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