exact ly one angle measurement eq ual to 90° • Acute have all ang les less th an 90 ; note that if all angles are equal, then the tr iangle is called equiangular • Isosceles triangles a] T he vertex angle has s ide s contain ing the two congru e nt s id es of th e triangle b] The base is the side w ith a differen t length than the other two sides; no t • rap zOld have exac tl y one pa ir of para ll e l si des ; there is never more than one pair of para ll e l sides a] Para ll el sides : ba b] No n-para llel sides : legs c] The a ngles wi th vertices th at arc the en dpo int s o f the same base arc call ed ba angl d]lsosceles trapeZOids have legs that arc the sam e le ngth • Parallelograms have pairs of parallel s id es a] Rectangl s have ri ght ang les b]Rhombus s (s ing rh omb us) have sides eq ua l in len gth c] Squares have equal sides an d equal an gles ; therefore , eve ry squ are is bo th a recta ngle an d a rhombus [CIRCLES] [1) The set of points in a pla ne eq uid is tan t from the center of the ci rcle , which lies in the inter ior of the circl e and is no t a point on the circl e ; 360° [2) A radius is a line segment whose endpoints are the center ofthe circle and any poi nt on the ci rcle; the length of a radius is th e distance of each point fro m th e center [3) A chord is a li ne seg me nt w ho se endpoints arc po ints on the ci rcle [4] A diameter is a cho rd that contains the center of the circle; the length of a diameter IS the distance from on e poin t to another on the circle, goin g through the center 15) A secant is a li ne intersecting a circle in two points [6] A tangent is a line that is co-p lanar w ith a [9] An inscribed polygon has vertices th at circle and intersects it at o ne point o n ly, call ed th e point of tangency a A cornmon tangent is a lin e th at is tangent to co-planar circles • Common internal anqents intersect between the two circles • Common external tangents not intersect between the circl es b Two circ les are tangent when they are co-planar and share the same tangent line at the same point of tangency; they may be externally or internally tange nt [7) Equal Circles have equal-length radii [8) Concentric circles lie in the same plane and have the same cen ter are po ints on the circl e ; in th is sa ill e si tua tion, the ci rcl e is c ircumscribed about the po lygon 110]A circumscribed polygor has sides that arc segme nts of tan gents to the ci rcle; i.e , the s id es of the polygon each ta in exac t ly one poi n t on the circ le ; in th is same situati o n, the circle is insc ribed in the po lygo n [11) An arc is part ofa ci rc le a A erni ircl is a n arc wh ose en d points are the endpo ints o f a diamete r; 180°; exac tl y th ree points m ust be u.' cd to na me a scm iei rcle; notati o n: AiJC wh ere A a nd C are the endpo ints o f th e d ia mete r Through a point not on a line, exactly on e perpen dicular can be drawn to the line The shortest distance from any point to a line or to a plane is the pcrpcndicular distance Through a poi nt not on a line , exactly one parallel can be drawn to the line Parallel lines are everywhere the same d istancc apart If three or more parallel lines cut off equal segments on one transversal, then thcy cut o ff cqual segmcnt s on cvcry transversal they sharc A linc and a plan c are parallel if they not to uch or intcrsect Two or more planes are parallel if they not touch or intersect If two parallcl plancs arc both intcrsccted by a third plane, thcn thc lines of intersection are parallel If a point lies on the perpendicular bisector of a line segment, then the point is equidistant (equal distances) from the cndpoints of the line segment If a point is equidistant from the endpoints of a line segment , then thc point lies on thc perpendicular biscctor of the line segment To trisect a line segment, separate it into three other congruent (equal in length) line segments, such that the sum of the lengths of the three segments is equal to the length of the original line segment Angles are measured using a protractor and degree mca surements: There are 360 in a circle; placing the center of a protractor at th e vcrtcx of an anglc and counting the degree measure is like putting the vertex of the an g le at the center of a circle anc! comparing th e angle measure to the degrees of the circle If two angles are supplements of congruent angles, th en they are congruent Vertical angles are cong ruent and ve equal measures If a point lies on the bisector of an angle, th en the point is (equal distances) from the sides of the angle Distance from a point to a line is always the length o f the perpen dic ular line segment that has the point as one endpoint and a point on th e line as the other If a point is equidistant from the sides of an angle, then the point lies on the biseetor of the angle An angle is trisected by rays or lines that contain the vertex of the angle and separate the angle into three adjacent angle s (in pairs) that all have equal measures 9() ~ 0 If two angl es are compl e ments of the same angl e , then they arc equal In meas ure (con gruent) If two an g le s arc compl ements of congruent