Geometry exam semester 2

standard equation of a circle

(x-h)² + (y-k)² = r²

properties of proportions (major props)

1) a/b=c/d is equivalent to b/a=d/c 2) a/b=c/d is equivalent to a/c=b/d 3) a/b=c/d is equivalent to a+b/b=c+d/d

area of a kite/rhombus

1/2(d1)(d2)

pythagorean primitives

3-4-5 5-12-13 7-24-25 8-15-17 9-40-41

what type are the triangles in a hexagon?

30-60-90 right triangles when the base is one half of the side equilateral triangles when the full side is a base

polyhedron

3D figure where the surfaces are polygons

solids

3D shapes

regular polygon area

A=1/2asn

area of a triangle

A=1/2bh

area of a trapezoid

A=1/2h(b1+b2)

sector area

A=a/360πr²

area of a parallelogram

A=bh

area of a circle

A=πr²

Thales' theorem

An angle inscribed in a semicircle is a right angle

minor arc

An arc of a circle whose measure is less than 180 degrees.

calculating ERA

ERA- earned run average baseball: x/9 innings softball: x/7 innings earned runs/innings = x/innings in a game

SAS ~ theorem

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar

SSS ~ theorem

If the corresponding sides of two triangles are proportional, then the triangles are similar.

pyramid LA formula

L.A.=1/2pl

cylinder LA formula

L.A.=2πrH

prism LA formula

L.A.=PH

cone LA formula

L.A.=πrl

pyramid SA formula

S.A.=1/2pl+B

cylinder SA formula

S.A.=2πrH+2πr²

hemisphere SA formula

S.A.=3πr²

sphere SA formula

S.A.=4πr²

prism SA formula

S.A.=PH+2B

cone SA formula

S.A.=πrl+πr²

pyramid volume formula

V=1/3BH

cone volume formula

V=1/3πr²H

hemisphere volume formula

V=2/3πr³

sphere volume formula

V=4/3πr³

prism volume formula

V=BH

cylinder volume formula

V=πr²H

ratio

a comparison of two quantities a:b

oblique cylinder

a cylinder whose side is not perpendicular to its base

right cylinder

a cylinder whose side is perpendicular to its base

lateral face

a face that is not a base

secant

a line that cuts through a circle and touches it at two points

tangent

a line that touches a circle in one point

rectangle

a parallelogram with four right angles

segment of a circle

a part of a circle bounded by an arc and the segment joining its endpoints

parallelogram

a quadrilateral with both pairs of opposite sides parallel

chord

a segment where endpoints meet on a circle

pythagorean triple

a set of non-zero whole numbers a, b, and c, that satisfy the equation a²+b²=c²

sector of a circle

a slice of a circle

proportion

a statement of equality between two ratios a/b=c/d

similarity ratio

a:b

cow (corollary 3)

ab=z(x+y)

rad theorem

all radii are congruent

central angle

an angle whose vertex is the center of the circle

major arc

an arc with a measure greater than 180

opposite angles

angles that don't share a side

adjacent arc

arcs of the same circle that have exactly one point in common

area ratio

a²:b²

fish (corollary 2)

a²=x(x+y)

volume ratio

a³:b³

concentric circles

circles that lie in the same plane and have the same center

congruent circles

circles with congruent radii

CSSTP

corresponding sides of similar triangles are proportional

cone slant height

distance from the vertex to a point on the edge of the base of the cone

face

each polyhedron on the polyhedron

prove converse of opposite sides theorem

given: AB≅CD, BC≅AD prove: ABCD is a parallelogram statements: reasons: 1) AB≅CD, BC≅AD 1) given 2) construct AC 2) line post 3) AC≅AC 3) refl POC 4) △ABC≅△CDA 4) SSS △ post 5) ∠CAD≅∠BCA 5) CPCTC ∠BAC≅∠DCA 6) AB||CD, BC||AD 6) conv AIA theo 7) ABCD is a paral. 7) Def parallelogram

prove converse of PDBT

given: AC and BD bisect each other at E prove: ABCD is a parallelogram statements: reasons: 1) AC and BD bisect 1) given each other at E 2) BE≅DE, AE≅CE 2) def seg bisector 3) ∠BEA≅∠CED 3) VA theo ∠AED≅∠BEC 4) △AED≅△CEB 4) SAS ≅ post △BEA≅△DEC 5) ∠BAE≅∠DCE 5) CPCTC ∠DAE≅∠BCE 6) AB||CD, BC||AD 6) conv AIA theo 7) ABCD is a par. 7) Def parallelogram

