Geometric Formulas qs

Prévia do material em texto

-	 [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 either 
plane 11 or pl ane ABC if points A , B, and 
C are on plane /I 
[DEFINED Terms] 
[GENERAL Terms] 
[1J i( C rlCJI Jl" ) Shapes are the same shape 
and s ize 
[21 (II 11"1' Shapes arc the same shape, but 
can be different sizes 
[3J i.q Jdl Sets of po ints or numerica l 
measurements arc exactly the same 
[4J JIllor! Describes the result when all of 
the points are put together 
[5J (II rs C 101) Describes the points 
where ind icated shapes touch 
[6J p c The set of all points 
111"'4'. 
[1J Collinear points are on the same line 
[2J Non colill c I points are not on the 
same line 
[3J Inter II 9 lines have one and on ly 
one point in common 
[4J PIp I dlcul, r lines intersect and form 
90° angles at the intersection; 1. 
[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 anar lines at diffe re nt points 
[7J P r II I lines are co-planar (in the same . 
plane) , share no points in common, do 
not intersect , go in the same direction 
and never touch; II 
[LINE Segments] 
[1J T he set of any 2 points on a line and al l 
of the collin ear poin ts between the m ; 
AB where A and B are the endpoi nts of 
the Iine segment 
[2J 	The I CJth is the distance between the 2 
endpoints; it is a numerical va lue; AB 
means the length of 
A B 
1'.'.[1J The set of collinea r points going in one 
d irection from onc poin t (the endpoin t of 
the ray) on a li ne ; notation: AS w here A is 
the endpoint; not ice lIB *BA because they 
have different endpoin ts a nd co nta in 
d ifferent points on the li ne 
[2J 	Oppo It r y are collinear, share only a 
common endpoin t and go in o ppos ite 
directions 
[ANGLES] 
[1J 	The union of two rays that share o ne and 
only on e point, the endpoin t of the rays 
a. The Sid of the angle are the rays and 
the v r x is the endpoint of the rays 
b. The Interrol is all the points be tween 
the two sides of the angle 
c. LABe where B is the vertex or simply IB 
if there is only one angle w ith vertex B 
[2J Ov Ilapping angl s share some 
common interio r points 
[3J An ac Jte angle measures 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 angles 
whose measures total 90° 
[8J Supplementary angles are two an g les 
whose measures tota l 1800 
[9J 	Vertical angl s are two ang les that share 
o nly a eommon vertex and whose s ides 
form lines 
[10JAdjacent angles are two angles that 
sha re exaetly one vertex and one side, 
but no common interior points ; i.e., they 
do not overlap 
[11J An ngle bisector is a ray or a line that 
contains the vertex of the angle , is in the 
interior, and separates the angle into two 
adjacent angles with equal measures 
[3] 	The n idpoint is a point exactly in the 
midd le of the two endpoints 
[4] 	The bisector intersects a line segment 
at its m idpoint 
[5J 	The perpendicular bls tor intersects 
a line segmen t at its midpoint and forms 
90° angl es at the intersection 
1 
[TRANSVERSAL LINE Angles] 
[1J 	 In erlor ilngll are form ed with thc rays 
from the 2 li nes and the tra nsversal, such 
that the interior regions of the angles are 
located between the 2 lines 
[2J Alt rn 1 e II trior mgl a rc inte r iorc 
a ng les w ith di fferent vertexes a nd 
interior regions on opposite sides of the 
tran sversa l 
[3J ,1m lel II t nor ngle are inter ior 
a ngl es wi th different vertexes an d 
inte rior reg ions on th e same s ide o f the 
t ransversal 
[4J 	Ex I lor, r gl are formed with rays 
from the 2 lines and the transversa l, such 
that thc in terior regio ns of the angles are 
not between the 2 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 iIIr.. 