angles, then they are congruent If tw o angl es are supplements of th e same ang le , then they are congru ent b A minor arc length is less tha n the lengt h o f the se m ic ircl e; only two points may be used to name a minor' arc ; no tatio n: DE where D and E are the endpoin ts of the arc c A major arc length is m ore than th e le ng th of the semi c irc le ; exac t ly three points a rc use d to namc a major arc ; notation: FCfj wh e re F and H are the endpoints of the arc [12J A central angle vertex is the center of the c ircle with si des th at contain radi i of the circle [13J A inscribed-angle vertex is on a circle with sides that contain chord s of the circle [POSTULATES] Statements that have been accepted without formal proof [1J A line contains at least points, and any points locate exactly one line [2J Any non-collinear points locate exactly one plane [3] A line and one point not on the line locate exactly one plane [4J Any points locate at least one plane [5J If points of a line are in a plane, then the line is in the plane [6] If points are in a plane, then the line containing the points is also in the plane [7] If planes intersect, then the intersection is a line If two rays not int e rsect , the n th e union of the rays is s imply a ll o f th e points on both rays If two rays intersect in on e and only o ne point, but not at th e endp o int, th e n th e union is a ll of the points on both ray s; th e intersection is that o ne po int where th ey touch If two rays intersect in on e and onl y one point, the e ndpoint, th en th e uni o n is an angle; the intersection is th e endpo int ~ If tw o rays intersec t In mo re point, th e n the union IS a inte rsection is a line segm en t A B AB AB tha n one lin e; th e BA BA If line s a re pa ll e l, th e n th e a lt e rna te inte r ior angl e s o f a tra nsve rsa l are co ngruent If the a ltern ate in te ri o r a ng les o f a tran sversal are cong ru en t th en th e lin es a re para ll e l If lin es are para ll e l, then th e sa me s idc inte ri o r a ngl es o f a tra ns ve rsa l a re s uppl e men tary If th e sa m e -s id e inte ri o r a ngl es of a tran s versa l are s uppl e me ntary, the n the Iin es are para ll e l are pa ll e l, th e n th e If lin es corre spo nd in g a ng les of a tra nsve rsa l are congruent If the corres pondin g an g les o f a tra ns ve rs a l a re co ngr uen t, th e n t he lines are parall e l If lin e s a re parall e l, th e n th e a lte rn ate exte rior angle s of a tra n sv e rsal are congrue nt If th e al ternat e exte rior a ng les o f a tran sversa l are g ru e nt, then th e lin es are parallel If a transv ersal is perpe ndi cular to on e o f two parall e l lines , th e n it is al so perpen dicular to the other n m alternat Jr : 4- 6; 5- same-Side interior 4- 5; 3- correspond ng': s: 1- 5; 4- 8; 3- 7; 2- alternate exterior 1- 7; 2- The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180 degrees To find the measure of each interior angle of a regular polygon, find the sum of all of the interior angles and divide by the number of interior angles, thus , the formula (TI - 2) 180 TI The sum of the measures of the exterior angles of any convex polygon, using one exterior angle at each vertex, is 360° I r The 3-angle total measurement= 180° If two angle measurements of one triangle=two angle measurements of another triangle, then the measurements of the third angles are also= Each angle of an is 60° There can be no more than one right or obtuse angle in anyone triangle The acute angles of a right triangle arc complementary The measurement of an exterior angle= the sum of the measurements ofthe two remote (not having the same vertex as the exterior angle) interior angles If two sides of a triangle are equal , then the angles opposite to those sides are also equal; and, if two angles are equal , then the sides opposite those angles are also equal If two sides of one triangle are equal in length to two sides of another, but the included angle of one triangle is larger than the included angle of the other triangle, then the longer third side of the triangles is opposite the larger included angle of the triangles If two sides of one triangle arc equal to two sides of another, but the third side of one is longer than the third side of the other, then the larger included angle (included between the two equal sides) is opposite to the longer third side of the triangles If a line is parallel to one side and intersects the other two sides, then it divides those two sides proportionally, and creates similar triangles If a an angle of a triangle, it divides the opposite side into segments proportional to the other two sides The line segment that joins the midpoints of two sides of a triangle has two properties: It is to the third side, and It is of the third side '1 The bisectors of the angles of a triangle intersect in one point, which is equidistant from the sides The of the sides of a triangle intersect in one point, equidistant from the vertices The medians (line segments whose endpoints are one vertex of the triangle and the midpoint of the side opposite that vertex) of a triangle intersect in