Final Review

circumference of a circle

C = 2πr or C = πd

the lateral area of a cylinder is equal to the product of the ____ and the ____ of the base ∴ L.A. of a cylinder = ____ = ____

Ch; 2πrh

if c² < a² + b², then the triangle is

acute

angles of elevation and depression in any one drawing are always congruent because they are

alternate interior angles

if two triangles are similar, the ratio of any pair of their corresponding ____, ____, or ____ ____ equals the ratio of their corresponding sides

altitudes; medians; angle bisectors

incenter: the point where the 3 ____ ____ of a triangle intersect

angle bisectors

if two chords of a circle (or of congruent circles) are congruent, then the corresponding ____ are congruent

arcs

congruent ____ ↔ congruent ____

arcs; chords

cylinders are/are not prisms

are not

every closed region has an ____

area

the ____ of a circle is the region inside the circle. it is measured in ____ ____

area; square units

if two closed figures are congruent then their ____ are equal

areas

if two figures are similar, then the ratio of their ____ equals the ____ of the ratio of corresponding segments (similar-figures theorem) ∴ A₁/A₂ =

areas; square; (s₁/s₂)²

the measure of the median of a trapezoid equals the ____ of the measures of the bases ∴ M =

average; ½(b₁ + b₂)

the 4 properties of a regular square pyramid are 1. the bottom of a pyramid is called the ____ 2. the very top of a pyramid is called the ____ 3. the altitude of a pyramid is ____ to the base at its ____ 4. the ____ ____ is perpendicular to a side of the base

base vertex perpendicular; center slant height

common internal tangent: a tangent that lies ____ the circles and intersects a segment joining the ____

between; centers

an area of a parallelogram is A =

bh

the area of a rectangle is

bh

if a radius is perpendicular to a chord, then it ____ ____ ____

bisects the chord

if an altitude is drawn to the hypotenuse of a right triangle, then each ____ is the geometric mean between the ____ ____ and the part of the hypotenuse ____ to that leg. the formula for this is:

c/a = a/y or a² = cy; c/b = b/x or b² = cx

lateral area of a cylinder 1. a cylinder is often visualized as a ____, in which the lateral area is the ____ and when spread out, looks like a ____ 2. the height of the rectangle is the ____ of the can 3. the base of the rectangle is the ____ of the can

can; label; rectangle height circumference

central angle: an angle whose vertex is at the ____ of a circle

center

concentric circles: two or more coplanar circles with the same ____

center

foot of the altitude: the ____ of the base

center

the center of an arc is the ____ of the circle of which it is a part

center

apothem of a regular polygon: a segment joining the ____ to the ____ of any side

center; midpoint

radius of a regular polygon: a segment joining the ____ to any ____

center; vertex

common-tangent procedure: 1. draw the segment joining the ____ 2. draw the ____ to the points of contact 3. through the center of the ____ circle, draw a line ____ to the common tangent 4. observe that this line will intersect the radius of the ____ ____ (extended if necessary) to form a ____ and a ____ ____ 5. use the ____ ____ and properties of a rectangle

centers radii smaller; parallel larger circle; rectangle; right triangle pythagorean theorem

major arc: an arc whose points are on or outside of a ____ ____

central angle

if two arcs of a circle (or of congruent circles) are congruent, then corresponding ____ ____ are congruent

central angles

if two chords of a circle (or of congruent circles) are congruent, then the corresponding ____ ____ are congruent

central angles

congruent ____ ____ ↔ congruent ____

central angles; arcs

congruent ____ ____ ↔ congruent ____

central angles; chords

segment of a circle: a region bounded by a ____ of the circle and its corresponding ____

chord; arc

diameter: a ____ that passes through the ____ of the circle

chord; center

a ____ is a line segment joining two points on a circle. a ____is the longest chord of a circle

chord; diameter

if two arcs of a circle (or of congruent circles) are congruent, then the corresponding ____ are congruent

chords

∴ congruent ____ ↔ congruent ____ ↔ congruent ____ ____

chords; arcs; central angles

if two ____ of a circle are equidistant from the center of a circle, then they are ____

chords; congruent

cone: a solid whose base is a ____ 1. resembles a pyramid but its base is a ____ 2. (in this case) cone means right circular cone - the ____ passes through the center of the circular base

