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Geometry, Chem > geometry total review > Flashcards

geometry total review Flashcards

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1
Q

the intersection of 2 lines

A

point

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2
Q

intersection of 2 planes

A

line

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3
Q

collinear points are points that lie on the same

A

line

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4
Q

points and lines are coplanar if they lie on the same

A

plane

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5
Q

midpoint formula

A

x1 + x2/2 y1+ y2/2

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6
Q

distance formula

A

d= square root of (x2-x1) squared + (y2-y1) squared

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7
Q

circumference of a circle

A

(pi)(d)

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8
Q

area of a circle

A

(pi)(r)squared

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9
Q

perimeter of a square

A

4s

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10
Q

perimeter of a rectangle

A

2l+ 2w

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11
Q

area of a square

A

s(squared)

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12
Q

area of a rectangle

A

lw

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13
Q

area of a trapezoid

A

b1+b2/2 (h)

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14
Q

after a single line reflection OR an odd number of line reflections…

A

a figure will have a different orientation

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15
Q

after an even number of line reflections

A

a figure will have the same orientation

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16
Q

if reflected over the x axis, (x, y) becomes

A

(x, -y)

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17
Q

if reflected in the y-axis, (x,y) becomes

A

(-x,y)

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18
Q

if reflected in the line y=x, (x,y) becomes

A

(y,x)

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19
Q

if reflected in the line y=-x, (x,y) becomes

A

(-x,-y)

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20
Q

positive rotation

A

counterclockwise

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21
Q

negative rotation

A

clockwise

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22
Q

after rotation of 90, (x,y) becomes

A

(-x,y)

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23
Q

after rotation of 180, (x,y) becomes

A

(-x,-y)

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24
Q

after rotation of 270, (x,y) becomes

A

(x,-y)

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25
Q

if a figure has point symmetry…

A

the figure is its own image when rotated 180

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26
Q

when applying a composition of transformations

A

apply 2nd one first

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27
Q

glide reflection

A

reflection & transformation

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28
Q

invariant point

A

under transformation the point is the same

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29
Q

congruency transformation

A

pre image and image are congruent

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30
Q

similarity transformation

A

pre image and image are similar

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31
Q

*in a diagram, the face that the prism sits on is not always the base

A

the base must be congruent polygons

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32
Q

solids with a height and slant height

A

pyramid and cone

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33
Q

height vs slant height

A

height is perpendicular to the base, slant is height of the face

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34
Q

how is a cross section formed

A

when you slice through a solid object

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35
Q

number of faces of a solid determines max number of sides when you slice through a solid object

A

ex. cube has 6 faces, the largest polygon cross section that can be cut from a square pyramid is a hexagon, since a square pyramid has 5 faces, the largest cross section is a pentagon, since a triangular pyramid has 4 faces, the largest polygons cross section that cut from a triangular pyramid is a quadrilateral

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36
Q

possible cross sections of a cylinder

A

circle, oval, rectangle

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37
Q

all cross sections of a sphere

A

circles

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38
Q

rotating a cross section about an axis forms a solid

A
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39
Q

-rotating a rectangle around an axis forms a cylinder
- rotating a right triangle along an axis forms a cone
- rotating a semicircle along an axis forms a sphere

A
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40
Q

density formulas

A

density= mass/volume
population density= population/land area (water volume)

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41
Q

if a composition of transformation includes dilation, then that transformations

A

it’s a similarity transformation

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42
Q

a composition of line reflections over intersecting lines is equivalent to

A

a rotation

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43
Q

a composition of line reflectoins over 2 parallel lines/any odd number of parallel lines is equal to

A

single reflection

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44
Q

a composition of line reflections over 3 parallel lines is equal to

A

single reflection

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45
Q

to rotate/dilate a figure about a center other than the origin

A

1) translate the center of rotation to origin
2) apply that translation to each point in the figure
3) apply the rotation/dilation to those points, then translate back

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46
Q

a translation/dilation can prove

A

all circles are similar

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47
Q

given the graph of a figure and its image after a composition of transformations, first check to see if the pre image and image are equal so you can determine if the composition included a dilation, then check orientation of the pre image and image to determine if there was a reflection, then look for possible translations/rotations

