6.1-7.5 Geometry PreAP (CUMULATIVE), PreAP Geometry CUMULATIVE (8 & 10), PreAP Geometry CUMULATIVE (11 & 12)
Inverse Arcsin Arccos Arctan
Calculations for finding the acute angles of a triangle are called the "______________" trig functions. They can also be expressed as: ______________, ______________, and ______________
No
Can ratios be made from two numbers with different units (without converting one of the units)?
No
Can there be a zero (0) in the denominator of a ratio?
C = 2πr
Circumference equation
3 (leg), 4 (leg), 5 (hyp.) 5 (leg), 12 (leg), 13 (hyp.) 7 (leg), 24 (leg), 25 (hyp.)
Common Pythagorean triples that you should know (3):
They are the same
Compare the tangents of ∠A and ∠B in 45-45-90 triangle ABC, where ∠C is right angle
Endpoints
Consecutive vertices are two vertices that are ______________ of the same side
Cosine = adjacent/hypotenuse
Cosine equation
Truncate
Cut-off or shortened (13 Archimedean Solids)
π / 180
Degrees to Radians unit conversion equation
d = √(x2 - x1)^2 + (y2 - y1)^2
Distance formula
No
Do ratios have units?
Face Vertex
Duality means the role of ____________ and ____________ have been reversed
Acute
Either __________ angle can be used to set up a trig ratio.
C = 2πr or C = πd
Equation for circumference
d = 2r r = d / 2
Equation for diameter and radius
Midsegment = (Length of base 1 + length of base 2) / 2
Equation for length of midsegment
y - y₁ = m (x - x₁)
Equation of a line: point-slope form
y = mx + b
Equation of a line: slope-intercept form
ax + by = c
Equation of a line: standard form
0.5 (the difference of the measures of the intercepted arcs)
If a tangents/secants intersect in the exterior of a circle, the measure of the angle formed is?
Base
If a trapezoid has one pair of congruent _____________ angles, then it is an isosceles trapezoid
Base angles
If a trapezoid is isosceles, then each pair of _____________ _____________ is congruent
Sides Congruent
If both pairs of opposite ______________ of a quadrilateral are ______________, then the quadrilateral is a parallelogram
Sides Parallel
If both pairs of opposite ______________ of a quadrilateral are ______________, then the quadrilateral is a parallelogram
Angles
If both pairs of opposite ______________ of a quadrilateral are congruent, then the quadrilateral is a parallelogram
cos^-1 (x) = m∠A
If cos A = x, then
Triangles
If each diagonal separates the quadrilateral into two congruent ______________, then the quadrilateral is a parallelogram.
Perpendicular Diameter
If one chord is a ______________ bisector of another chord, then the first chord is a ______________
Bisects Opposite Rhombus
If one diagonal of a parallelogram _____________ a pair of _____________ angles, then the parallelogram is a _____________
Congruent
If one pair of consecutive sides of a parallelogram are ____________, the parallelogram is a rhombus
Parallel Congruent
If one pair of opposite sides of a quadrilateral are ______________ and ______________, then the quadrilateral is a parallelogram
Right triangle
If one side of an inscribed triangle is a diameter of the circle, then the triangle is a ____________ ____________ and the angle opposite the diamter is the right angle
sin^-1 (x) = m∠A
If sin A = x, then
tan^-1 (x) = m∠A
If tan A = x, then
circle P or ʘP
If the center of the circle if P, then the circle is denoted by
Rhombus
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a _____________
Bisect
If the diagonals of a quadrilateral ______________ each other, then the quadrilateral is a parallelogram
Congruent Rectangle
If the diagonals of a rectangle are _____________, then the parallelogram is a _____________
Volume Congruence Postulate
If two polyhedra are congruent, they have the same volume
a^2 : b^2 a^3 : b^3
If two solids are similar with a scale factor of a:b, then corresponding areas have a ratio of ____________ and corresponding volumes have a ratio of ____________
Angle bisectors
If two triangles are similar, the lengths of corresponding ___________ ___________ are proportional to the lengths of corresponding sides
Altitudes
If two triangles are similar, the lengths of corresponding ___________ are proportional to the lengths of corresponding sides
Medians
If two triangles are similar, the lengths of corresponding ___________ are proportional to the lengths of corresponding sides
No.
If two triangles have same base and height, must they be congruent?
Yes.
If two triangles have same base and height, must they have the same area?
