Geometry H Final Exam

Rectangle Area Formula

*A* = *bh*

Parallelogram Area Formula

*A* = *bh*

Triangle Area Formula

*A* = ½*bh*

Trapezoid Area Formula

*A* = ½(*b*₁+b₂)*h*

Kite Area Formula

*A* = ½*d*₁*d*₂

apothem

a perpendicular segment from the center of a regular polygon to a side

Regular Polygon Area Formula

*A* = ½*asn*

Circle Area Formula

*A* = π*r*²

Sector Area Formula

*A* = (*θ*/360)π*r*²

Segment of a Circle Area Formula

*A* = (*θ*/360)π*r*² − ½*bh*

annulus

the region between two concentric circles

annulus area formula

A = π*R*² - π*r*²

prism surface area formula

*SA* = 2*B* + *Ph*

Surface area of a cylinder

*SA* = 2π*r*² + 2π*rh*

pyramid surface area formuoa

*SA* = ½*Ps*

cone surface area formula

*SA* = π*rl* + π*r*²

surface area of a sphere

4π*r*²

sine

opposite over hypotenuse

cosine

adjacent over hypotenuse

tangent

opposite over adjacent

angle of elevation

the angle formed by a horizontal line and the line of sight to an object above the horizontal line

angle of depression

the angle formed by a horizontal line and the line of sight to an object below the horizontal line

Law of Sines

sin*A*/*a* = sin*B*/*b* = sin*C*/*c*

law of cosine

*c*²=*a*²+*b*²-2*ab*cos*C*

non-right triangle area formula

*A* = ½*ab*sin*C*

similar polygons

have the same shape but not necessarily the same size

angles of similar polygons

congruent

sides of similar polygons

proportional

dilated polygons

similar

Angle-Angle similartiy

if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar

side-side-side similarity

if the corresponding side lengths of two triangles are proportional, then the triangles are similar

side-angle-side similarity

if an angle of a triangle is congruent to an angle of another triangle and if the included sides of these angles are proportional, then the two triangles are similar

proportional parts

if two triangles are similar, ten the corresponding altitudes, medians and angle bisectors are proportional to the corresponding sides

triangle angle bisector theorem

if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides

areas of similar polygons

if corresponding side lengths of two similar polygons or radii of two circles compare in the ratio *m*/*n*, then their areas compare in the ratio *m*²/*n*²

surface area of similar solids

if corresponding side lengths of two similar polygons or radii of two circles compare in the ratio *m*/*n*, then their surface areas compare in the ratio *m*²/*n*²

volume of similar solids

if if the corresponding edge lengths, radii, or heights of two similar solids compare in the ratio *m*/*n,* then their volumes compare in the ratio *m*³/*n*³

triangle proportionality theorem

if a line || to one side of a triangle intersects the other two sides, then it divides the two sides proportionallly

extended parallel/proportionality

if two or more lines pass through two sides of a triangle || to the third side, then they divide the two sides proportionally

three parallel lines theorem

if three parallel lines intersect two transversals, then they divide the transversals proportionally

polyhedron

a solid formed by polygonal surfaces that enclose a single region of space

regular polyhedron

a polyhedron enclosed by congruent regular polygons that meet at each vertex exactly in the same way

tetrahedron

a polyhedron with four triangular faces

cube

a prism with six square faces

octahedron

polyhedron with 8 triangular faces

dodecahedron

polyhedron with 12 pentagonal faces

icosahedron

polyhedron with 20 triangular faces

prism

a special type of polyhedron with two faces (bases) that are congruent, parallel polygons that are connected by parallelograms (lateral faces with lateral edges)

altitude

any perpendicular segment from one base to the plane of the other base (height of the prism)

right prism

each lateral edge is perpendicular to both bases

oblique prism

a prism with lateral edges not perpendicular to the bases

axis

an imaginary line about which a body rotates

apex

the highest point

vertex

each angular point of a polygon, polyhedron, or other figure

right pyramid

a pyramid whose faces are isosceles triangles

great circle

the circle that encloses the base of a hemisphere

hemisphere

half of a sphere

volume

the measure of the amount of space contained in a solid

prism volume formula

*V* = *Bh*

cylinder volume formula

*V* = π*r*²*h*

pyramid volume formula

*V* = ⅓*Bh*

cone volume formula

*V* = ⅓π*r*²*h*

sphere volume formula

*V* = ⁴⁄₃π*r*³

sphere surface area formula

*V* = 4π*r*²

displacement

the fluid that rises above the original fluid line when a solid object is submerged in the fluid

density

the ratio of the mass of an object to its volume

Pythagorean theorem

in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenusec

converse of the Pythagorean theorem

if the lengths of the three sides of a triangle satisfy the Pythagorean equation, then the triangle is a right triangle

Pythagorean Inequalities theorem

If c²>a²+b², then the triangle is obtuse. If c²

45-45-90 triangle

if the legs have the same length, then the hypotenuse is √2 times as long as either leg.

30-60-90 triangle

If the shorter leg is "*a*" the hypotenuse is 2*a* and the longer leg is a times square root of 3

distance formula

*d* = √\[( *x*₂ - *x*₁)² + (*y*₂ - *y*₁)²\]

equation of a circle

(*x*-*h*)² + (*y*-*k*)² = r²

radius

a segment whose endpoints are the center and any point of the circle

chord

a segment whose endpoints are on a circle

diameter

a chord that contains the center of the circle

secant

a line that intersects a circle in two points

tangent

a line that intersects a circle in exactly one point

point of tangency

the point where a circle and a tangent intersect

arc

a portion of the edge of a circle

minor arc

arc measuring less than 180 degrees

major arc

an arc that measures greater than 180 degrees

central angle

an angle whose vertex is the center of the circle

inscribed angle

an angle whose vertex is on the circle and whose sides pass through the endpoints of the arc

intercepted arc

the arc that lies in the interior of an inscribed angle and has endpoints on the angle

arc measure

the measure of the central angle that intercepts an arc, measured in degrees

tangent segment

a line segment that lies on a tangent line with one endpoint at the point of tangency

tangent circles

circles that are tangent to the same line at the same point

inscribed polygon

a polygon in which all of the vertices lie on a circle

circumscribed circle

the circle that contains the vertices of an inscribed polygon

cyclic quadrilateral

a quadrilateral that can be inscribed in a circle

tangent is ___ to a radius connected to the point of tangency

perpendicular

congruent chords form

congruent arcs

if the diameter of the circle is perpendicular to a chord

the diameter bisects the chord and its arc

two chords are congruent if they are ___ from the center

equidistant

perpendicular bisector of a chord

passes through the center of the circle

if two chords in a circle are congruent

their central angles are congruent

the measure of an inscribed angle

half the measure of the intercepted arc

inscribed angles intercepting the same arc

congruent

angles inscribed in a semicircle

right angles