geometry total review Flashcards
1
Q
the intersection of 2 lines
A
point
2
Q
intersection of 2 planes
A
line
3
Q
collinear points are points that lie on the same
A
line
4
Q
points and lines are coplanar if they lie on the same
A
plane
5
Q
midpoint formula
A
x1 + x2/2 y1+ y2/2
6
Q
distance formula
A
d= square root of (x2-x1) squared + (y2-y1) squared
7
Q
circumference of a circle
A
(pi)(d)
8
Q
area of a circle
A
(pi)(r)squared
9
Q
perimeter of a square
A
4s
10
Q
perimeter of a rectangle
A
2l+ 2w
11
Q
area of a square
A
s(squared)
12
Q
area of a rectangle
A
lw
13
Q
area of a trapezoid
A
b1+b2/2 (h)
14
Q
after a single line reflection OR an odd number of line reflections…
A
a figure will have a different orientation
15
Q
after an even number of line reflections
A
a figure will have the same orientation
16
Q
if reflected over the x axis, (x, y) becomes
A
(x, -y)
17
Q
if reflected in the y-axis, (x,y) becomes
A
(-x,y)
18
Q
if reflected in the line y=x, (x,y) becomes
A
(y,x)
19
Q
if reflected in the line y=-x, (x,y) becomes
A
(-x,-y)
20
Q
positive rotation
A
counterclockwise
21
Q
negative rotation
A
clockwise
22
Q
after rotation of 90, (x,y) becomes
A
(-x,y)
23
Q
after rotation of 180, (x,y) becomes
A
(-x,-y)
24
Q
after rotation of 270, (x,y) becomes
A
(x,-y)
25
if a figure has point symmetry...
the figure is its own image when rotated 180
26
when applying a composition of transformations
apply 2nd one first
27
glide reflection
reflection & transformation
28
invariant point
under transformation the point is the same
29
congruency transformation
pre image and image are congruent
30
similarity transformation
pre image and image are similar
31
*in a diagram, the face that the prism sits on is not always the base
the base must be congruent polygons
32
solids with a height and slant height
pyramid and cone
33
height vs slant height
height is perpendicular to the base, slant is height of the face
34
how is a cross section formed
when you slice through a solid object
35
number of faces of a solid determines max number of sides when you slice through a solid object
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
36
possible cross sections of a cylinder
circle, oval, rectangle
37
all cross sections of a sphere
circles
38
rotating a cross section about an axis forms a solid
39
-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
40
density formulas
density= mass/volume
population density= population/land area (water volume)
41
if a composition of transformation includes dilation, then that transformations
it's a similarity transformation
42
a composition of line reflections over intersecting lines is equivalent to
a rotation
43
a composition of line reflectoins over 2 parallel lines/any odd number of parallel lines is equal to
single reflection
44
a composition of line reflections over 3 parallel lines is equal to
single reflection
45
to rotate/dilate a figure about a center other than the origin
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
46
a translation/dilation can prove
all circles are similar
47
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
48
parallel line have equal slopes
perpendicular lines have opposite negated slope
49
slope formula
y2-y1/x2-x1
50
slope intercept: y=mx + b
point slope: y-y1= m(x-x1)
horixontal line: y= #
vertical line: x= #
51
i think you find perpendicular bisector between 2 points by finding midpoint
find midpoint between the 2 points
52
corresponding angles are equal if 2 ll lines are cut by a transversal
53
if 2 lines cut by a transversal form supplementary same side exterior/interior angles
the lines are II
54
if 2 lines are perpendicular to the same line
the lines are ll
55
if a line is perpendicular to the same line
it is perpendicular to the other line
56
if 2 lines are II to the same line
the lines are II to each other
57
the construction of parallel lines can be justified using the theorem
If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.
