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