LA = π(R1 + R2) √(R2 - R1)^2 + h^2 or: LA = π(R1 + R2) * l (Everything from R2 - to h^2 is being square rooted)
Lateral Area of a Frustum
LA = (1/2)Pl
Lateral Area of a Regular Pyramid
LA = (1/2)Cl
Lateral Area of a Right Cone
LA = Ch
Lateral Area of a Right Cylinder
LA = Ph
Lateral Area of a Right Prism
LA = Cl
Lateral Area of an Oblique Cylinder
1 : √3 : 2
Side-length ratios of leg : leg : hyp in a 30° - 60° - 90°
1 : 1 : √2
Side-length ratios of leg : leg : hyp in a 45° - 45° - 90° triangle
fraction circumference
Since an arc is a portion of a circle, its length is a _____________ of its _____________
Sine = opposite/hypotenuse
Sine equation
l^2 = h^2 + r^2 -l = slant height -h = height -r = radius
Slant height equation
SA = LA + B1 + B2
Surface Area of a Frustum
SA = 3π(r^2)
Surface Area of a Hemisphere
SA = B + (1/2)Pl -B = area of base -P = perimeter of base -l = slant height
Surface Area of a Regular Pyramid
SA = B + (1/2)Cl -B = area of base -C = circumference of base -l = slant height
Surface Area of a Right Cone
SA = 2B + Ch or: SA = 2(π(r^2)) + (2πr)h -B = area of base -C = circumference of base -h = height between bases
Surface Area of a Right Cylinder
SA = 2B + Ph -B = area of base -P = perimeter of base -h = height between bases
Surface Area of a Right Prism
SA = 4π(r^2)
Surface Area of a Sphere
SA = 2B + Cl -l = slant height
Surface Area of an Oblique Cylinder
If △ABC ~ △DEF, then △DEF~ △ABC
Symmetric Property of Similarity
Base
The midsegment of a trapezoid is parallel to each _____________ and its length is half the sum of the lengths of the bases
Legs
The midsegment of a trapezoid is the segment that connects the midpoints of the trapezoid's _____________
Legs
The nonparallel sides of a trapezoid are called _____________
Volume
The number of cubic units contained in the interior of a solid
Bases
The parallel sides of a trapezoid are called _____________
Faces
The skeletal view shows all ____________ from a certain perspective
Lengths Angle Trig.
To "Solve a Right Triangle" means to find all unknown side ______________ and all unknown ______________ measures. You can use ______________ for all measures, or sometimes you can use Pythagorean Theorem for a missing side length.
If △ABC ~ △DEF, and △DEF~ △XYZ, then △ABC ~ △XYZ
Transitive Property of Similarity
Acute
Trig functions can be used not only to find the unknown leg or hypotenuse lengths, they can also be used to find the ______________ angles of the triangle.
Proportion
Two equal ratios are called a __________
Equivalent ratios
Two ratios that have the same simplified form are __________ __________
Net
Two-dimensional drawing of a 3-dimensional object
similar
all circles are ____________
Lateral surface
all the segments that connect the vertex with the points on the edge of the base (OF A CONE)
arc Two
an _____________ is a measure in degrees between _____________ points on a circle
Radius (of a sphere)
any segment whose endpoints are the center point and a point on the sphere
A=(d1⋅d2)/2 d=diagonal
area of a kite formula
A=bh b=base h=height
area of a parallelogram
A=bh b=base h-height
area of a rectangle
A=(d1⋅d2)/2 d=diagonal
area of a rhombus formula
A=s^2 s=side
area of a square
A=0.5h(b1 + b2)
area of a trapezoid formula
A=(1/2)bh b=base h=height
area of a triangle
A=(s²√3)/4 s=side
area of an equilateral triangle
concentric circles
coplanar circles that have the same center but different radii
Slant height of a cone
distance between the vertex and a point on the edge of the base
common external tangent
does not intersect the segment joining the centers
Regular pyramid
has a regular polygon as a base, segment joining the vertex and center of the base is perpendicular to the base. The lateral faces are congruent isosceles triangles.
Cavalieri's Principle
if two solids have the same height and same cross sectional area at every level, then they have the same volume.