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 the " 
same positions relative to the lines and m 
the transversa l; one of th e pair of Z 
corresponding angles is a n interi or ang lc 
and the other is an exterior angle ~ 
[POLYGONS] 
[1J 	Polygons arc plan (flat), closed s hapes 
that are formed by l ine segments tha t 
inter ect only at thcir endpoints 
a. 	Not T hcy are na m ed by listing the 
endpoints of the l ine segm ents in 
order, goin g either clockwise or 
co untercl oc kwisc, sta rt ing at anyone 
of the endpoi nts 
b. The sides are li ne egments 
c. 	The int riar is all of the points 
enclosed by the s ides 
d. The xterior is all o f the point · on the 
plane of the polygo n, but neithe r on 
the sides nor in the interio r 
e. The vertic (or vertexes) are the iIIr.. 
endpoi nts of the li ne segments ,.. 
f. 	 Inc ludc a ll the points on the s ides ( line = 
segments) and the vertices ... 
g. The int rior angl o r a po lygon have " 
the same vert ices as the verticcs of the m 
polygon, have sides that contain the Z 
sides o r the po lygo n, and have in teri or 
reg ions that conta in the interior of the ~ 
polygo n- every polygon has as many 
interior ang les as it has vertices 
Polygons (continued) 
h. Consecutive interior angles have 
ve rt ices th at are endpoi nts of the same 
side of the polygon 
I. 	 The exterior angles are formed when 
the sides of the po lygon are extended; 
each has a ve rtex and one si de that are 
also a vertex and co ntain one side of 
the pol ygon; the second si d e of the 
exterior ang le is the ex te nsion of th e 
other polygon side contain ing the angle 
vertex; the interior of the exterior angle 
is part of the exterior region of the 
polygon; ex teri o r an g les are supp le­
ments of their adjacent interior angles 
j. 	Diagonals of a polygon are line 
segments wit h endpo ints that are 
vertices of th e po lygon , but the 
di agonals are not sides of the polygon 
[2) CONCAVE polygons have at least one 
interio r angle measuring more than 1800 
• 	 [3) CONVEX polygon s have no inter ior 
angles more than 180 0 and all interior 
angles each measure less than 180° 
[4) 	REGULAR polygons have all si de lengths 
equa l and a ll interi or an gl e meas ures 
equa l 
[5) CLASSIFICATIONS OF POLYGONS 
a. Classified by the number of si de s; 
eq ua l to the nu mber of vertices 
b. Th~ side len gths an d ang le measures 
arc not necessa rily eq ual un les s the 
word " regular" is also used to na me the 
po lygon 
c. Categories 
• Triangles have three sides 
• Q uadrilate ra ls have four sides 
• Pentagons have five sides 
• Hexagons have six si des 
• Heptagons have seven sides 
• Oc tagons have e ight sides 
• Nonagons have nine sides 
• Decagons have ten si des 
• n-gons have n s ides 
[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 not 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 the 
distance of each point from the center 
[3) 	A chord is a li ne seg ment who se 
endpoints arc 2 points on the ci rcle 
[4] 	A diameter is a chord that contains the 
center of the circle; the length of a diameter 
IS the distance from one poin t 
to another on the circle, going through 
the center 
15)A secant is a li ne intersecting a circle in 
two points 
r~ ,,= ~~ 
[6) SPECIAL POLYGONS 
a. Triangles 
• Po lygons with 3 sides and 3 vertices; 
the symbol fo r a triangle is ~; tri angle 
ABC is written ~ABC 
• 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 the 
triangle where the altitude is perpen­
d icular to that line; every triangle has 3 
altitudes 
• A base is a side of the triangle on the 
line perpendicular to an altitude; every 