58
triangle angle sum theorem
<1 + <2 + <3 =180
59
exterior angle theorem
<1 + <2 + <4
60
point of congruency of angle bisectors of a triangle
incenter, always inside the circle
61
the point of concurrecy of the perpendicular bisectors of a triangle is
circumcenter
62
point of concurrency of the medians of a triangle is the
centroid
63
point of concureency of the altitudes of a triangle is
orthocenter
64
center of gravity/balance point of triangle
centroid
65
points of concurrency always inside a triangle
incenter, centroid
66
points of concruuency that could be inside, outside, or on the triangle
circumcenter, orthocenter
66
points of concruuency that could be inside, outside, or on the triangle
circumcenter, orthocenter
67
if the triangle is acute
circumcenter & orthocenter lie inside the triangle
68
if the triangle is obtuse
circumcenter and orthocenter is outside the triangle
69
if the triangle is right
the circumcenter lies midpoint of the hypotenuse and the orthocenter lies on the vertex that's a right angle
70
centroid divides median of a triangle into 2 parts in
2/1 ratio
71
median
connected vertex of a triangle to the midpoint of the opposite side
72
altitude
segement from the vertex of a triangle perpendicular to the opposite side
73
shortest side is opposite angle and longest side is opposite the largest angle
74
in a triangle, the third side must be
more than difference for the other 2 sides and less than the sum of the other 2 sides
75
regular polygon
all sides and angles are equal
76
sum of the interior angles of a polygon
180 (n-2) (n = # of sides)
77
each interior angle of a regular polygon
180 (n-2)/n
78
sum of the exterior angles of a polygon
360
79
each exterior angle of a regular polygon
360/n
80
opposite sides of a parallelogram
equal and II
81
5 ways to show a quadrilateral is a parallelogram
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
properties of a rectangle
equal right angles, equal bisectors
83
properties of a rhombus
equal sides, diagonals bisect the angles and are perpendicular to each other
84
properties of isosceles trapezoids
base angles are equal, diagonals are equal
85
diagonals of a kite are perpendicular
86
to prove the triangle is a right triangle
find the slopes of all sides and show 2 slopes are perpendicular
87
coordinate geometry- prove a quadrilateral is a parallelogram
find slopes of all 4 sides and show that both pairs of opposite sides are II (same slope)
88
how to show diagonals bisect each other
find midpoints of both midpoints
89
to prove a quad. is a rhombus
1) show it is a parallelogram
2) 2) shoe either diagonals are perpendicular or adjaceecnt sides are equal
90
to prove a quadrilateral is a rectangle
1) show it is a parallelogram
2) show adjacent sides are perpendicular or diagonals are equal
91
to prove a quadrilateral is a square
- show it's a parallelogram
- show that adjacent sides are perpendicular AND diagonals are equal
92
to prove a quadrilateral is a trapezoid,
-find slopes of all 4 sides
-show one pair of sides are parallel
93
to prove a quadrilateral is an isosceles trapezoid
-show that either non II sides are equal or diagonals are equal
94
how to write the equation of an altitude to a side of a triangle
- find slope of the opposite side, flip and negate
-use vertex and perpendicular slope to write equation
95
how to write equation of median to a side of a triangle
-find midpoint of opposite side
-then use vertex and mispoint to write equation
96
in similar figures
-corresponding angles are congruent and corresponding sides are in proportion
97
ratio of corresponding sides in similar figures are
congruent
98
in similar figures, the ratio of corresponding altitudes, medians, diagonals, and angle bisectors is equal to the ratio of corresponding sides
99
in similar figures, the ratio of the perimeters...
are equal to the ratio of corresponding sides
100
in similar figures, the ratio of the areas is
the square of the ratio of the corresponding sides
101
in similar figures, the ratio of the volumes are
the cube of the ratio of the corresponding sides
102
if 2 triangles are similar to the same triangle...