Height of pyramid
perpendicular distance between base and vertex
Lateral faces
rectangular faces between two bases
Lateral edges
segments connecting two lateral faces
s=(a+b+c)/2 a, b, c=triangle sides
semiperimeter formula
(x-h)^2 + (y-k)^2 = r^2
standard equation of a circle
360°
sum of the central angles of a circle
tangent line one
the ____________ is a ____________ in the plane of a circle that intersects the circle in exactly ____________ point
minor arc shortest equal
the _____________ _____________ is the _____________ arc connecting two endpoints on a circle; less than 180°; named with two letters; _____________ to the measure of its related central angle
major arc longest equal
the _____________ _____________ is the _____________ arc connecting two endpoints on a circle; more than 180°; named with 3 letters; _____________ to the measure of its related central angle
circumscribed circle
the circle that contains the vertices of an inscribed polygon
circumference
the distance around a circle
arc length
the distance between the endpoints along an arc measured in linear units
Great Circle
the intersection formed when a plane contains the center of a sphere
Lateral edge of pyramid
the intersection of two lateral faces
arc addition postulate
the measure of an arc formed by two adjacent arcs is the *sum of the measures* of the two arcs
center
the point from which all points of a circle are equidistant
point of tangency
the point where a tangent intersects the circle
interior
the points *inside* the circle
exterior
the points *outside* the circle
l / 2πr = x / 360 or l / πr = x / 180
the ratio of the lengths of an arc 'l' to the circumference of the circle is equal to the ratio of the degree measure of the arc to 360°
Sphere
the set of all points in space equidistant from a given point called the center
Similarity statement
△ABC ~ △DEF is an example of a _________ _________
Oblique prism
"leaning prism", lateral sides are not perpendicular to bases
Height
(h), altitude
Altitude
(h), segment perpendicular to both bases
Slant height
(l), length of the oblique side
external (whole) = external (whole)
*segments of secants theorem* if two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment
Quadrilateral
4 sides
Regular polyhedron Faces
A ____________ ____________ is a polyhedron with all faces that are regular polygons. The same number of ____________ meet at each vertex in the same way.
Cross Section Plane
A ____________ ____________ is an intersection of a ____________ and a solid
Polyhedron Polygons
A ____________ is a solid bounded by ____________ that enclose a single region of space
circle points equidistant center
A ____________ is the set of all ____________ *in a plane* that are ____________ from a given point called the_____________ of the circle
Trapezoid
A _____________ is a quadrilateral with one pair of parallel sides which are called bases.
Kite
A _____________ is a quadrilateral with two pairs of consecutive congruent sides, but opposite sides are not congruent
Vertices
A diagonal is a segment that connects 2 nonconsecutive ______________
Consecutive Opposite
A kite is a quadrilateral with two pairs of _____________ congruent sides, but _____________ sides are not congruent
Parallelogram Sides Right angles
A square is a _____________ with 4 congruent _____________ and 4 _____________ _____________
Isosceles trapezoid
A trapezoid in which the *legs* are congruent
Similar Value Measure Size
Because all right triangles that have a given measure for ∠A are __________, the __________ of a trig. ratio depends only on the __________ of ∠A. It does not depend on the __________ of the triangle.
Rhombi
Bent or turned (13 Archimedean Solids)
Supplementary
Interior angles and exterior angles of a regular polygon are ______________
Supplementary
Opposite angles of inscribed quadrilaterals are?
Leg 1 Leg 2 Hypotenuse
Parts of a right triangle
P = number of sides * side length
Perimeter of polygon equation
Altitude of a cone
Perpendicular distance between the vertex and the plane containing the base
Parallel Half
*Midsegment Theorem* The segment connecting the midpoints of two sides of a triangle is _________ to the third side and is _________ as long as that side.
Midpoints
*Midsegment of a triangle* A segment that connects two _________
sum = 360
*Polygon Exterior Angles Sum Theorem* The sum of the measures of the exterior angles, one from each vertex of a convex polygon, is
Sum = ( n - 2 ) 180
*Polygon Interior Angles Theorem* The sum of the measures of the interior angles of a *convex* n-gon is
Right
*Properties of a rectangle:* All four angles are _____________ angles
Supplementary
*Properties of a rectangle:* Consecutive angles are _____________
Bisect
*Properties of a rectangle:* Diagonals _____________ each other
Congruent
*Properties of a rectangle:* Opposite angles are _____________
Parallel Congruent
*Properties of a rectangle:* Opposite sides are _____________ and _____________
At least two numbers
How many numbers do ratios have to have?