triangle has 3 bases 
• Formula for area A =tllb or iI=thb 
where a=altitude , b=base or 
where h=height (altitude), b=base 
• Class~fied in 2 ways, by side length s 
and by angle measurements 
a] When classified by side lengths: 
• Scalene have no side lengths=, 
• Isosceles have at least 2 side 
lengths equal, 
• Equilateral have all 3 side lengths 
equal; note it is also an isoscel es 
triangle 
b]When classified by angle measure ­
ments: 
• Obtuse have ex actly one angle 
measurement more than 90 0 
• Right have exact ly one angle 
measurement equal to 90° 
90
• Acute have all 3 ang les less than 
0 ; note that if all 3 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 congruent s id es 
of the triangle 
b] The base is the side with a differen t 
length than the other two sides; no t 
[6] 	A tangent is a line that is co-p lanar w ith a 
circle and intersects it at one point on ly, 
call ed the point of tangency 
a. A cornmon tangent is a line that is 
tangent to 2 co-planar circles 
• Common internal anqents intersect 
between the two circles 
• Common external tangents do not 
intersect between the circles 
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 tangent 
[7) Equal Circles have equal-length radii 
[8) Concentric circles lie in the same plane 
and have the same cen ter 
2 
necessarily the side on the bottom 
of the triangl e 
c] The base angles of an isosceles 
triangle have the base contai ned in 
one of th eir sides ; they are always 
equal in measu re 
• Right Triangles 
a] The hypotenuse is opposi te 
the ri gh t angle and is the longest 
side 
b ]Th e legs arc th e 2 sides that a rc 
not the hy poten use; the line 
segments contained in the sides o f 
the right angle 
b. Quadrilaterals 
• 4-s ided polygons 
• Have 2 d iagonal s and 4 vert ices 
• 	 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] Non-para llel sides : legs 
c] The 2 angles wi th vertices that arc 
the endpo ints o f the same base arc 
call ed ba angl 
d]lsosceles trapeZOids have 
legs that arc the same length 
• Parallelograms have 2 pairs of 
parallel s ides 
a] Rectangl s have 4 ri ght angles 
b]Rhombus s (s ing . rhombus) have 
4 sides equa l in len gth 
c] Squares have 4 equal sides an d 4 
equal angles ; therefore , eve ry 
square is bo th a recta ngle and a 
rhombus 
[9] 	An inscribed polygon has vertices that 
are po ints on the circl e ; in th is sa ill e 
si tuation, the ci rcl e is c ircumscribed about 
the po lygon 
110]A circumscribed polygor has sides that 
arc segments of tangents to the ci rcle; i.e ., 
the s id es of the polygon each con ta in 
exac t ly one poi nt on the circ le ; in th is 
same situati on, the circle is insc ribed in 
the po lygon 
[11) An arc is part ofa ci rc le 
a. A erni ircl is a n arc whose 
en dpoints are the endpo ints o f a 
diamete r; 180°; exac tl y th ree points 
must be u.' cd to na me a scm iei rcle; 
notati on: AiJC wh ere A and C are the 
endpoints o f the d iamete r 
Through a point not on a line, exactly 
one perpendicular 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 
distancc apart 
If three or more parallel lines cut off 
equal segments on one transversal, then 
thcy cut off cqual segmcnts on cvcry 
transversal they sharc 
A linc and a planc are parallel if they do 
not touch or intcrsect 
Two or more planes are parallel if they 
do not touch or intersect 
Angles are measured using a protractor and 
degree mcasurements: There are 3600 in a 
circle; placing the center of a protractor at 
the vcrtcx of an anglc and counting the 
degree measure is like putting the vertex of 
the ang le at the center of a circle anc! 