A
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48
Q

parallel line have equal slopes
perpendicular lines have opposite negated slope

A
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49
Q

slope formula

A

y2-y1/x2-x1

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50
Q

slope intercept: y=mx + b
point slope: y-y1= m(x-x1)
horixontal line: y= #
vertical line: x= #

A
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51
Q

i think you find perpendicular bisector between 2 points by finding midpoint

A

find midpoint between the 2 points

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52
Q

corresponding angles are equal if 2 ll lines are cut by a transversal

A
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53
Q

if 2 lines cut by a transversal form supplementary same side exterior/interior angles

A

the lines are II

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54
Q

if 2 lines are perpendicular to the same line

A

the lines are ll

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55
Q

if a line is perpendicular to the same line

A

it is perpendicular to the other line

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56
Q

if 2 lines are II to the same line

A

the lines are II to each other

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57
Q

the construction of parallel lines can be justified using the theorem

A

If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.

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58
Q

triangle angle sum theorem

A

<1 + <2 + <3 =180

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59
Q

exterior angle theorem

A

<1 + <2 + <4

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60
Q

point of congruency of angle bisectors of a triangle

A

incenter, always inside the circle

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61
Q

the point of concurrecy of the perpendicular bisectors of a triangle is

A

circumcenter

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62
Q

point of concurrency of the medians of a triangle is the

A

centroid

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63
Q

point of concureency of the altitudes of a triangle is

A

orthocenter

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64
Q

center of gravity/balance point of triangle

A

centroid

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65
Q

points of concurrency always inside a triangle

A

incenter, centroid

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66
Q

points of concruuency that could be inside, outside, or on the triangle

A

circumcenter, orthocenter

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66
Q

points of concruuency that could be inside, outside, or on the triangle

A

circumcenter, orthocenter

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67
Q

if the triangle is acute

A

circumcenter & orthocenter lie inside the triangle

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68
Q

if the triangle is obtuse

A

circumcenter and orthocenter is outside the triangle

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69
Q

if the triangle is right

A

the circumcenter lies midpoint of the hypotenuse and the orthocenter lies on the vertex that’s a right angle

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70
Q

centroid divides median of a triangle into 2 parts in

A

2/1 ratio

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71
Q

median

A

connected vertex of a triangle to the midpoint of the opposite side

72
Q

altitude

A

segement from the vertex of a triangle perpendicular to the opposite side

73
Q

shortest side is opposite angle and longest side is opposite the largest angle

A
74
Q

in a triangle, the third side must be

A

more than difference for the other 2 sides and less than the sum of the other 2 sides

75
Q

regular polygon

A

all sides and angles are equal

76
Q

sum of the interior angles of a polygon

A

180 (n-2) (n = # of sides)

77
Q

each interior angle of a regular polygon

A

180 (n-2)/n

78
Q

sum of the exterior angles of a polygon

A

360

79
Q

each exterior angle of a regular polygon

A

360/n

80
Q

opposite sides of a parallelogram

A

equal and II

81
Q

5 ways to show a quadrilateral is a parallelogram

A

1) show both pairs of opposite sides are II
2) both pairs of opposite sides are equal
3) both pairs of opposite angles are equal
4) diagonals bisect each other

82
Q

properties of a rectangle

A

equal right angles, equal bisectors

83
Q

properties of a rhombus

A

equal sides, diagonals bisect the angles and are perpendicular to each other

84
Q

properties of isosceles trapezoids

A

base angles are equal, diagonals are equal

85
Q

diagonals of a kite are perpendicular

A
86
Q

to prove the triangle is a right triangle

A

find the slopes of all sides and show 2 slopes are perpendicular

87
Q

coordinate geometry- prove a quadrilateral is a parallelogram

A

find slopes of all 4 sides and show that both pairs of opposite sides are II (same slope)

88
Q

how to show diagonals bisect each other

A

find midpoints of both midpoints

89
Q

to prove a quad. is a rhombus

A

1) show it is a parallelogram
2) 2) shoe either diagonals are perpendicular or adjaceecnt sides are equal

90
Q

to prove a quadrilateral is a rectangle

A

1) show it is a parallelogram
2) show adjacent sides are perpendicular or diagonals are equal

