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