Diameter
If 1 chord is an angle bisector of another chord, 1st chord is a?
0.5 (sum of measures of the arcs intercepted by the angle and its vertical angle)
If 2 chords intersect in interior of circle, the measure of the angles formed is?
Chord and the arc
If a diameter is perpendicular to a chord, then the diameter bisects the?
Perpendicular Arc
If a diameter of a circle is ______________ to a chord, then the diameter bisects the chord and its ______________
Four
If a parallelogram has one right angle, then it has ______________ right angles
Congruent
If a parallelogram is a rectangle then its diagonals are _____________
Bisects
If a parallelogram is a rhombus, then each diagonal _____________ a pair of opposite angles
Diagonals
If a parallelogram is a rhombus, then its _____________ are perpendicular
Parallelogram
If a quadrilateral is a ______________, then its opposite angles are congruent
Angles
If a quadrilateral is a kite, then exactly one pair of opposite _____________ are congruent
Perpendicular
If a quadrilateral is a kite, then its diagonals are _____________
Congruent triangles
If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two ______________ ______________.
Opposite sides
If a quadrilateral is a parallelogram, then its ______________ ______________ are congruent
Consecutive
If a quadrilateral is a parallelogram, then its ______________ angles are supplementary
Diagonals
If a quadrilateral is a parallelogram, then its ______________ bisect each other
Square
If a quadrilateral is both a rectangle and a rhombus, then it is a _____________
Right prism
lateral sides are perpendicular to both bases
Opposite side Proportional
If a ray bisects an angle of a triangle, then it divides the ___________ ___________ into segments whose lengths are ___________ to the lengths of the other two sides of the triangle
Inscribed Hypotenuse
If a right triangle is ____________ in a circle, then the ____________ is a diamteter of the circle
Consecutive
Polygons can be named their shape (like pentagon) and their vertices (like ABCDE, DEABC, etc.) [pentagon ABCDE, for example], *as long as it is done in* ______________ order
Parallelogram
Quadrilateral with both pairs of opposite sides parallel
a : b
Scale factor of similar solids
Bases
two parallel congruent sides
common tangent
a line, ray, or segment that is tangent to two circles in the same plane
Polygon
a plane figure that is formed by 3 or more segments called sides
Regular polygon
a polygon where all angles and all sides are congruent
circumscribed polygon
a polygon whose sides are tangent to a circle
inscribed polygon
a polygon whose vertices all lie on a circle
Pyramid
a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex
Prism
a polyhedron with two parallel congruent sides, it is classified according to the shape of the base
Consecutive vertices
two vertices that are endpoints of the same side
Square Less Acute
*Pythagorean Inequality Theorem #1* If the _____________ of the length of the longest side of a triangle is _____________ than the sum of the squares of the lengths of the other two sides, then the triangle is an _____________ triangle.
Acute
*Pythagorean Inequality Theorem #1: Simplified* If c² < a² + b², then the triangle is an _____________ triangle
Square More Obtuse
*Pythagorean Inequality Theorem #2* If the _____________ of the length of the longest side of a triangle is _____________ than the sum of the squares of the lengths of the other two sides, then the triangle is an _____________ triangle.
Obtuse
*Pythagorean Inequality Theorem #2: Simplified* If c² > a² + b², then the triangle is an _____________ triangle
a² + b² = c²
*Pythagorean Theorem Simplified* ____ + ____ = ____
Legs Hypotenuse
*Pythagorean Theorem* In a right triangle, the sum of the squares of the lengths of the _____________ is equal to the square of the length of the _____________
a² + b² = c²
*Pythagorean Triple* A set of three nonzero while numbers a, b, and c such that _____________
Multiples Triples
*Pythagorean triples: don't forget:* _____________ of the _____________ are also possibilities
a/b a:b
*Ratio* If a and b are two numbers or quantities, measured in the same units, and b ≠ 0, then the ratio of a to b can be written as __________ or __________
Scale factor
*Scale Factor of Two Similar Polygons* If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the _________ _________.
Angle Angle Sides
*Side-Angle-Side SAS Similarity Theorem* If an __________ of one triangle is congruent to an __________ of a second triangle and the lengths of the __________ including these angles are proportional, then the triangles are similar
Diagonal
a segment that connects 2 nonconsecutive vertices
Sides
*Side-Side-Side SSS Similarity Theorem* If corresponding __________ of two triangles are proportional, then the two triangles are similar
Chord
a segment whose endpoints are on the sphere
Congruent Proportional
*Similar Polygons* Two polygons are similar if their corresponding angles are _________ and the lengths of corresponding sides are _________.