they are similar to each other
103
side splitter theorem
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
the mid segment of a triangle joins
the midpoints of two sides of a triangle
105
the mid segment of a trapezoid joins
of the non parallel opposite sides
106
the mid segment of a trapezoid
is II to the bases of a trapezoid and its length is 1/2 sum of the bases
107
partitioning
1) add the numbers
2) x2-x1 and y2-y1 for points
3) multiply by ratio
108
*in trig problems, the angle of elevation and angle of depression always have a horizontal ray
109
law of sine
a/sin A = b/sin B = c/sin C
110
in a right triangle, sin A + cos B, but when solving for x in an equation, set equal to 90
111
45-45-90
hypotenuse is s(square root 2) and legs are s
112
30-60-90
small side: s (cube root)
side: s
hypotenuse: 2s
113
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
use seg 1/alt = alt/seg 2
114
if a2 + b2 < c2 the triangle is obtuse
if a2 + b2 >c2 the triangle is acute
115
if a2 + b2= c2, it's a right angle
116
general form of a circle
x - h )^2 + ( y - k )^2 = r^2
117
center radius form of a circle
x2 + y2 = r2
118
if a circle is given in general form
complete the squaer to get it into center-radius form
119
how to write the equation of a circle given the diameter endpoints
use the midpoint to find the center of the circle and dostance between the center and one endpoint to find the radius
120
angle formed by a chord and tangent
1/2 intercepted arc
121
angle formed by 2 chords intersecting inside a circle
(a+b/2)
122
angle formed by 2 secants or 2 tangents (or one of each) equal (a-b/2)
123
if a quadrilateral is inscribed in a circle
its opposite angles is supplementary
124
a radius or diameter perpendicular to a chord bisect the chord and its arc
125
parallel chords create equal arcs, equal chords have equal arcs
126
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
127
tangents to acircle from the the same external point arre equal
128
if 2 chords intersect in a circle
the product of their segments are equal
129
if 2 secants are drawn from the same external point, pow +pow
130
secant and tangent drawn from same point= tangent2 = pw
131
an angle inscribed in a semi cricle is a right angle
132
adjacent circles
have 3 common tangents
133
non adjacent circles have
4 common tangents
134
circumference of the circle
2pir
135
dissection area circle thingy
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
dissection area circle thingy
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
dissection area circle thingy
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
convert angles in degrees and raedians
radian to degrees- 180/pi
degrees to radian- pi/180
139
arc length formula
n/360 (2pir)
140
sector area
x/360(pirsquared)
141
*when given an equation given arc length or sector area, convert to degrees first if it's in radian
180(pi)
142
3d closed spatial figure
solid
143
insection of 2 faces of a solid
edge
144
intersection of 2 or more edges of a sollid
vertex
145
2 II congruent faces on top and bottom of solid
bases
146
a prism is named by
its base, which can be any polygon, the other sides are called lateral faces
147
if thre base of a solid like a prism is perpendicular to the lateral edges , it's called a right prism
148
if the base of a solid such a prism and the lateral edges are not perpendicular, it's nan oblique prism
149
3D solid with 2 II parallel congruent faces
prism
150
the lateral faces of a prism are all of the faces except for the
bases
151
3D solid with congruent circular bases in a pair of II planes is
cylinder
152
3d solid with a circular base and a vertex
cone
153
polyhedron with all faces )except for one) intersecting at one vertex aka apex
pyramid
154
set of all points in space equidsitant from a given point called the center is
sphere
155
lateral area
sum of the aras of the lateral faces of a solid
156
solids with 2 bases
prisms and cylinders
157
solids with only one base
cone, pyramid
158
lateral faces of a regular pyramid
isosceles triangles
159
lateral area of a prism
ph
160
lateral area of a cylinder
2 pirh
161
lateral area of a pyramid
1/2 pl
162
lateral area of a cone
pirl
163
for figures with 2 bases
sa= la + 2B
164
for a figure with one base
SA= LA + B
165
SA cube
6s*squared
166
SA of a sphere
4pir2
LA is not calculated because there are no faces in a sphere
167
every crross section of a sphere is a circle because the intersection of a plane and a sphere is a circle
168
every crross section of a sphere is a circle because the intersection of a plane and a sphere is a circle
169
cavalieri's principle
if 2 solids have the same height and all cross sections have the same area, then the two solids have the same volume
170
if a prism and pyramid have the same base area and the same height, then the volume of the prism is
3x volume of pyramid
171
if a cylinder and a cone have thee same base area and same height
the volume of the cylinder is 3x volume of thee cone
172
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
173
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
174
law of cosine to find side c2
c2 (side)= a2 +b2- 2ab (cos C)
175
law of cosine to find angle cosC
cos C =c2-a2-b2/-2ab
176
other useful equations
speed = distance/time, average speed= change in distance/change in time
177
same side exterior/interior angles are supplementary
