geometry total review Flashcards
the intersection of 2 lines
point
intersection of 2 planes
line
collinear points are points that lie on the same
line
points and lines are coplanar if they lie on the same
plane
midpoint formula
x1 + x2/2 y1+ y2/2
distance formula
d= square root of (x2-x1) squared + (y2-y1) squared
circumference of a circle
(pi)(d)
area of a circle
(pi)(r)squared
perimeter of a square
4s
perimeter of a rectangle
2l+ 2w
area of a square
s(squared)
area of a rectangle
lw
area of a trapezoid
b1+b2/2 (h)
after a single line reflection OR an odd number of line reflections…
a figure will have a different orientation
after an even number of line reflections
a figure will have the same orientation
if reflected over the x axis, (x, y) becomes
(x, -y)
if reflected in the y-axis, (x,y) becomes
(-x,y)
if reflected in the line y=x, (x,y) becomes
(y,x)
if reflected in the line y=-x, (x,y) becomes
(-x,-y)
positive rotation
counterclockwise
negative rotation
clockwise
after rotation of 90, (x,y) becomes
(-x,y)
after rotation of 180, (x,y) becomes
(-x,-y)
after rotation of 270, (x,y) becomes
(x,-y)
if a figure has point symmetry…
the figure is its own image when rotated 180
when applying a composition of transformations
apply 2nd one first
glide reflection
reflection & transformation
invariant point
under transformation the point is the same
congruency transformation
pre image and image are congruent
similarity transformation
pre image and image are similar
*in a diagram, the face that the prism sits on is not always the base
the base must be congruent polygons
solids with a height and slant height
pyramid and cone
height vs slant height
height is perpendicular to the base, slant is height of the face
how is a cross section formed
when you slice through a solid object
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
possible cross sections of a cylinder
circle, oval, rectangle
all cross sections of a sphere
circles
rotating a cross section about an axis forms a solid
-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
density formulas
density= mass/volume
population density= population/land area (water volume)
if a composition of transformation includes dilation, then that transformations
it’s a similarity transformation
a composition of line reflections over intersecting lines is equivalent to
a rotation
a composition of line reflectoins over 2 parallel lines/any odd number of parallel lines is equal to
single reflection
a composition of line reflections over 3 parallel lines is equal to
single reflection
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
a translation/dilation can prove
all circles are similar
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
parallel line have equal slopes
perpendicular lines have opposite negated slope
slope formula
y2-y1/x2-x1
slope intercept: y=mx + b
point slope: y-y1= m(x-x1)
horixontal line: y= #
vertical line: x= #
i think you find perpendicular bisector between 2 points by finding midpoint
find midpoint between the 2 points
corresponding angles are equal if 2 ll lines are cut by a transversal
if 2 lines cut by a transversal form supplementary same side exterior/interior angles
the lines are II
if 2 lines are perpendicular to the same line
the lines are ll
if a line is perpendicular to the same line
it is perpendicular to the other line
if 2 lines are II to the same line
the lines are II to each other
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.
triangle angle sum theorem
<1 + <2 + <3 =180
exterior angle theorem
<1 + <2 + <4
point of congruency of angle bisectors of a triangle
incenter, always inside the circle
the point of concurrecy of the perpendicular bisectors of a triangle is
circumcenter
point of concurrency of the medians of a triangle is the
centroid
point of concureency of the altitudes of a triangle is
orthocenter
center of gravity/balance point of triangle
centroid
points of concurrency always inside a triangle
incenter, centroid
points of concruuency that could be inside, outside, or on the triangle
circumcenter, orthocenter
points of concruuency that could be inside, outside, or on the triangle
circumcenter, orthocenter
if the triangle is acute
circumcenter & orthocenter lie inside the triangle
if the triangle is obtuse
circumcenter and orthocenter is outside the triangle
if the triangle is right
the circumcenter lies midpoint of the hypotenuse and the orthocenter lies on the vertex that’s a right angle
centroid divides median of a triangle into 2 parts in
2/1 ratio
median
connected vertex of a triangle to the midpoint of the opposite side
altitude
segement from the vertex of a triangle perpendicular to the opposite side
shortest side is opposite angle and longest side is opposite the largest angle
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
regular polygon
all sides and angles are equal
sum of the interior angles of a polygon
180 (n-2) (n = # of sides)
each interior angle of a regular polygon
180 (n-2)/n
sum of the exterior angles of a polygon
360
each exterior angle of a regular polygon
360/n
opposite sides of a parallelogram
equal and II
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
properties of a rectangle
equal right angles, equal bisectors
properties of a rhombus
equal sides, diagonals bisect the angles and are perpendicular to each other
properties of isosceles trapezoids
base angles are equal, diagonals are equal
diagonals of a kite are perpendicular
to prove the triangle is a right triangle
find the slopes of all sides and show 2 slopes are perpendicular
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)
how to show diagonals bisect each other
find midpoints of both midpoints
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
to prove a quadrilateral is a rectangle