Similar
*Theorem 8.1* If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are _____________ to the original triangle and to each other
Perimeter Sides
*Theorem: Perimeters of Similar Polygons* If two polygons are similar, then the ratio of their _________ is equal to the ratio of their corresponding _________.
Cone
a solid that has a circular base and a vertex that is not in the same plane as the base
1/2 Intercepted arc
If a tangent and a chord intersect, the measure of each angle formed is ___________ of the measure of the ______________ ___________
Corresponding chords
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their ______________ ______________ are congruent
1/2 the measure of the intercepted arc
*Theorems for inscribed angles:* If an angle is inscribed in a circle, then the measure of the angle equals _________________________________________
The extremes
a..... c — = — b..... d What are a and d called?
The angles are congruent
*Theorems for inscribed angles:* If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then _______________________________________________________
The means
a..... c — = — b..... d What are b and c called?
Proportionally Third side
*Triangle Proportionality Converse* If a line divides two sides of a triangle _________, then it is parallel to the _________ _________.
Parallel Proportionally
*Triangle Proportionality Theorem* If a line __________ to one side of a triangle intersects the other sides, then it divides the two sides _________
sum areas
*area addition postulate* the area of a region is the ____________ of the ____________ of its non-overlapping parts
area
*area congruence postulate* if two figures are congruent, then they have the same ____________
square
*area of a square postulate* The area of a square is the ____________ of the length of its side
(part x part) = (part x part)
*segments of chords theorem* if two chords intersect in a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the second chord
tangent^2 = external (whole)
*segments of secants and tangents theorem* if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment equals the product of the lengths of the secant segment and its external secant segment
Decagon
10 sides
ab √ab
*Geometric Mean* The geometric mean of two positive numbers a and b is the positive number x that satisfies: a..... x — = — x..... d So x² = __________ and x = __________
Similar
*Angle-Angle Similarity Postulate* If 2 angles of one triangle are congruent to 2 angles of another triangle, then the two triangles are __________
a : b a² : b²
*Area of Similar Polygons* If two polygons are similar with the lengths of corresponding sides in the ratio of a : b, then the ratio of their perimeters is ____________ and the ratio of their areas is ____________
Square Shortest Equal Square
*Converse of the Pythagorean Theorem* If the sum of the _____________ of the lengths of the _____________ sides of a triangle is _____________ to the _____________ of the length of the longest side, then the triangle is a right triangle
Right
*Converse of the Pythagorean Theorem: Simplified* If a² + b² = c², then the triangle is a _____________ triangle
Parallel lines Transversals
*Corollary 1* If three _________ _________ intersect two _________, then they divide the transversals proportionally
Three Every
*Corollary 2* If _________ or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on _________ transversal.
Exterior angle = 360 / n
*Corollary of Polygon Exterior Angles Sum Theorem* The measure of each exterior angle of a regular polygon is
Int. angle = ( n - 2 ) 180 / n
*Corollary to Polygon Interior Angles Theorem* The measure of each interior angle of a regular n-gon is
Equals
*Cross Product Property of Proportions* In a proportion, the product of the extremes __________ the product of the means
F + V = E + 2
*Euler's Theorem* The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by ____________ (equation)
Altitude Geometric mean Segments
*Geometric Mean (Altitude) Theorem* The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of this _____________ is the _____________ _____________ between the lengths of these two _____________ of the hypotenuse.
PAAP
*Geometric Mean (Altitude) Theorem: Simplified* Part 1 ....Alt. ----- = ----- Alt. .....Part 2
Leg Geometric mean Hypotenuse Segment
*Geometric Mean (Leg) Theorem* The altitude drawn to the hypotenuse of a right triangle separates the hypotenuse into two segments. The length of a _____________ of this triangle is the _____________ _____________ between the length of the _____________ and the _____________ of the hypotenuse adjacent to that leg.