one point two-thirds of the distance from each vertex to the midpoint of the opposite side If two sides of a triangle are unequal in length , then the opposite angles arc unequal and the larger angle is opposite to the longer side; and conversely, if two angles of a triangle arc unequal , then the sides opposite those angles are unequal and the longer side is opposite the larger angle The sum of the lengths of any two sides is greater than the length of the third side; the difference of the lengths of any two sides is less than the length of the third side duct rn a l When an altitude is drawn to the hypotenuse of a right triangle The two triangles formed are similar to each other and to the original right triangle The altitude is the between the lengths of the two segments of the hypotenuse Each leg is the geometric mean between the hypotenuse and the length of the segmcnt of the hypotenuse adjacent (touches ) to the leg t In a right triangle, , where , and ' are the lengths of the legs and is the length of the hypotenuse If the square of the hypotenuse is equal to the sum of the squares of the other two sides, then the triangle is a If three sides of one triangle are congruent to three sides of another, then the triangles are congruent If two sides and the included angle of one triangle are congruent to two sides and the included angle of another then the triangles are congruent If two angles and the included side of one triangl e are congruent to two angl es and the included side of another, then the triangles are congruent If two angles and a non-included side of one triangle are congruent to the two corresponding angles and non included side of another, then the triangles are congruent If the hypotcnuse and one leg of a right triangle are congruent to the hypotenuse and the corresponding leg of another then the two right triangles are congruent If the square of the longest side is greater than the sum of the squares of the other two sides, then it is an triangle; if it is less than the sum of the squares of the other two sides, then it is an triangle In a 45-45-90 triangle, the legs have equal lengths and the length of the hypotenuse is 12 times the length of one of the legs In a 30-60-90 triangle, the length of the shortest leg is 1/ the length of the hypotenuse, and the length of the longer leg is 13 times the length of the shortest leg The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices If two angles of one triangle arc congruent to two angles of another then the triangles are similar (same sh ape but not necessarily the same size) If th e sides of one triangle are propor tional to th e corresponding sides of another, then the triangles are similar If two sides of one tri a ngle are propor tional to two sides of another and the included anglcs of each triangle are congruent, then the triangl es are similar An equilateral triangle is also equian gular; and, an equiangular triangle is also equilateral An equilateral triangle has three 60 degree angles The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base of the triangle sum mber idcs ~ ATERALS Z III D C T o P d e t The (the line segment whose endpoints are the midpoints of the non-parallel sides) is parallel to the bases, and its length is equal to half the sum of the lengths of the bases The may be calculated by averaging the length of the bases and mUltiplying by the height (altitude that is the length of the line segment that form s 90-degree angles with the bases); thus , the formula: A (hi ~h, )h (hi ~b, )h =~(hl +h, )h where the bases are b and b and the height is h Two angles with vertices that are the endpoints of the same le g of a trapezoid are All interior angle measures of all trapezoids total 360 The base ang les are congruent (has congruent leg s) Opposite angles are suppl e me ntary Opposite sides are parallel and congruent Opposite ang les arc congruent All inte rior a ngles total 360 C onsecutive interior an g les (thei r vertic es are endpoint s for th e same side) arc suppl eme ntary Di agonal s bi sect each o ther A quadril atera l is a para lle logram if: If interior angle s each eq ual 90 , th e n th e rhombu s is more spec ifi cally called a square The diagonal s arc perpendicular bi sectors of each othe r Each diagonal bi sects the pair of oppo s ite a ngl es who s e vertice s a rc th e endpo ints of the diagonal One pair of opposite sides is congruent and parallel Both pairs of opposite sides are congruent Both pairs of opposite angles are congruent The diagonals bisect each other The can be calculated by multiplying the base and the height; that is , A=bh=hb Since opposite sides are both parallel and equal, any side can be the base; the height (altitude) is any line segment perpendicular to the base whose endpoints arc on the base and the side opposite the base equal s ides and equal an g le s; every square I both a rectangle and a rhombu s The diagonal s are congruent, bi sec t e ach o ther, arc perp e ndicul a r to each other and bi sect the inte rior a ng le s Parallelograms with right angles Diagonals are congruent and bisect each other The equal s Iw or hb where 1= length , w = width , h = height, and b = base If the sides a re all equal, then the rectangle is more sp ecifi cally called a square Parallelograms with congruent sides Oppos ite angles are congruent All a ngle measure s total A F Thi s in d icate the re lati o ns hips of qu adri la te ral s A= Quadrilaterals B=Rhombi C = Rectangles D=Squares E =Trap e zoids F =Parallelog rams 360 Any co nsecuti v e angles arc su pplementary [CIRCLES] If a lin e is to a circle , th en it is perpendicular to the radiu s whose e ndpoint is the point of tangen cy (the point where the tangen t line intersects the circle) / I f t wo tang e nts to the same circle intersect in the exterior region, then the line segments whos e endpoints are the poin t of intersection of the tangent lines and the two points of tangency are equal in length ; or, line segments drawn from a co-planar exterior point of a circle to points o f tangency on the circle are congruent If a line in the p lane of a circl e is pe rpendicular to a radius at its oute r e ndpoint then the line is tangent to the circle The measure of a is equal to the measure of its central angle The measure of a is 180 Th e measure of a is equal to 360 mlllus the measure of its corresponding minor arc In the same circle or in equal circl e s , equa I chords have equal arcs and equal arc s have equal chords A perpendicular to a cho rd bisects the chord and its arc In the same circle or in equal circle s, congruent chords are the same distanc e from the center, a nd chords the sa m e distance from the center arc congruent An is equal to hal f of its intercepted arc (the arc which li es in th e inter io r of the in sc ri bed a ngle a nd wh ose end po int are on the sides o f the ang le) m MPN If tw o r- mMN intercept th e sam e are , th e n th e an g le s are gr uent ]f a is in scribed in a circle, t hen o ppo s ite an g les are upplemen ta ry An an g le in sc ri bed in a e l11i e irc le is alway s a ri g ht angl e Circles (coi/lilllled) equal to half the difference of the intercepted arcs When two chords intersect inside a circle, the product of the segment lengths of one chord = to the product of the segment lengths of the other chord When two secant line segments are drawn to a circle from the same exterior endpoint, the product of one secant and An angle formed by a and a is equal to half of the measure of its intercepted arc An angle formed by two chords intersecting inside a circle = to half the sum of the intercepted arcs An angle formed by two secants , or two tangents, or a secant and a tangent, that intersect at a point outside of the circle is The area, A, o f a two-di m en s ional s pe is the number of square units that c a n be put in the re g ion enclosed by the sides A rea is obtain ed throu gh som e combination of multiplying hei g hts a nd base s, which always form 90° angles with ea ch other, except in circles its e xternal segment length = the product of the other secant and its e xternal segment length Wh e n a tangent and a seca nt lin e segment a re dra wn to a c ircle fro l11 the same exterior point, th e square of the length of the tange nt segment = to the product o f th e secant and its externa l seg ment length T he perimeter, P, of a two-dim e ns io na l s hape is the sum of all s ide lengths The volum e, Y, of a th ree-dim e ns iona l s hape is the number of cubic units th at can be put in th e space en closed by a ll th e s ides V=e Each edge leng th, e, is equal to the other edge in a cu be; if e= 8, th en: V=( 8)(8)(8), V= 512 cubi c units Ifb=8, then : A = 64 square units A =lrh, or A = /w If II =4 and b= 12, the n: A =(4)(12), A =48 squa re units V=1rr 2h If radius r= and h =8, the n: V= n:(9 )2(8), V= (3.l 4)(8 l )( 8) , A= !hh ~ Ifll= 8andb= 12,th~n: Z A = 1(8 )(12), A =48 square units iU D h V= 203 72 eubie units 2h V=!nr Ifr=6 and h =8 , th en: A=hh C V= * n: (6)2(8 ), V= * (3 14)(36)( 8) , If h =6 and h = 9, th e n: A = (6)(9), A = 54 sq uare units ~ V= 30 1.44 cu bic un i ts V= (area of triangle)" A=1h (h,+h2) If h = 9, hi = and h2 = 12, then : If -""' -~2 u o- has an area equal to (5 )( 12), then: bl A = (9)(8+ 12), A = (9)(20), A = 90 square units V =30h and if 1r= 8, then: V=(30) (8), V =240 c ubic units ~ '-' : -" A=lrt· A = 1rr2; If r = 5, then: A =n:5 =(3.14)25=78.5 squa re units V = t(a rea of rectan gle)" If / = and 1V =4, the rec tan g le has an area of 20, th e n: C=21tr If r=5, th en: V= * (2 0)11 and if h=9, then: C=(2 )(3 l4)(5 )= 10(3 14) = 1.4 units V= * (20)(9), V=60 cu bic uni ts V =±1rr3 I f a right trian gle has hypotenuse c and legs a an d h , then: c =a +h ~ Z V=/wlr If/ = 12, w = and Ir = 4, then: V= ( 12)(3 )(4 ), V= 144 c ub ic un its b !f radi us r=5, the n: V= j- (3.1 )(5)3, V= 523 cubic units "o;p w Algebra Part I, Algebra Part Algebrai.: r quati(ln' e" leulus I ('a lculu, 2, C:.iculu > Methods Geometry Part I (icolll ctry Part Linear Algebra Math Rc\ icw Trig0nometry I iUr -Aa free nundf~ads re "OTE: TO STlJIlE~TS Due IU ils c{Ondcn~d fonnal plcn'>C D fl "igncu d;\s~\\ C l'lcI 'lronu.: m 111 cc hallil· ~!1 including phOhlCUPY _rc :ordlllg orl.; 1\11 r ig hl\ r es cn ecl No pan or rc p rt K! u