circle circle altitude

total area of a sphere: equals four times the area of a ____ with the same ____ ∴ T.A. =

circle; radius; 4πr²

cylinder: a solid figure whose bases are ____ 1. resembles a prism - has two congruent ____ ____ 2. (in this case) a cylinder will be a right circular cylinder - the line containing the centers of the bases is ____ to each base

circles parallel bases perpendicular

common external tangent: a tangent that does not lie between the ____ and does not intersect the segment joining the ____

circles; centers

the center of a circle circumscribed about a polygon is the ____ of the polygon

circumcenter

the length of an arc is equal to the ____ of its circle times the ____ part of the circle determined by the arc (measured in ____) ∴ length of an arc =

circumference; fractional; degrees; (measure of arc/360)πd

a polygon is ____ ____ a circle if each of its sides is tangent to the circle

circumscribed about; tangent

the area of a ____ region is the number of square units of space within the boundary of the region

closed

if two closed regions intersect only along a ____ ____, then the are of their union is equal to the ____ of their individual regions

common boundary; sum

if two inscribed or tangent-chord angles intercept congruent arcs, then they are ____

congruent

if two inscribed or tangent-chord angles intercept the same arc, then they are ____

congruent

if two tangent segments are drawn to a circle from an external point, then those segments are ____ (two-tangent theorem)

congruent

important observations about apothems and radii: 1. all apothems of a regular polygon are ____ 2. only regular polygons have ____ 3. an apothem is the ____ of a circle ____ in a polygon 4. an apothem is a ____ ____ of a side 5. a radius of a regular polygon is a radius of a circle ____ about the polygon a radius of a regular polygon ____ an angle of the polygon

congruent apothems radius; inscribed perpendicular bisector circumscribed bisects

if 2 chords of a circle are ____ then they are ____ from the center of the circle

congruent; equidistant

concentric circles: two or more ____ circles with the same ____ (they have different ____)

coplanar; center; radii

if two central angles of a circle (or of congruent circles) are congruent, then the ____ ____ are congruent

corresponding chords

unlike the ____ ____ or a prism or a cylinder, a ____ ____ of a pyramid or a cone is not congruent to the figure's base. it is ____ to the base and ____ to the base

cross section; cross section; parallel; similar

the volume of a prism or a cylinder is equal to the product of the figure's ____ ____ and its ____ ∴ V of a prism or cylinder =

cross-sectional area; height; ⊄h

volume of a solid: the number of ____ units of space contained by a solid

cubic

the volume of a cone is related to the volume of a ____ having the same base and height. it is equal to ____ the product of the height and the area of the base ∴ V of a cone = ____ = ____

cylinder; ⅓; ⅓βh; ⅓; ⅓πr²h

semicircle: an arc whose endpoints are the endpoints of a ____

diameter

curved arrows are used to indicate ____ ____

directed angles

the volume of a cylinder is equal to the product of the ____ and the area of the ____ ∴ V of a cylinder = ____ = ____

height; base; βh; πr²

angle of depression: the angle formed by the ____ and line of sight looking downward

horizontal

angle of elevation: the angle formed by the ____ and the line of sight looking upward

horizontal

if an altitude is drawn to the ____ of a ____ triangle, then the two triangles formed are similar to the given triangle and to each other

hypotenuse; right

the ____ of an arc is a fractional part of the circle's circumference. it is expressed in ____ ____ such as feet, centimeters, or inches

length; linear units

the volume of a right rectangular prism is equal to the product of its ____, its ____, and its ____ ∴ V =

length; width; height; lwh

the measure of a ____ ____ is 360 minus the measure of the minor arc with the same endpoints

major arc

the ____ of an arc is equivalent to the number of ____ it occupies (a complete circle occupies how many degrees?)