HLLP
*Geometric Mean (Leg) Theorem: Simplified* Hyp. ....Leg ----- = ----- Leg ......Part
Hendecagon *or* Undecagon
11 sides
Dodecagon
12 sides
N-gon Ej: Fifteen-gon
13-19 and 21+ sides
Icosagon
20 sides
Triangle
3 sides
Pentagon
5 sides
Hexagon
6 sides
Septagon *or* Heptagon
7 sides
Octagon
8 sides
Nonagon
9 sides
Opposite sides
A parallelogram is a quadrilateral with both pairs of ______________ ______________ parallel
Segments
A polygon is a plane figure that is formed by 3 or more ______________ called sides
Inscribed polygon Vertices
A polygon is an ____________ ____________ if all of its ____________ lie on a circle.
Supplemetary
A quadrilateral can be inscribed in a circle if and only if its opposite angles are ____________
Trig. ratio
A ratio of the lengths of two sides of a right triangle
Extended ratio
A ratio that compares more than two quantities and is expressed in form a : b : c : d is called an __________ __________
Parallelogram Right angles
A rectangle is a _____________ with 4 _____________ _____________
Angles Sides
A regular polygon is a polygon where all ______________ and all ______________ are congruent
Parallelogram Sides
A rhombus is a _____________ with 4 congruent _____________
Diagonals
A trapezoid is isosceles if and only if its _____________ are congruent
Proportionality statement
AB..... BC.... AC —— = —— = —— DE..... EF..... DF is an example of a _________ _________
Parallelograms Rectangles Rhombi
All of the properties of _____________, _____________, and _____________ apply to squares
Inscribed angle Vertex Chords
An ____________ ____________ is an angle whose ____________ is on the circle and whose sides contain ____________ of the circle
Intercepted arc Interior Endpoints
An ____________ ____________ is the arc that lies in the ____________ of an inscribed angle and has ____________ on the angle
Legs
An isosceles trapezoid has congruent _____________
Radian measure. θ = l/r
Angles can also be measured in units based on arc length. The _____________ _____________, θ, of a central angle is the ratio of the arc length to the radius of the circle. *θ = ___________*?
Convex polyhedron Inside Face
Any two points on the surface of a ____________ ____________ can be connected by a segment that lies entirely ____________ or on the ____________ of the polyhedron
(Arc length / 2πr) = (m<angle / 360)
Arc length proportion
A = (1/2)Pa P = perimeter a = apothem
Area formula for a regular polygon
Area(segment) = A(sector) - A(triangle)
Area of a Segment of a circle
A = π r²
Area of a circle
A = (m of arcAB / 360°) (π * r²)
Area of a circle sector equation
A / (π * r²) = m of arcAB / 360°
Area of a circle sector proportion
Area(whole) - Area(unshaded) = Area(shaded)
Area of shaded region formula
180 degrees
For polygon to be convex, polygon cannot have interior angles bigger than
45-45-90 triangle
For what angle are sine and cosine equal
>0 and <45 degrees
For what angle measures is cosine bugger than sine
>45 and <90 degrees
For what angle measures is sine bigger than cosine
convex
Formulas work for ______________ polygons only
√s(s-a)(s-b)(s-c) s=*semiperemeter*
Heron's Formula (for triangles where 3 side lengths are known)
They are reciprocals
How does the tangent of one acute angle of a right triangle compare to a tangent of other acute angle of the triangle
Twice
In a 30° - 60° - 90° triangle, the hypotenuse is __________ as long as the shorter leg
√3
In a 30° - 60° - 90° triangle, the longer leg is __________ as long as the shorter leg
√2
In a 45° - 45° - 90° triangle, the hypotenuse is __________ times as long as each leg
They're equidistant from the center
In same or congruent circles, 2 chords are congruent iff?
Two chords Equidistant
In the same circle, or in congruent circles, ______________ ______________ are congruent if and only if they are ______________ from the center
Tetrahedron (4 △) Cube (6 ◻ - also called hexahedron) Octahedron (8 △) Dodecahedron (12 ⬠) Icosahedron (20 △)
List the 5 platonic solids (from smallest to largest) + their face shape and number of faces
360 / n n = number of sides of inscribed polygons
Measure of central angle equation
((x1 + x2) / 2), ((y1 + y2)/2)
Midpoint formula
Corresponding chords are congruent
Minor arcs are congruent iff?
