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geometry regents
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Gravity
A way not to fail the regents
Terms in this set (135)
perpendicular lines
intersect to form right angles
diameters/radii and tangents
intersect on a circle to form right angles
corresponding sides of similar triangles are
in proportion
legs in an isosceles shape are
equal
if 2 parallel lines are cut by a transversal the alternate interior angles are
congruent
an angle inscribed in a semi circle
is a right angle
base angles of any isosceles shape
are congruent
SOH
Opposite/Hypotenuse
CAH
Adjacent/Hypotenuse
TOA
Opposite/Adjacent
Cofunctions
Cosine and Sine are equal
Complementary angles
angles that add up to 90 degrees
Supplementary angles
angles that add up to 180
inscribed angles
are equal to half of their intercepted arc
central angles
equal to their intercepted arc
altitude
the length of the perpendicular line from a vertex to the opposite side of a figure.
bisect
divide (a line, angle, shape, etc.) into two equal parts.
Transformations: r over y=x
(y,x)
Transformations: R -90* (x,y)
(y,-x)
R 180 or -180 (x,y)
(-x,-y)
r over x-axis (x,y)
(x,-y) Reflect across the "X"-axis
r over y-axis (x,y)
(-x,y) reflect across the y-axis
T a,b (x,y)
(x+a,y+b) Move it "A" units horizantoly and "B" units verticlly
r over y=-x
(-y,-x)
R 90* (x,y)
(-y,x)
line
an infinitely long set of points that has no width or thickness
ray
a portion of a line with one end point and including all points on one side of the end point
segment
a portion of a line bounded by two end points
plane
a flat set of points with no thickness that extends infinatelely in all directions
angle
a figure formed by two rays with a common endpoint
angle measure
the of opening of an angle, measured in degrees or radians
linear pair
two adjacent angles that form a straight line (180* line)
vertical angles
the congruent opposite angles formed by intersecting lines
angle bisectors
divide angles into two congruent angles.
alternate interior angles
are congruent when between parallel lines on a transversal
same side interior angles
are supplementary between parallel lines on a transversal
corresponding angles
are congruent when on parallel lines with a transversal
perpendicular bisector
a line, segment, or ray that is perpendicular to and bisects a segment.
interior angle (of a polygon)
the angle inside the polygon formed by two adjacent sides
exterior angle (of a polygon)
the angle formed by a side and the extension of an adjacent side in a polygon
dilation
not a rigid transformation (stretch)
median
a segment from the vertex to the midpoint of the opposite side.
scalene triangles
no sides are congruent
isosceles
two sides are congruent
equilateral triangle
three sides are congruent
All Of My Children Are Bringing In Peanut Butter Cookies
Altitudes/orthocenter. Medians/centroid. Angle bisectors/incenter. Perpendicular bisectors/circumcenter
Angle sum theorem
the sum of the measures of the interior angles = 180*
exterior angle theorem
the measure of any exterior angle of a triangle = the sum of the measures of the nonadjacent interior angles
isosceles triangle theorem and its converse
if two sides of a triangle are congruent, then the angles opposite them are congruent.
if two angles in a triangle are congruent, then the sides opposite them are congruent
equilateral triangle theorem
all interior angles of an equilateral triangle measure 60*
Pythagorean theorem
In a right triangle, the sum of the squares of the legs equlals the square of the hypotenuse
congruence
if two figures can be mapped onto another by a sequence of rigid motions.
reflexive property of equality
Any quality is equal to itself . For figures, any figure is congruent to itself
slope formula
(Y2-Y1)/(X2-X1) OR (RISE/RUN)
Distance formula
√(X1-X2)squared+(Y2-Y1)squared
How to divide a segment proportionally
By using the two proportion ratios. (X-X1)/(X2-X).
(Y-Y1)/(Y2-Y). and setting each of the proportions = to what ever the ratio that divides the line segment
how to find the area of a polygon on a graph
sketch a rectangle around the shape. find the areas of the triangles between the polygon and the rectangle. find all the areas of the triangles, add them up, and subtract them from the area of the rectangle.
collinear
three points are collinear if the slopes between any two pairs are equal
slope of a line= slope of perpendicular line
2/3 and -3/2
segment parallel to a side theorem
a segment parallel to a side of a triangle forms a triangle similar to the O.G. triangle. If a segment intersects two sides of a triangle such that a triangle similar to the O.G. is formed, the segment is parallel to the third side of the O.G. triangle
side splitter theorem
a segment parallel to a side in triangle divides the two sides intersect proportionally
centroid theorem
the centroid of a triangle divides each median in a 1:2 ratio, with the longer segmant having a vertex as one of its endpoints
midsegment theorem
a segment joining the midpoints of two sides of a triangle (a midsegmant) is parallel to the opposite side, and it's length is equal to 1/2 the length of the opposite side
altitude to the hypotenuse of a right triangle theorem
the altitude to the hypotenuse of a right triangle forms two triangles that are similar to the O.G. triangle
parallelogram properties
opposite sides are congruent. opposite angles are congruent. adjacent angles are supplemantry. the diagonols bisect each other. the diagonals each divide the prallelogram into two congruant triangle
trapezoid properties
one pair of parallel sides. opposite angles are supplemantry
rectangle properties
any one of the parallelogram properties. one right angle, diagonols are congruent
rhombus properties
any one of the parallelogram properties. diagonals are perpindicular. one pair of consecutive sides is congruent. a diagonol bisects one of the angles. DIAGNOLS FORM FOUR CONGRUENT ISOSCELES RIGHT TRIANGLES
square properties
any one of the parallelogram + square + rhombus properties
two lines are parallel
if the alternate interior angles formed are congruent
radius
a segment with one endpoint at the center of the circle and one endpoint on the circle
chord
a segment with both endpoints on the circle
diameter
a chord that passes through the center of the circle
secant
a line that intersects a circle at exactly two points
tangent
a line that intersects a circle at exactly at one point
point of tangency
the point at which a tangent intersects a circle
radii properties
all radii of a given circle are congruent, two circles are congruent if and only if their radii are congruent
central angle theorem
the angle measure of an arc equals the measure of the central angle that interceps the arc
inscribed angle theorem
the angle measure of an arc equals twice the measure of the inscribedangle that interceps the arc
congruence chord theorem
congruent chords intercept congruent arcs on a circle. congruent arcs on a circle are intercepted by congruent chords
parallel chord theorem
the two arcs formed between a pair of parallel chords are congruent. if the two arcs formed between a pair of chords are congruent then the chords are parallel.