comparing the angle measure to the 
degrees of the circle 
9() 
~ 
1 0 	 0 
If two angl es are compl ements of the 
same angl e , then they arc equal In 
measure (congruent) 
If two angles arc compl ements of 
congruent angles, then they are congruent 
If two angl es are supplements of th e 
same ang le , then they are congruent 
b . 	 A minor arc length is less than the 
length o f the sem ic ircl e; only two 
points may be used to name a minor' 
arc ; notation: DE where D and E are 
the endpoints of the arc 
c. 	 A major arc length is more than th e 
length of the semi c irc le ; exac t ly 
three points a rc used to namc a 
major arc ; notation: FCfj where F 
and H are the endpoints of the arc 
[12J A central angle vertex is the center 
of the c ircle with si des that contain 
radi i of the circle 
[13J A inscribed-angle vertex is on a 
circle with sides that contain chords 
of the circle 
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 
If two angles are supplements of 
congruent angles, then they are congruent 
Vertical angles are congruent and have 
equal measures 
If a point lies on the bisector of an angle, 
then the point is (equal 
distances) from the sides of the angle 
Distance from a point to a line 
is always the length of the perpen­
dic ular line segment that has the point 
as one endpoint and a point on the 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 angles (in 
pairs) that all have equal measures 
[POSTULATES] 
Statements that have been accepted 
without formal proof 
[1J A line contains at least 2 points, and any 2 
points locate exactly one line 
[2J Any 3 non-collinear points locate exactly 
one plane 
[3] 	A line and one point not on the line locate 
exactly one plane 
[4J Any 3 points locate at least one plane 
[5J If 2 points of a line are in a plane, then the 
line is in the plane 
[6] If 2 points are in a plane, then the line 
containing the 2 points is also in the plane 
[7] 	If 2 planes intersect, then the intersection 
is a line 
3 
If two rays do not int e rsect , then th e 
union of the rays is s imply a ll of th e 
points on both rays 
If two rays intersect in one and only one 
point, but not at th e endpo int, th en the 
union is a ll of the points on both rays; the 
intersection is that onepo int where they 
touch 
If two rays intersect in one and only one 
point, the endpoint, then the uni on is an 
angle; the intersection is the endpo int 
~ 
If two rays intersec t In mo re tha n one 
point, th en the union IS a lin e; the 
inte rsection is a line segmen t 
A B 	 AB BA 
AB BA 
If lines a re pa ra ll e l, th en th e a lt e rna te 
inte r ior angl es of a tra nsve rsa l are 
congruent 
If the a ltern ate in te ri o r ang les of a 
transversal are congruen t. then th e lines 
a re para ll e l 
If lines are para ll e l, then th e same s idc 
inte ri o r angl es of a transve rsa l a re 
suppl emen tary 
If th e sa me-s ide inte ri o r angl es of a 
tran sversa l are s uppl eme ntary, the n the 
I ines are para ll e l 
If lin es are pa ra ll e l, th en the 
correspond ing ang les of a transve rsa l are 
congruent 
If the correspondin g an g les o f a trans­
ve rsa l a re co ngruen t, th en the lines are 
parall e l 
If lines a re parall e l, th en th e a lte rn ate 
exte rior angles of a tra nsv e rsal are 
congruent 
If th e al ternate exte rior ang les o f a 
tran sversa l are congruent, then the lines 
are parallel 
If a transv ersal is perpendi cular to one o f 
two parall e l lines , then it is al so perpen ­
----n 
dicular to the other 
------m 
alternat Jr .: 4- 6; 5- 3 
same-Side interior 4- 5; 3- 6 
correspond ng': s: 1- 5; 4- 8; 3- 7; 2- 6 
alternate exterior 1- 7; 2-8 
duct 
rn a l 
The sum of the measures of the interior '1 The 3 bisectors of the angles of a When an altitude is drawn to the 
angles of a convex polygon with n sides is triangle intersect in one point, which is hypotenuse of a right triangle 
(n-2)180 degrees equidistant from the 3 sides The two triangles formed are 
To find the measure of each interior The of the similar to each other and to the 
angle of a regular polygon, find the sum of sides of a triangle intersect in one original right triangle 
all of the interior angles and divide by the point, equidistant from the 3 vertices The altitude is the 
number of interior angles, thus , the The medians (line segments whose between the lengths of 
formula (TI - 2) 180 endpoints are one vertex of the triangle the two segments of theTI and the midpoint of the side opposite 
hypotenuseThe sum of the measures of the exterior that vertex) of a triangle intersect in 
Each leg is the geometric mean angles of any convex polygon, using one one point two-thirds of the distance 
between the hypotenuse and theexterior angle at each vertex, is 360° from each vertex to the midpoint of the sum
length of the segmcnt of theI r opposite side 
hypotenuse adjacent (touches) 
The 3-angle total measurement= 180° If two sides of a triangle are unequal in mber 
If two angle measurements of one length, then the opposite angles arc to the leg idcs 
triangle=two angle measurements of unequal and the larger angle is opposite t . 