91
Q

to prove a quadrilateral is a square

A
  • show it’s a parallelogram
  • show that adjacent sides are perpendicular AND diagonals are equal
92
Q

to prove a quadrilateral is a trapezoid,

A

-find slopes of all 4 sides
-show one pair of sides are parallel

93
Q

to prove a quadrilateral is an isosceles trapezoid

A

-show that either non II sides are equal or diagonals are equal

94
Q

how to write the equation of an altitude to a side of a triangle

A
  • find slope of the opposite side, flip and negate
    -use vertex and perpendicular slope to write equation
95
Q

how to write equation of median to a side of a triangle

A

-find midpoint of opposite side
-then use vertex and mispoint to write equation

96
Q

in similar figures

A

-corresponding angles are congruent and corresponding sides are in proportion

97
Q

ratio of corresponding sides in similar figures are

A

congruent

98
Q

in similar figures, the ratio of corresponding altitudes, medians, diagonals, and angle bisectors is equal to the ratio of corresponding sides

A
99
Q

in similar figures, the ratio of the perimeters…

A

are equal to the ratio of corresponding sides

100
Q

in similar figures, the ratio of the areas is

A

the square of the ratio of the corresponding sides

101
Q

in similar figures, the ratio of the volumes are

A

the cube of the ratio of the corresponding sides

102
Q

if 2 triangles are similar to the same triangle…

A

they are similar to each other

103
Q

side splitter theorem

A

if 3 II lines intersect 2 transversals, then those transversals are divided proportionally
- if a line is II to one side of a triangle sand intersects the other 2 sides, then it divides those sides proprtionally

104
Q

the mid segment of a triangle joins

A

the midpoints of two sides of a triangle

105
Q

the mid segment of a trapezoid joins

A

of the non parallel opposite sides

106
Q

the mid segment of a trapezoid

A

is II to the bases of a trapezoid and its length is 1/2 sum of the bases

107
Q

partitioning

A

1) add the numbers
2) x2-x1 and y2-y1 for points
3) multiply by ratio

108
Q

*in trig problems, the angle of elevation and angle of depression always have a horizontal ray

A
109
Q

law of sine

A

a/sin A = b/sin B = c/sin C

110
Q

in a right triangle, sin A + cos B, but when solving for x in an equation, set equal to 90

A
111
Q

45-45-90

A

hypotenuse is s(square root 2) and legs are s

112
Q

30-60-90

A

small side: s (cube root)
side: s
hypotenuse: 2s

113
Q

if the 2 means of a proportion are equal, that number is called the mean proportional/geometric mean, in a/x = x/b, x is the geometric mean and always positive

A

use seg 1/alt = alt/seg 2

114
Q

if a2 + b2 < c2 the triangle is obtuse
if a2 + b2 >c2 the triangle is acute

A
115
Q

if a2 + b2= c2, it’s a right angle

A
116
Q

general form of a circle

A

x - h )^2 + ( y - k )^2 = r^2

117
Q

center radius form of a circle

A

x2 + y2 = r2

118
Q

if a circle is given in general form

A

complete the squaer to get it into center-radius form

119
Q

how to write the equation of a circle given the diameter endpoints

A

use the midpoint to find the center of the circle and dostance between the center and one endpoint to find the radius

120
Q

angle formed by a chord and tangent

A

1/2 intercepted arc

121
Q

angle formed by 2 chords intersecting inside a circle

A

(a+b/2)

122
Q

angle formed by 2 secants or 2 tangents (or one of each) equal (a-b/2)

A
123
Q

if a quadrilateral is inscribed in a circle

A

its opposite angles is supplementary

124
Q

a radius or diameter perpendicular to a chord bisect the chord and its arc

A
125
Q

parallel chords create equal arcs, equal chords have equal arcs

A
126
Q

chords equidistant from the center of the circle is equal
*longer chords are closer to the center of the circle and shorter chords are further from the center