measure; degrees; 360°

median of a trapezoid: the line segment joining the ____ of the nonparallel sides of a trapezoid

midpoints

principle of the reduced triangle: 1. reduce the difficulty of the problem by ____ or ____ the three lengths by the same number to obtain a similar, but simpler triangle in the same family 2. solve the missing ____ of this easier triangle 3. ____ back to the original problem

multiplying; dividing side convert

the number of diagonals in a polygon =

n(n-3)/2

clockwise rotations are

negative

to compute any ____, ____ length drawn in the coordinate plane, we could draw a right triangle and use the pythagorean theorem; however, it is easier to use the ____ ____, which is derived from the pythagorean theorem

non-vertical; horizontal; distance formula

radicand

number underneath the radical

if c² > a² + b², then the triangle is

obtuse

minor arc: an arc whose points are ____ or ____ the sides of a central angle

on; between

a point is ____ a circle if its distance from the ____ is equal to the radius

on; center

pyramid: a polyhedron that has only ____ ____ 1. lateral edges are not ____ 2. lateral edges meet at a point called the ____ 3. ____ ____ can be any type of polygon 4. lateral faces will always be ____ 5. named by the ____

one base parallel vertex the base triangles base

exterior points of a circle: points ____ of a circle

outside

externally tangent circles: two tangent circles that lie ____ of each other

outside

secant-secant angle: an angle whose vertex is ____ of a circle and whose sides are determined by two ____

outside; secants

tangent-tangent angle: an angle whose vertex is ____ a circle and whose sides are determined by two ____

outside; tangents

prism: one type of polyhedron 1. bases: the ____ and ____ faces 2. lateral edges: the parallel edges joining the ____ ____ of the ____ 3. lateral faces: the faces of the prism that are not ____ (always ____ ) 4. named by their ____

parallel; congruent vertices; bases bases; parallelograms bases

a tangent line is ____ to the radius drawn to the point of contact

perpendicular

if a radius of a circle bisects a chord that is not a diameter, then it is ____ to that chord

perpendicular

the ____ ____ of a chord passes through the center of the circle

perpendicular bisector

circumcenter: the point where 3 ____ ____ of a triangle meet

perpendicular bisectors

the distance from the center of a circle to a chord is the measure of the ____ ____ from the center to the chord

perpendicular segment

cross section: the intersection of a solid with a ____. unless otherwise noted, cross sections will be ____ to the base

plane; parallel

tangent segment: the part of a tangent line between the ____ ____ ____ and a point ____ ____ ____

point of contact; outside the circle

circle: the sets of all ____ in a ____ that are a given distance from a given point in the plane

points; plane

polyhedra ("many faces"): solids with flat faces 1. the faces are ____ 2. the lines where they intersect are called ____

polygons edges

counterclockwise rotations are

positive

if p = (x₁, y₁) and q = (x₂, y₂) are two points, then the distance between them can be found with the formula:

pq = √(x₂-x₁)² + (y₂+y₁)² or pq = √(∆x)² + (∆y)²

a radical sign asks for the ____ square root; therefore, the answer will not be ____

principal; negative

the volume of a pyramid is related to the volume of a ____ having the same base and height. it is equal to ____ the product of the height and the area of the base ∴ V of a pyramid =

prism; ⅓; ⅓βh

the area of a trapezoid is the ____ of the median and the height ∴ A =

product; Mh

if two chords of a circle intersect inside the circle, then the ____ of the measures of the segments of one chord is equal to the ____ of the measures of the segments of the other chord

product; product

one way to determine the ratio of the areas of two figures is to calculate the ____ of the two ratios

quotient

congruent circles: two circles that have congruent ____

radii

sector of a circle: the region bounded by two ____ and an ____ of the circle

radii; arc

if a parallelogram is inscribed in a circle, it must be a ____

rectangle

if the lateral edges are perpendicular to the bases, then the lateral faces will be ____

rectangles

cube: a right ____ ____ wiht congruent edges (all of its faces are squares)

rectangular prism

regular pyramid: 1. has a ____ ____ as its base 2. has congruent ____ ____ 3. lateral faces are congruent ____ ____

regular polygon lateral edges isosceles triangles

the area formula for a kite can also be applied to a ____ and a ____

rhombus; square

if c² = a² + b²

right

angle measures are generated from a ____

rotation

congruent arcs: arcs that have the same measure and are parts of the ____ ____ or ____ ____

same circle; congruent circles

secant segment: the part of a ____ line that joins a point outside the circle to the ____ intersection point of the secant and the circle