another triangle, then the measurements to the longer side; and conversely, if If three sides of 
of the third angles are also= two angles of a triangle arc unequal , one triangle are congruent to three 
Each angle of an 
 then the sides opposite those angles are sides of another, then the triangles 
is 60° unequal and the longer side is opposite are congruent 
There can be no more than one right or the larger angle I f two sides and 
obtuse angle in anyone triangle The sum of the lengths of any two sides the included angle of one triangle 
The acute angles of a right triangle arc is greater than the length of the third are congruent to two sides and the 
complementary side; the difference of the lengths of included angle of another. then the 
The measurement of an exterior angle= any two sides is less than the length of triangles are congruent 
the sum of the measurements ofthe two the third side If two angles and 
remote (not having the same vertex as the included side of one triangl e 
the exterior angle) interior angles An equilateral triangle is also equian­
 are congruent to two angles and 
I f two sides of a triangle are equal , then gular; and, an equiangular triangle is the included side of another, then 
the angles opposite to those sides are also equilateral the triangles are congruent 
also equal; and, if two angles are equal , An equilateral triangle has three 60­ If two angles and a then the sides opposite those angles are degree angles 
non-included side of one triangle 
also equal The bisector of the vertex angle of an 
are congruent to the twoIf two sides isosceles triangle is the perpendicular 
corresponding angles and non­of one triangle are equal in length to bisector of the base of the triangle 
included side of another, then the 
two sides of another, but the included 
triangles are congruent 
angle of one triangle is larger than the In a right 
If the hypotcnuse
included angle of the other triangle, triangle, , where ., and ' 
and one leg of a right triangle are 
then the longer third side of the are the lengths of the legs and is the 
triangles is opposite the larger included length of the hypotenuse congruent to the hypotenuse and 
the corresponding leg of another.
angle of the triangles If the square of the hypotenuse is 
If two sides equal to the sum of the squares of the then the two right triangles are 
of one triangle arc equal to two sides of other two sides, then the triangle is a congruent 
another, but the third side of one is 
longer than the third side of the other, If the square of the longest side is If two 
then the larger included angle (included greater than the sum of the squares of angles of one triangle arc 
between the two equal sides) is the other two sides, then it is an congruent to two angles of 
opposite to the longer third side of the triangle; if it is less than the 
 another. then the triangles are 
triangles sum of the squares of the other two similar (same shape but not 
sides, then it is an triangle necessarily the same size) 
I f a line is parallel to one side and In a 45-45-90 
 If th e 
intersects the other two sides, then it triangle, the legs have equal lengths sides of one triangle are propor­
divides those two sides proportionally, and the length of the hypotenuse is tional to the corresponding sides of 
and creates 2 similar triangles 12 times the length of one of the legs 
 another, then the triangles are