Tetrahedron > Tetrahedron Cube > Octahedron Dodecahedron > Icosahedron
Name the 5 platonic solids by their duals
180 / π
Radians to Degrees unit conversion equation
a² : b²
Ratio of areas for similar polygons
a : b
Ratio of perimeters for similar polygons
a^2 : b^2
Ratio of the surface area of similar solids
a^3 : b^3
Ratio of the volume of similar solids
△ABC ~ △ABC
Reflexive Property of Similarity
Segment
Region bounded by a chord and the arc intercepted by the chord
Snub
Rounded (13 Archimedean Solids)
1
Scale Factor of Congruent Solids is?
Shorter angle = shorter opposite sides
Tangent (A) decreases as angle (m∠A) decreases because
Tangent = opposite/adjacent
Tangent equation
Congruent
Tangent segments from common external points are?
13 Archimedean Solids Semi-regular
The ____________ ____________ ____________ are ____________ with faces that consist of more than one type of regular polygon. The same number of each of type of face meet at each vertex
Central angle Radii
The ____________ ____________ of a regular polygon is the angle formed by the two ____________ used to make the triangle that contains the apothem
Edges Segments
The ____________ are the ____________ formed by the intersection of the faces
Faces Polygons
The ____________ are the ____________ of the polyhedron
Vertices Edges
The ____________ are the points where 3 or more ____________ intersect (also known as a corner)
Altitude Apothem
The ____________ from the center of the figure to any side of the polygon is called the ____________
Surface Polygon
The ____________ is the set of all points on the ____________ of a polyhedron
Radii Radii Same
The ____________ of an inscribed polygon and the ____________ of the circle are the ____________ length
Geometric mean A, B X
The _____________ _____________ of two positive numbers _____________ and _____________ is the positive number _____________ such that: .A...... X --- = --- .X...... B
Height Isosceles Radii
The apothem is the ____________ of an ____________ triangle that has two ____________ as legs
Match up
The center of an inscribed polygon and the circle ____________ ____________
Circumscribed circle
The circle that contains the vertices of an inscribed polygon is a ____________ ____________
Volume of a Cube
V = s^2
V = (1/3)h(B1 + B2 + √B1 • B2) (B1 • B2 is being square rooted)
Volume of a Frustum
V = (2/3)π(r^3)
Volume of a Hemisphere
V = (1/3)Bh -B = area of base -h = height
Volume of a Regular Pyramid
V = (1/3)Bh -B = area of base -h = height
Volume of a Right Cone
V = Bh -B = area of base -h = height between bases
Volume of a Right Cylinder
V = Bh -B = area of base -h = height between bases
Volume of a Right Prism
V = (4/3)π(r^3)
Volume of a Sphere
V = Bh
Volume of an Oblique Cylinder
Platonic solids
What are the five regular polygons called?
Similar
What does the symbol ~ mean?
Regular
What type of polygon can be inscribed into a circle?
Johannes Kepler
Who came up with the Platonic Solids *duals*?
Plato
Who came up with the Platonic Solids?
Polygons
____________ form the faces of a polyhedron
diameter center
a ____________ is a *chord* passing through the ____________ of the circle
secant two
a ____________ is a *line* that intersects a circle in exactly ____________ points
radius center
a ____________ is a *segment* with the ____________ as one endpoint and a point on the circle as the other endpoint
chord on
a ____________ is a segment whose endpoints are ____________ the circle
central angle vertex
a _____________ _____________ is an angle whose _____________ is the center of the circle
semicircle diameter
a _____________ is an arc with endpoints that are the endpoints of a _____________; 180°; named with 3 letters
Diameter
a chord that contains the center
Perpendicular
a line is a tangent iff the line is ___________ to the radius of the circle
x^2 + y^2 = r^2
equation of a circle with center at origin
Right cone
figure in which the vertex lies directly above the center of base
Frustum of a cone
formed by cutting the top off of a cone with a cut parallel to the base.
Slant height of a pyramid
in a regular pyramid it is the height of the lateral face
Base edge
intersection of the base and lateral face
common internal tangent
intersects the segment joining the centers
Hemisphere
the two congruent halves of a sphere which contain the center
Volume Addition Postulate
the volume of a solid equals the sum of the volume of the nonoverlapping parts
adjacent arcs one
these are arcs in a circle that have exactly _____________ point in common
congruent arcs measure
two arcs are _____________ _____________ if they have the same _____________ and are in the *same* circle or in *congruent* circles
congruent circles radius
two circles are _____________ _____________ if they have the same _____________
no one two
two circles can intersect in ____________ points, ____________ point, or ____________ points
congruent circles
two circles that have the same radius or diameter