prove kona ice theorem

given: AX and BX are tangent to circle O prove: AX≅BX statements: reasons: 1) AX and BX are tans 1) given 2) Construct OX 2) line post 3) AO≅BO 3) rad theo 4) OX≅OX 4) refl POC 5) AO⟂AX, OB⟂OX 5) surfer dude theo 6) ∠A and ∠B are rt 6) def perp 7)△OAX & △OBX are 7) def rt △ rt triangles 8) △OAX≅△OBX 8) muHL theo 9) AX≅BX 9) CPCTC

prove congruent and parallelogram theorem

given: BC||AD, BC||AD prove: ABCD is a parallelogram statements: reasons: 1) BC||AD, BC≅AD 1) given 2) construct AC 2) line post 3) AC≅AC 3) refl POC 4) ∠CAD≅∠ACB 4) AIA theo 5) △CAD≅△ACB 5) SAS ≅ post 6) ∠ACD≅∠CAB 6) CPCTC 7) AB||CD 7) conv AIA theo 8) ABCD is a paral. 8) Def parallelogram

prove the baseball theorem

given: circle O, OE≅OF, OE⟂AB, OF⟂CD Prove: AB≅CD Statements: Reasons: 1) OE≅OF, OE⟂AB 1) GivenOF⟂CD 2) OA≅OB≅OC≅OD 2) Rad theo 3) ∠AEO & ∠CFO are rt 3) def ⟂ 4) △AEO≅△CFO 4) muHL Theo 5) ∠A≅∠C 5) CPCTC 6) ∠B≅∠A, ∠C≅∠D 6) ITT 7) ∠B≅∠D 7) Trans POC 8) ∠AOB≅∠COD 8) 3rd angles theo 9) AB≅CD 9) conv CACTus

prove opposite sides theorem and opposite angles theorem

given: parallelogram ABCD prove: AB≅CD, ∠B≅∠C statements: reasons: 1) construct AC 1) line post 2) AB||CD, BC||AD 2) def parallelogram 3) ∠BAC≅∠DCA 3) AIA theo ∠BCA≅∠DAC 4) AC≅AC 4) refl POC 5) △BAC≅△DCA 5) ASA ≅ post 6) AB≅CD, ∠B≅∠D 6) CPCTC

prove PDBT

given: parallelogram PDBT prove: AC and BD bisect each other at E statements: reasons: 1) AB||CD, AD||BC 1) def parallelogram 2) ∠ABD≅∠BDC 2) AIA theo ∠BAC≅∠DCA 3) AB≅CD 3) opp sides theo 4) △ABE≅△CDE 4) ASA ≅ post 5) AE≅CE, BE≅DE 5) CPCTC 6) AC and BD bisect 6) def seg bisector each other at E

prove envelope theorem

given: rectangle ABCD prove: AC≅BD statements: reasons: 1) ABCD is a parallelog. 1) def rect. ∠ABC & ∠DBC are rt 2) BC≅BC 2) refl POC 3)∠ABC≅∠DCB 3) all rt theo 4) AB≅CD 4) opp sides theo 5) △ABC≅△DCB 5) SAS≅post 6) AC≅BD 6) CPCTC

prove taco theorem

given: rhombus ABCD prove: AC bisects ∠BAD and ∠BCD statements: reasons: 1) AB≅BC≅CD≅AD 1) def rhomb 2) AC≅AC 2) refl poc 3) △ABC≅△ADC 3) SSS≅ post 4) ∠BAC≅∠DAC 4) CPCTC ∠BCA≅DCA 5) AC bisects ∠BAD 5)def∠bisector and ∠BCD

prove diamond theorem

given: rhombus ABCD prove: BD⟂AC statements: reasons: 1) rhombus ABCD 1) given 2) ABCD is a parallelog. 2) def rhom. 3) AE≅CE 3) PDBT 4) BE≅BE 4) refl POC 5) △AEB≅△CEB 5) SSS≅post 6) ∠1≅∠2 6) CPCTC 7) ∠1 and ∠2 are supp 7) LPP 8) ∠1 and ∠2 are rt ∠s 8) ≅ and supp theo 9) BD⟂AC 9) def ⟂