I f a an angle of a triangle, it In a 30-60-90 similardivides the opposite side into segments triangle, the length of the shortest leg If twoproportional to the other two sides is 1/ 2 the length of the hypotenuse, 
sides of one tri angle are propor­The line segment that joins the and the length of the longer leg is 13 
tional to two sides of another and 
midpoints of two sides of a triangle has times the length of the shortest leg 
the included anglcs of each 
two properties: The midpoint of the hypotenuse of a 
triangle are congruent, then theIt is to the third side, and right triangle is equidistant from the 
triangles are similar It is of the third side three vertices 
4 
I 
~ 
ATERALS4 
The (the line segment 
whose endpoints are the midpointsZ of the 2 non-parallel sides) is parallel 
III to the bases, and its length is equal to 
half the sum of the lengths of the 2D­ bases 
C The may be calculated by 
averaging the length of the bases and 
T mUltiplying by the height (altitude 
o that is the length of the line segment 
P that forms 90-degree angles with the 
d bases); thus , the formula: 
A (hi ~h, )h (hi ~b, )h =~(hl +h, )h 
e 
t where the 2 bases are b1 and b2 and 
the height is h 
Two angles with vertices that are the 
endpoints of the same leg of a 
trapezoid are 
All 4 interior angle measures of all 
trapezoids total 360 0 
The base angles are congruent (has 
congruent legs) 
Opposite angles are suppl e ­
mentaryOpposite sides are parallel and 
congruent 
Opposite angles arc congruent 
All 4 inte rior angles total 3600 
C onsecutive interior an g les (thei r 
vertic es are endpoints for the same 
side) arc suppl eme ntary 
Di agonal s bi sect each o ther 
A quadril atera l is a para lle logram if: 
[CIRCLES] 
If a lin e is to a circle , then it 
is perpendicular to the radiu s whose 
endpoint is the point of tangen cy (the 
point where the tangen t line intersects 
the circle) 
/. 
I f two tangents to the same circle 
intersect in the exterior region, then the 
line segments whose endpoints are the 
point 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 
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 
Parallelograms with 4 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 4 sides a re all equal, then 
the rectangle is more specifi ­
cally called a square 
Parallelograms with 4 congruent 
sides 
Oppos ite angles are congruent 
All 4 angle measures total 
3600 
Any 2 consecuti v e angles arc 
supplementary 
If a line in the p lane of a circl e is 
pe rpendicular to a radius at its oute r 
endpoint. 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 1800 
The measure of a is equal to 
360 0 mlllus the measure of its 
corresponding minor arc 
In the same circle or in equal circl es , 
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 circles, 
congruent chords are the same distance 
from the center, and chords the same 
distance from the center arc congruent 
5 
If 4 interior angles each 
eq ual 90 0 , 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 angl es 
whose vertices a rc th e 
endpo ints of the diagonal 
4 equal s ides and 4 equal 
an g le s; every square 
both a rectangle and a 
rhombus 
The diagonal s are 
congruent, bi sec t each 
o ther, arc perpe ndicul a r 
to each other and bi sect 
the inte rior ang les 
A 
F 
Thi s ind icate 
the re lati onships of quadri la te ral s 
A= Quadrilaterals 
B=Rhombi 
C = Rectangles 
D=Squares 
E =Trape zoids 
F =Parallelog rams 
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 
angle and whose end po int are on 
the sides o f the angle) 
r-... 