secant; farther

external secant segment: a secant segment is the part of a ____ line that joins the outside point to the ____ intersection point

secant; nearer

if two ____ segments are drawn from an external point to a circle, then the product of the measures of one ____ segment and its ____ ____ is equal to the product of the measures of the other ____ segment and its ____ part (secant-secant power theorem)

secant; secant; external part; secant; external

secant-tangent angle: an angle whose vertex is outside a circle and whose sides are determined by a ____ and a ____

secant; tangent

a ____ of a circle is a region bounded by two radii and an arc of the circle. it is a fractional part of a circle and measured in ____ ____

sector; square units

chord: a ____ joining any ____ ____ on the circle

segment; 2 points

an angle inscribed in a ____ is a right angle

semicircle

corresponding segments can be any segments associated with the figures, such as ____, ____, ____, ____, or ____

sides; altitudes; medians; diagonals; radii

the three basic trigonometric ratios are ____, ____, and ____. these ratios involve the sides of a ____ triangle

sine (sin); cosine (cos); tangent (tan); right

the lateral area of a cone is equal to one-half the product of the ____ ____ and the ____ of the base ∴ L.A. = ____ = ____

slant height; circumference; ½Cl; πrl

the three ratios can be remembered by the anagram

sohcahtoa

pythagorean theorem: the ____ of the measure of the hypotenuse of a right triangle is equal to the ____ of the ____ of the measures of the legs. the formula for this is:

square; sum; squares; c² = a² + b²

in a pyramid or a cone, the ratio of the area of a cross section to the area of the base equals the ____ of the ratio of the figure's respective distances from the vertex ∴ the ratio is

square; ⊄/β = (k/h)²

an angle is in ____ ____ (with respect to the x,y coordinate system if...) 1. its vertex is at the ____ (0,0) 2. its initial side coincides with the ____ ____

standard position; origin; positive x-axis

if a quadrilateral is inscribed in a circle, its opposite angles are ____

supplementary

the area of a square is

s²

common tangent: a line ____ to two circles (aka. tangent line)

tangent

if a line is perpendicular to a radius at its outer endpoint, then it is ____ to the circle

tangent

if a ____ segment and a ____ segment are drawn from an ____ point to a circle, then the ____ segment is the geometric mean between the whole ____ segment and the ____ ____ segment ∴ the ____ of the measure of the ____ segment is equal to the ____ of the measure of the ____ segment and the external secant segment

tangent; secant; external; tangent; secant; external secant; square; tangent; product; secant

coterminal angles: angles that have their ____ ____ coincide when they are in ____ ____. coterminal angles start and stop in the same place but ____ ____ ____ ∴ coterminal angle =

terminal sides; standard position; have different measures; measure of the given angle + n(360°)

β =

the area of the base

circumference

the distance around a circle

the measure of a minor arc or semicircle (in degrees) is ____ ____ ____ the measure of the ____ ____ that intercepts the arc

the same as; central angle

the volume of a hemisphere =

²/₃πr³

the measure of an inscribed angle is ____ the measure of its intercepted arc

½

the measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle (vertex outside a circle) is ____ ____ ____ of the measures of the intercepted arcs

½ the difference

area of a triangle =

½bh

the volume of any prism is equal to the product of the ____ and the ____ of the base ∴ V of a prism =

βh

volume of a rectangular box =

βh

total area of a cone: the sum of the lateral area and the area of the base ∴ T.A. of a cone =

πrl + πr²

the volume of a sphere is equal to ____ of the product of π and the cube of the radius ∴ V of a sphere =

⁴/₃; ⁴/₃πr³

radical

√

area of a cyclic quadrilateral (brahmagupta's formula) =

√(s - a)(s - b)(s - c)(s - d)

area of a triangle (hero's/heron's formula) =

√s(s - a)(s - b)(s - c)

how many pythagorean triple families are there?

∞ many

how many radii in a circle?

