given two points (x 1 ,y 1 ) and (x 2 ,y 2 )
DESCRIPTION
Given two points (x 1 ,y 1 ) and (x 2 ,y 2 ). Transversals. Equal angles: corresponding (A and E, B and F, C and G, D and H) alternate interior (C and F, D and E) alternate exterior (A and H, B and G). a. b. c. d. e. f. g. h. Same side interior - angles add up to 180 - PowerPoint PPT PresentationTRANSCRIPT
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Given two points (x1,y1) and (x2,y2)
•
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Transversalsa b
c d
e f
g h
Equal angles:• corresponding
o (A and E, B and F, C and G, D and H)
• alternate interioro (C and F, D and
E)• alternate exterior
o (A and H, B and G) Same side interior
- angles add up to 180(C and E, D and F)
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Circles
- set of all points in a plane equidistant from a center
d = 2r
C = dπ
A = πr2
Oradius
chord
diameter
tangent
secant
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Theorems• the perpendicular from the center of a circle to a
chord bisects the chord (AB bisects CD)
• the segment from the center of the circle to the midpoint of a chord is perpendicular to it (AB is perpendicular to CD)
• the perpendicular bisector of a chord passes through the center (the line bisecting CD will pass A)
C
D
A B
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• chords equidistant from the center are congruent (AB and CD)
• congruent chords have congruent arcs (AB and CD)
A
B
C D
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• in a quadrilateral inscribed within a circle ("cyclic quadrilateral"), the opposite angles are supplementary (A and C, B and D)
• parallel secants have congruent arcs (EF and GH)
A
B
C
D
E
F
G
H
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Power Theorems
• Two tangents
AC = BC
• Secant and tangent
(AB)(AC) = (DA)2
A C
B
A
C
B
D
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• Two secants
(AB)(AC) = (AD)(AE)
• Two chords
(AB)(BC) = (DB)(BE)
A
B
C
D
E
A
BC
D
E
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• the measure of the angle of two chords intersecting within a circle is equal to half the sum of their intercepted arcs
• ½ (a + b) = xa
x
b
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•
b
x
a
x
ba
xa
b
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- vertex is at the center
• x = a
Central Angle Inscribed Angle
- vertex is on the circle
• x = a/2
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Sectors
- region enclosed by two radii and the intercepted arc
• A = x/360 πr2
• Segment of a circle
- region between a cord and intercepted arc
- A = (Asector ) - (Atriangle)
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Polygons
Convex polygons - all diagonals lie entirely inside the polygon
Regular polygons - equilateral and equiangular
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Diagonals - segments connecting nonconsecutive vertices of the polygon
• number of diagonals =
• Sum of the measures of interior angles: (n-2)180
Measure of each interior angle:
Sum of the measures of exterior angles: 360
Measure of each exterior angle: 360/n
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Squares
P = 4s
A = s2 = d2/2
d2 = s2
Cube:
V = s2
S = 6s2
d2 = 3s2
Rectangles
P = 2 (l + w)
A = lw
d2 = (l2 + w2)
(look pythagorean theorem again)
Rectangular Box:
V = lwh
S = 2lw + 2lh + 2wh
d = l2 + w2 + h2
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Other cool shapes
Parallelogram
A = bh
* consecutive angles are supplementary
Rhombus
A = bh = ½ (d1d2)
* equilateral parallelogram
Trapezoid
A = ½ h(b1+b2)
midsegment = (b1+b2)/2
Kite
A = ½ (d1d2)
* diagonals are perpendicular
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Cone
V = 1/3 πr2h
Sphere
V = 4/3 πr3
S = 4πr2
Cylinder
V = πr2h
S = 2πr2 + 2πrh
Pyramid
V = (s2h)/3
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Apothem
= in a regular polygon, is the perpendicular distance from the center of each of the sides (it's like the radius of a polygon)
• A = ½ (apothem)(perimeter)
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Triangles
- angles add up to 180
Sides: c < a + b (triangle inequality theorem)
P = a + b + c
A = ½ (bh)
• Equilateral Triangleso A = s2 sqrt ¾o P = 3s
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Right Triangles
- Pythagorean Theorem: c2 = a2 + b2
Special Right Triangles
30°
60°90°
45°
45°
90°
x√3
x
2x √2 x
x
x
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Similarity
- if the corresponding angles are equal, and the corresponding sides are proportional
- AAA, SSS, and SAS Similarity Theorems
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Basic Proportionality Theorem
- if a line/segment parallel to one side of a triangle intersects the other two sides in distinct points, then cuts off segments proportional to those sides
- DE ║ BC; AB/AD = AC/AE and BD/AD = CE/AE
A
B C
D E
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Midline Theorem
- The segment connecting the midpoints of the two sides of a triangle is equal to half the side it is parallel to.
- MN = ½ BCA
B C
M N
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- The altitude to the hypotenuse of a right triangle divides it into two similar right triangles (both are similar to the original triangle)
Triangles ABC, ACD, and CBD are similar
A
B
CD
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Congruence
- triangles are congruent if the corresponding sides and angles are congruent
- SAS, ASA, SSS, SAA postulates
- Hypotenuse-Leg theorem
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Exterior Angle Theorem
- the measure of an exterior angle is equal to the sum of the measures of its two remote interior angles
1 = a + b
1a
b
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- If the two sides are congruent, the two angles opposite them are also congruent (the converse is also trueee)
Side-Angle Inequality Theorem
- If the angles aren't congruent, the sides aren't either (the converse is also trueee)
Isosceles Triangle Theorem
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Degree and Radian Measures
2 π (radians) = 1 revolution = 360°
So the Quadrantal Angles are at
• 0°
• 90° (½π, ¼ of a revolution)
• 180° (π, ½ of a revolution)
• 270° (3/2 π, 3/4 of a revolution)
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Trigonometric Ratios
sin = opposite/hypotenuse
cos = adjacent/hypotenuse
tan = opposite/adjacent
csc = hypotenuse/opposite
sec = hypotenuse/adjacent
cot = adjacent/opposite
CofunctionsSine and Cosine [sinA = cos (90-A) and cosA = sin (90-A)]Secant and CosecantTangent and Cotangent
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Trigonometric Identities
• sin2 θ + cos2 θ = 1
• tan2 θ + 1 = sec2 θ
• cot2 θ + 1 = csc2 θ
• sin (– θ ) = –sin θ • cos (– θ ) = cos θ
• tan (– θ ) = –tan θ • csc (– θ ) = –csc θ
• sec (– θ ) = sec θ
• cot (– θ ) = –cot θ
** Basically everything becomes negative except cos and sec
• sin θ = 1/csc θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ
• tan θ = sin θ / cos θ• cot θ = cos θ /sin θ