chord- perpindicular bisector theorem
the perpendicular bisector of any chord passes through the center of the circle. a diameter or radius that is perpindicular to a chord bisects the chord. A diameter or radius that bisects a chord is perpindicular to the chord
tangent radius theorem
a diameter or radius to a point of tangency is perpindicular to the tangent. A line perpindicular to a tangent at the point of tangency passes through the center of the circle
congruent tangent theorem
given a circle and external point Q, segments between the external point and the two points of tangency are congruent
radian
π/180. A unit of angle measure. 2π is = to one complete revolution around a circle
Radian area
1/2Rsquared(measure of central angle)
major arc
an arc with a measurement greater than 180º
minor arc
an arc with a measurement less than 180º
sem-circular arc
an arc with a measurement of exactly 180º
angle of depression
the angle formed by the horizontal and the line of sight when looking downward to an object
angle of elevation
the angle formed by the horizontal and line of sight when looking upward an object
angle of rotation
the angle measure by which a figure or point spins around a center point
apex
the tip of a pyramid or cone or triangle
cavalieri's principle
if two solids are contained between two parallel planes, and every parallel plane between these two planes intercepts regions of equal area, then the solids have equal volume. Also, any two parallel planes intercept two solids of equal volume
center-radius equation of a circle
(x - h)squared + (y - k)squared = rsquared
coincide (coincedent)
figure that lay entirely on one another
corresponding parts
a pair of parts (usually points, sides, or angles) of two figures that are paired together through a specified relationship, such as a congruence or similarity statement or a transformation function
concave polygon
a polygon with at least one diagonal outside the polygon
concentric circles
circles with the same center
convex polygon
a polygon whose diagonals all lie within the polygon
CPCTC
corresponding parts of congruent triangles are congruent
direct transformation
a transformation that preserves oriontation
equiangular
a figure whose angles all have the same measure
equidistant
the same distance from two or more points
equilateral
a figure whose sides all have the same length
glide reflection
the composition of a line reflection and a translation along a vector parallel
identity transformation
a transformation in which the pre-image and image are coincident
isometry/rigid motion
a transformation that preserves distance. the image and pre-image are congruent under a rigid motion. translations, reflections, and rotations are isometries
mean proportional (geometric mean)
the square root of the product of two numbers, a and b. If a/m=m/b, then m is the geometric mean
opposite transformation
a transformation that changes the orientation of a figure
polyhedron
a solid figure in which each face is a polygon
postulate
a statement that is accepted to be true without proof. EX: Subtraction postulate
right circular cone/cylinder
a cylinder/cone with a circular base and whose altitudes pass through the center of the base
right pyramid
a pyramid whose faces are isosceles triangles
vector
a quantity that has both magnitude and direction; represented geometrically by a directed line segment. It's symbol is an arrow towards the right
intersecting chord theorem
a relation of the four line segments created by two intersecting chords in a circle. It states that the products of the lengths of the line segments on each chord are equal.
A dialation produces
parallel sides
point-slope formula
Y-Y1=M(X-X1)
Of what triangle is it's orthocenter outside the triangle?
Obtuse triangle
What does the centroid do?
It divides each median of the triangle into segments with a 2:1 ratio
How do you find the centroid of a triangle?
By averaging the x coordinates (just the x) and the y coordinates (just the y) of all three vertices of the triangle
If two triangles are similar
their sides are in proportion
In a proportion, the product of the means...
is equal to the product of the extremes
Cavalieri's principle
If two solid are contained between two parallel planes, and every parallel plane between these two planes intercept regions of equal area, then the solids have equal volume. Also, any two parallel planes intercept two solids of equal volume
Sector area of a circle
(Measure of arc)/360xpieRsquared
If you have two secants originating from the same point outide the circle, how do you solve for lengths of the secants?
INNERxOUTER=INNERxOUTER
If you have one secant and one tangent that originate at the same point, How do you solve for the lengths of the line
Tangent length squared=outer time inner
How do you find out what rotation about the center of a shape will carry it onto itself?
Divide 360* by the number of sides in the shape, and keep adding the number of degrees you get until you reach one of the answers
how many degrees are in a quadrilateral?
360*
If a quadrilateral is inscribed in a circle...
Than the opposite angles are supplementary
HYLLS method, to find a segment of a triangle when you have the
HYpotenuse Leg Leg Segment
weight=
DENSITYxVOLUME
DENSITY=
weight/volume
SAAS, to solve segments of a triangle using an altitude (In the image, AD/CD=CD/DB)
Segment/Altitude=Altitude/Segment
parallel chords intercept
congruent cords
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