prove side splitter theorem

given: △ABC with DE||BC prove: DB/AD=EC/AE statements: reasons: 1) DE||BC 1) given 2) ∠1≅∠3, ∠2≅∠4 2) CA post 3) △ABC △ADE 3) AA~ post 4) AB/AD=AC/AE 4) CSSTP 5) AB=AD+DB 5) seg add post AC=AE+EC 6) AD+DB/AD= 6) substi prop AE+EC/AE 7) DB/AD=EC/AE 7) major props

semicircle

half of a circle

side splitter theorem (ROFL)

if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally

taco theorem

if a quadrilateral is a rhombus, then each diagonal bisects a pair of opposite angles

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

pythagorean theorem

if a triangle is right, then c²=a²+b²

cross products property

if a/b=c/d where b≠0 and d≠0, then ad=bc

altitude to hypotenuse theorem

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

converse of opposite sides theorem

if both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram

converse of the pythagorean theorem

if c²=a²+b², then the triangle is right

converse of taco theorem

if one diagonal of a parallelogram bisects a pair of opposite angle, then the parallelogram is a rhombus

congruent and parallelogram theorem

if one pair of opposite sides is both congruent and parallel, then it is a parallelogram

converse of envelope theorem

if the diagonals of a parallelogram are congruent then the parallelogram is a rectangle

converse of diamond theorem

if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus

converse of PDBT

if the diagonals of a quadrilateral bisect each other, then it is a parallelogram

corollary to side splitter theorem

if three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

??? theorem

in a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc in a circle, if a diameter bisects a chord that is not a diameter, then it is perpendicular to the chord in a circle, the perpendicular bisector of a chord contains the center of the circle

death star theorem

in a circle, the perpendicular bisectors of chords meet at the center of the circle

consecutive angles theorem

in a parallelogram, consecutive angles are supplementary

parallelogram diagonal bisector theorem (PDBT)

in a parallelogram, diagonals bisect each other

opposite angles theorem

in a parallelogram, opposite angles are congruent

opposite sides theorem

in a parallelogram, opposite sides are congruent

means and extremes

in the proportion a/b = c/d, b and c are the means, and a and d are the extremes

arc length

m=a/360(2πr)

inscribed angle theorem

m∠B=1/2(mAC)

base

one of the two parallel faces of a prism

pi ratio

pi=c/d

vertex

point where 3 or more edges intersect

prism

polyhedron with exactly two parallel, congruent bases

derive the equation of a circle

r=√(x-h)²+(y-k)² r²=(x-h)²+(y-k)² (x-h)²+(y-k)²=r²

similar polygons

same shape, congruent angles, proportional sides

congruent polygons

same size, same shape, congruent sides, congruent angles

edge

segment formed by the intersection of two faces

circle

set of all points in a plane at a given distance from a given point

opposite sides

sides that don't share a vertex

kona ice theorem

tangent segments that share a common endpoint are congruent

envelope theorem

the diagonals of a rectangle are congruent

diamond theorem

the diagonals of a rhombus are perpendicular

pyramid slant height

the length of the altitude of a lateral face of the pyramid

trapezoid midsegment theorem

the midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases

cyclic quadrilateral theorem (CQT)

the opposite angles of a cyclic quadrilateral are supplementary

apothem

the perpendicular distance from the center to a side

surfer dude theorem

the radius of a circle is perpendicular to the tangent at the point of tangency

annulus of a circle

the region between two concentric circles

AA ~ postulate

triangles are similar if 2 angles are congruent

corollary to IAT

two inscribed angles that intercept the same arc are congruent

indirect measurement

used to find lengths that are hard to measure directly

baseball theorem

within a circle, chords equidistant from the center are congruent.

CACTus theorem

within a circle, congruent central angles have congruent arcs.Within a circle, congruent central angles have congruent chords.Within a circle, congruent chords have congruent arcs

butterfly theorem

within a circle, congruent chords have congruent arcs

30-60-90 triangles

x, x√3, 2x

isosceles right triangles (45-45-90 triangles)

x. x, x√2

geometric mean

x=√ab a/x=x/b

volcano (corolalry 1)

z²=xy