∞ many

s = semiperimeter for brahmagupta =

(a + b + c + d)/2

s = semiperimeter for hero =

(a + b + c)/2

each exterior angle =

360/n

total area of a cylinder: the sum of the cylinder's ____ ____ and the areas of the ____ ____ ∴ T.A. of a cylinder =

lateral area; two bases; 2πrh + 2πr²

total surface area of a prism: the sum of the prism's ____ ____ and the areas of the two ____

lateral areas; bases

sphere: a solid figure with no ____ ____ and no ____ ____ (a ____)

lateral edges; lateral area; ball

slant height: the height of a ____ ____

lateral face

lateral surface area of a prism: the sum of the areas of the ____ ____

lateral faces

radius

a segment drawn from the center to a point on the circle

diameter

a segment that passes through the center of the circle with both endpoints on the circle

____° = one counterclockwise revolution

360

sum of exterior angles =

360

sum of interior angles =

(n-2)180

the area of an equilateral triangle is A =

(s²/4)(√3)

the area of a trapezoid is

(½h)(b₁ + b₂)

geometry: angle measures are between ____ and ____ trigonometry: angles can have ____ measures, both ____ and ____

0°; 180° arbitrary; positive; negative

____° = ¹/₃₆₀ (one counterclockwise revolution)

1

the sum of the measures of a tangent-tangent angle and its minor arc is ____

180

each interior angle =

180 - (360/n)

tangent: a line that intersects a circle at exactly ____. this point is called the ____ ____ ____ or ____ ____ ____

1; point of tangency; point of contact

arc: consists of ____ points on a circle and all points on the circle needed to connect the points by a single path

2

secant: a line that intersects a circle at exactly ____ points (every secant contains a ____ of the circle)

2; chord

major arcs are written with ____ letters

3

the basic pythagorean triple is ____ and any ____ of it is also a pythagorean triple

3-4-5; multiple

in any triangle whose angles have the measures ____, ____, and ____, the lengths of the sides opposite these angles can be represented by ____, ____, and ____. (____ triangle theorem)

30; 60; 90 x; x√3; 2x

other common pythagorean triple familes are

5-12-13; 7-24-25; 8-15-17

the 4 properties of a rectangular solid are 1. there are ____ rectangular faces 2. there are ____ edges 3. there are ____ diagonals 4. a ____ is a rectangular solid in which all ____ are congruent

6 2 4 cube; edges

the area of a sector of a circle is

A = (measure of arc in degrees/360)πr²

the area of a kite is

A = ½ d₁ d₂

the area of a regular polygon is

A = ½ap

area of a circle

A = πr²

arc

a curve with both endpoints on the circle (a part of the circle)

a median of a triangle divides the triangle into two triangles with ____ ____

equal areas

areas are ____ rather than ____

equal; congruent

tangent circles: circles that intersect each other at ____ ____ ____ ____

exactly at 1 point

if an altitude is drawn to the hypotenuse of a right triangle, then that altitude is the ____ ____ (____ ____) between the two parts of the hypotenuse. the formula for this is:

geometric mean; mean proportional; x/h = h/y or h² = xy

the area of the triangle is ____ the area of the parallelogram

half

the center of a circle inscribed in a polygon is the ____ of the polygon

incenter

angles rotate from the ____ ____ to the ____ ____

initial side; terminal side

cyclic quadrilaterals: quadrilaterals that can be ____ in a circle

inscribed

a polygon is ____ ____ a circle if all of its vertices lie on the circle

inscribed in

interior points of a circle: points ____ of a circle

inside

internally tangent circles: two tangent circles that lie ____ each other

inside

the ____ ____ is the part of the circle that lies in the interior of the given angle

intercepted arc

if two central angles of a circle (or of congruent circles) are congruent, then their ____ ____ are congruent

intercepted arcs

regular polygon: all ____ ____ are congruent and ____ are congruent

interior angles; sides

the center is/is not part of a circle

is not

an ____ ____ triangle theorem has measures 45, 45, and 90, the lengths of the sides opposite these angles can be represented by ____, ____, and ____ (45°-45°-90° triangle theorem)

isosceles right x; x; x√2

an inscribed angle is an angle whose ____ is on the circle and whose sides are chords of the circle

vertex

altitude of a regular pyramid: perpendicular segment from the ____ to the base

vertex; base

cubic unit: the ____ of a cube with edges ____ unit long

volume; one

pythagorean triple: any three ____ numbers that satisfy the equation ____

whole; c² = a² + b²