m MPN mMN 
If two intercept 
the sam e are , th en th e an g les are 
congruent 
] f a is inscribed in a 
circle, then o ppos ite an g les are 
upplemen ta ry 
An an g le insc ri bed in a e l11ie irc le 
is always a ri ght angle 
An 
Circles (coi/lilllled) 
An angle formed by a and a equal to half the difference of the its e xternal segment length = the product 
is equal to half of the measure intercepted arcs of the other secant and its e xternal 
of its intercepted arc When two chords intersect inside a segment length 
angle formed by two chords circle, the product of the segment Wh e n a tangent and a secant lin e 
intersecting inside a circle = to half the lengths of one chord = to the product of segment a re dra wn to a c ircle fro l11 the 
sum of the intercepted arcs the segment lengths of the other chord same exterior point, the square of 
An angle formed by two secants , or two When two secant line segments are the length of the tangent 
tangents, or a secant and a tangent, that drawn to a circle from the same exterior segment = to the product o f th e 
intersect at a point outside of the circle is endpoint, the product of one secant and secant and its externa l segment length 
iUr-----------------------------------------------------------------Aa 
The area, A, of a two-di mens ional shape is the number of 
square units that can be put in the reg ion enclosed by the sides 
A rea is obtained through some combination of multiplying 
heights and bases, which always form 90° angles with each other, 
except in circles 
Ifb=8, then : 
A = 64 square units 
A =lrh, or A = /w 
If II =4 and b= 12, then: 
A =(4)(12), A =48 square units 
A= !hh ~ Ifll= 8andb= 12,th~n: 
ZA = 1(8)(12), A =48 square unitsiU 
A=hh 
D. If h =6 and h=9, then: 
CA = (6)(9), A = 54 sq uare units 
~ A=1h(h,+h2) 
If h = 9, hi = 8 and h2= 12, then : 
A = 1 (9)(8+ 12), A = 1 (9)(20), A = 90 square units 
A=lrt· 2 
A = 1rr2; If r =5, then: 
A =n:5 2=(3.14)25=78.5 square units 
b l 
~----'-'--:---" 
C=21tr 
If r=5, then: 
C=(2)(3. l4)(5)= 10(3. 14) = 3 1.4 units 
b 
legs a and h , then: c2=a2+h2 
If a right triangle has hypotenuse c and 
V=/wlr ~ If/ = 12, w = 3 and Ir = 4, then: 
"o;pZ V= ( 12)(3 )(4 ), V= 144 cub ic units w I 
"OTE: TO STlJIlE~TS Due IU ils c{Ondcn~d fonnal. plcn'>CD. fl .. "igncu d;\s~\\ orl.; 
1\11 r ighl\ r escn ecl. No pan or thi ~ pll hli ..:ation may he 
rc p rt K! u <.: l',1 01- trall" lllll tcu In J II )' fi)Tm. l lr by any means 
l' lcI..' lronu.: m 111 cchallil· ~!1. including phOhlCUPY_rc..:ordlllg . (If 
<III)' lIlfllTmuli on stora ge and retTie ' al ~ ) ~ll·1ll . V. ilhnUl wrHlcn 
pcnni'~l()n frol11lhc pub li sher 
2011 2. 1UII"&. 20llS Ra rC h arl.s. lllr . 1I1 0M 
ISBN-13: 978-157222909-9 
ISBN-10: 157222909-8 
J~ 111,1l1Will!~~~~Illl lllil1llfllilllI 
T he perimeter, P, of a two-dime ns iona l shape is the sum 
of all s ide lengths 
The volume, Y, of a th ree-dimens iona l shape is the number 
of cubic units that can be put in the space enclosed by a ll th e s ides 
V=e3 
Each edge length, e, is equal to the 
other edge in a cube; if e= 8, then: 
V=( 8)(8)(8), V= 512 cubic units 
V=1rr 2h 
If radius r=9 and h =8, the n: 
V=n:(9 )2(8), V= (3.l 4)(8 l )( 8) , h 
V=2034.72 eubie units 
V=!nr 2h3 
Ifr=6 and h =8 , then: 
V= * n: (6)2(8 ), V= * (3. 14)(36)( 8) , 
V= 30 1.44 cu bic uni ts 
V= (area of triangle)" 
If -""'---~2.u----o- has an area equal to 1 (5 )( 12), then: 1 
V=30h and if 1r= 8, then: 
V=(30) (8), V=240 c ubic units 
V= t(area of rectangle)" 
If / = 5 and 1V=4, the rec tangle has 
an area of 20, then: 
V= * (2 0)11 and if h=9, then: 
V= * (20)(9), V=60 cubic uni ts 
V=±1rr3 3 
!f radius r=5, then: 
V= j- (3.1 4)(5)3, V= 523 .3 cubic units 
Algebra Part I, Algebra Part 2. Algebrai.: rquati(ln'. e" leulus I. ('a lculu, 2, C:.iculu > 
Methods. Geometry Part I. (icolllctry Part 2. Linear Algebra. Math Rc\ icw. Trig0nometry 
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