A
127
Q

tangents to acircle from the the same external point arre equal

A
128
Q

if 2 chords intersect in a circle

A

the product of their segments are equal

129
Q

if 2 secants are drawn from the same external point, pow +pow

A
130
Q

secant and tangent drawn from same point= tangent2 = pw

A
131
Q

an angle inscribed in a semi cricle is a right angle

A
132
Q

adjacent circles

A

have 3 common tangents

133
Q

non adjacent circles have

A

4 common tangents

134
Q

circumference of the circle

A

2pir

135
Q

dissection area circle thingy

A

if you dissect a circle and rearrange the sections, the shape will approach a parallelogram where 1;/2 the circumference is on the top and half on the bottom

136
Q

dissection area circle thingy

A

if you dissect a circle and rearrange the sections, the shape will approach a parallelogram where 1;/2 the circumference is on the top and half on the bottom

137
Q

dissection area circle thingy

A

if you dissect a circle and rearrange the sections, the shape will approach a parallelogram where 1;/2 the circumference is on the top and half on the bottom

138
Q

convert angles in degrees and raedians

A

radian to degrees- 180/pi
degrees to radian- pi/180

139
Q

arc length formula

A

n/360 (2pir)

140
Q

sector area

A

x/360(pirsquared)

141
Q

*when given an equation given arc length or sector area, convert to degrees first if it’s in radian

A

180(pi)

142
Q

3d closed spatial figure

A

solid

143
Q

insection of 2 faces of a solid

A

edge

144
Q

intersection of 2 or more edges of a sollid

A

vertex

145
Q

2 II congruent faces on top and bottom of solid

A

bases

146
Q

a prism is named by

A

its base, which can be any polygon, the other sides are called lateral faces

147
Q

if thre base of a solid like a prism is perpendicular to the lateral edges , it’s called a right prism

A
148
Q

if the base of a solid such a prism and the lateral edges are not perpendicular, it’s nan oblique prism

A
149
Q

3D solid with 2 II parallel congruent faces

A

prism

150
Q

the lateral faces of a prism are all of the faces except for the

A

bases

151
Q

3D solid with congruent circular bases in a pair of II planes is

A

cylinder

152
Q

3d solid with a circular base and a vertex

A

cone

153
Q

polyhedron with all faces )except for one) intersecting at one vertex aka apex

A

pyramid

154
Q

set of all points in space equidsitant from a given point called the center is

A

sphere

155
Q

lateral area

A

sum of the aras of the lateral faces of a solid

156
Q

solids with 2 bases

A

prisms and cylinders

157
Q

solids with only one base

A

cone, pyramid

158
Q

lateral faces of a regular pyramid

A

isosceles triangles

159
Q

lateral area of a prism

A

ph

160
Q

lateral area of a cylinder

A

2 pirh

161
Q

lateral area of a pyramid

A

1/2 pl

162
Q

lateral area of a cone

A

pirl

163
Q

for figures with 2 bases

A

sa= la + 2B

164
Q

for a figure with one base

A

SA= LA + B

165
Q

SA cube

A

6s*squared

166
Q

SA of a sphere

A

4pir2
LA is not calculated because there are no faces in a sphere

167
Q

every crross section of a sphere is a circle because the intersection of a plane and a sphere is a circle

A
168
Q

every crross section of a sphere is a circle because the intersection of a plane and a sphere is a circle

A
169
Q

cavalieri’s principle

A

if 2 solids have the same height and all cross sections have the same area, then the two solids have the same volume

170
Q

if a prism and pyramid have the same base area and the same height, then the volume of the prism is

A

3x volume of pyramid

171
Q

if a cylinder and a cone have thee same base area and same height

A

the volume of the cylinder is 3x volume of thee cone

172
Q

if a prism and pyramid have the same base area and the same height, and you fill the pyramid with water and then fill the prism with that water, it will fill up 1//3 of the prism z

A
173
Q

if a cylinder and a cone have the same base are aand height and you filll the cone with water and then pour water into the cylinder it will fill by 1/3 of the cylinder you would need to fill the cone with water 3x to fill the cylinder

A
174
Q

law of cosine to find side c2

A

c2 (side)= a2 +b2- 2ab (cos C)

175
Q

law of cosine to find angle cosC

A

cos C =c2-a2-b2/-2ab

176
Q

other useful equations

A

speed = distance/time, average speed= change in distance/change in time

177
Q

same side exterior/interior angles are supplementary

A

Geometry, Chem (25 decks)

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