Given two points (x1,y1) and (x2,y2)
Midpointy = (), ()
Distanced =
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)
Circles
- set of all points in a plane equidistant from a center
d = 2r
C = dπA = πr2
Oradius
chord
diameter
tangent
secant
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
• chords equidistant from the center are congruent (AB and CD)
• congruent chords have congruent arcs (AB and CD)
A
B
C D
• 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
Power Theorems• Two tangents
AC = BC
• Secant and tangent(AB)(AC) = (DA)2
A C
B
A
C
B
D
• Two secants(AB)(AC) = (AD)
(AE)
• Two chords(AB)(BC) = (DB)(BE)
A
B
C
D
E
A
BC
D
E
• 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
•
b
xa
xb
a
xa
b
- vertex is at the center
• x = a
Central Angle Inscribed Angle
- vertex is on the circle
• x = a/2
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)
Polygons
Convex polygons - all diagonals lie entirely inside the polygon
Regular polygons - equilateral and equiangular
Diagonals - segments connecting nonconsecutive vertices of the polygon• number of diagonals = • Sum of the measures of interior angles:
(n-2)180Measure of each interior angle: Sum of the measures of exterior angles: 360Measure of each exterior angle: 360/n
Squares
P = 4sA = s2 = d2/2d2 = s2
Cube:V = s2
S = 6s2
d2 = 3s2
Rectangles
P = 2 (l + w)A = lwd2 = (l2 + w2)(look pythagorean theorem
again)
Rectangular Box:V = lwhS = 2lw + 2lh + 2whd = l2 + w2 + h2
Other cool shapes
ParallelogramA = bh* consecutive angles
are supplementary
RhombusA = bh = ½ (d1d2)* equilateral
parallelogram
TrapezoidA = ½ h(b1+b2)midsegment = (b1+b2)/2
KiteA = ½ (d1d2)* diagonals are
perpendicular
ConeV = 1/3 πr2h
SphereV = 4/3 πr3
S = 4πr2
CylinderV = πr2hS = 2πr2 + 2πrh
PyramidV = (s2h)/3
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)
Triangles
- angles add up to 180Sides: c < a + b (triangle inequality theorem)
P = a + b + cA = ½ (bh)• Equilateral Triangles
o A = ¾ s2
o P = 3s2
Right Triangles
- Pythagorean Theorem: c2 = a2 + b2
Special Right Triangles30°
60°90°
45°
45°
90°
√3/2 x
x
2x √2 x
x
x
Similarity
- if the corresponding angles are equal, and the corresponding sides are proportional
- AAA, SSS, and SAS Similarity Theorems
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
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
- 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
AB
CD
Congruence
- triangles are congruent if the corresponding sides and angles are congruent
- SAS, ASA, SSS, SAA postulates- Hypotenuse-Leg theorem
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
- 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
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)
Trigonometric Ratiossin = opposite/hypotenusecos = adjacent/hypotenusetan = opposite/adjacent
csc = hypotenuse/oppositesec = hypotenuse/adjacentcot = adjacent/opposite
CofunctionsSine and Cosine [sinA = cos (90-A) and cosA = sin (90-A)]Secant and CosecantTangent and Cotangent
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
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