1) show it is a parallelogram
2) show adjacent sides are perpendicular or diagonals are equal
to prove a quadrilateral is a square
- show it’s a parallelogram
- show that adjacent sides are perpendicular AND diagonals are equal
to prove a quadrilateral is a trapezoid,
-find slopes of all 4 sides
-show one pair of sides are parallel
to prove a quadrilateral is an isosceles trapezoid
-show that either non II sides are equal or diagonals are equal
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
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
in similar figures
-corresponding angles are congruent and corresponding sides are in proportion
ratio of corresponding sides in similar figures are
congruent
in similar figures, the ratio of corresponding altitudes, medians, diagonals, and angle bisectors is equal to the ratio of corresponding sides
in similar figures, the ratio of the perimeters…
are equal to the ratio of corresponding sides
in similar figures, the ratio of the areas is
the square of the ratio of the corresponding sides
in similar figures, the ratio of the volumes are
the cube of the ratio of the corresponding sides
if 2 triangles are similar to the same triangle…
they are similar to each other
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
the mid segment of a triangle joins
the midpoints of two sides of a triangle
the mid segment of a trapezoid joins
of the non parallel opposite sides
the mid segment of a trapezoid
is II to the bases of a trapezoid and its length is 1/2 sum of the bases
partitioning
1) add the numbers
2) x2-x1 and y2-y1 for points
3) multiply by ratio
*in trig problems, the angle of elevation and angle of depression always have a horizontal ray
law of sine
a/sin A = b/sin B = c/sin C
in a right triangle, sin A + cos B, but when solving for x in an equation, set equal to 90
45-45-90
hypotenuse is s(square root 2) and legs are s
30-60-90
small side: s (cube root)
side: s
hypotenuse: 2s
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
if a2 + b2 < c2 the triangle is obtuse
if a2 + b2 >c2 the triangle is acute
if a2 + b2= c2, it’s a right angle
general form of a circle
x - h )^2 + ( y - k )^2 = r^2
center radius form of a circle
x2 + y2 = r2
if a circle is given in general form
complete the squaer to get it into center-radius form
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
angle formed by a chord and tangent
1/2 intercepted arc
angle formed by 2 chords intersecting inside a circle
(a+b/2)
angle formed by 2 secants or 2 tangents (or one of each) equal (a-b/2)
if a quadrilateral is inscribed in a circle
its opposite angles is supplementary
a radius or diameter perpendicular to a chord bisect the chord and its arc
parallel chords create equal arcs, equal chords have equal arcs
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
tangents to acircle from the the same external point arre equal
if 2 chords intersect in a circle
the product of their segments are equal
if 2 secants are drawn from the same external point, pow +pow
secant and tangent drawn from same point= tangent2 = pw
an angle inscribed in a semi cricle is a right angle
adjacent circles
have 3 common tangents
non adjacent circles have
4 common tangents
circumference of the circle
2pir
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
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
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
convert angles in degrees and raedians
radian to degrees- 180/pi
degrees to radian- pi/180
arc length formula
n/360 (2pir)
sector area
x/360(pirsquared)
*when given an equation given arc length or sector area, convert to degrees first if it’s in radian
180(pi)
3d closed spatial figure
solid
insection of 2 faces of a solid
edge
intersection of 2 or more edges of a sollid
vertex
2 II congruent faces on top and bottom of solid
bases
a prism is named by
its base, which can be any polygon, the other sides are called lateral faces
if thre base of a solid like a prism is perpendicular to the lateral edges , it’s called a right prism
if the base of a solid such a prism and the lateral edges are not perpendicular, it’s nan oblique prism
3D solid with 2 II parallel congruent faces
prism
the lateral faces of a prism are all of the faces except for the
bases
3D solid with congruent circular bases in a pair of II planes is
cylinder
3d solid with a circular base and a vertex
cone
polyhedron with all faces )except for one) intersecting at one vertex aka apex
pyramid
set of all points in space equidsitant from a given point called the center is
sphere
lateral area
sum of the aras of the lateral faces of a solid
solids with 2 bases
prisms and cylinders
solids with only one base
cone, pyramid
lateral faces of a regular pyramid
isosceles triangles
lateral area of a prism
ph
lateral area of a cylinder
2 pirh
lateral area of a pyramid
1/2 pl
lateral area of a cone
pirl
for figures with 2 bases
sa= la + 2B
for a figure with one base
SA= LA + B
SA cube
6s*squared
SA of a sphere
4pir2
LA is not calculated because there are no faces in a sphere
every crross section of a sphere is a circle because the intersection of a plane and a sphere is a circle
every crross section of a sphere is a circle because the intersection of a plane and a sphere is a circle
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
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
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
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
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
law of cosine to find side c2
c2 (side)= a2 +b2- 2ab (cos C)
law of cosine to find angle cosC
cos C =c2-a2-b2/-2ab
other useful equations
speed = distance/time, average speed= change in distance/change in time
same side exterior/interior angles are supplementary
