Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

30
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 Presentation

Transcript of Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Page 1: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Given two points (x1,y1) and (x2,y2)

•  

Page 2: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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)

Page 3: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Circles

- set of all points in a plane equidistant from a center

d = 2r

C = dπ

A = πr2

Oradius

chord

diameter

tangent

secant

Page 4: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 5: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

• chords equidistant from the center are congruent (AB and CD)

• congruent chords have congruent arcs (AB and CD)

A

B

C D

Page 6: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

• 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

Page 7: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Power Theorems

• Two tangents

AC = BC

• Secant and tangent

(AB)(AC) = (DA)2

A C

B

A

C

B

D

Page 8: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

• Two secants

(AB)(AC) = (AD)(AE)

• Two chords

(AB)(BC) = (DB)(BE)

A

B

C

D

E

A

BC

D

E

Page 9: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

• 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

Page 10: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

•  

b

x

a

x

ba

xa

b

Page 11: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

- vertex is at the center

• x = a

Central Angle Inscribed Angle

- vertex is on the circle

• x = a/2

Page 12: Given two points (x 1 ,y 1 ) and (x 2 ,y 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)

Page 13: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Polygons

Convex polygons - all diagonals lie entirely inside the polygon

Regular polygons - equilateral and equiangular

Page 14: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 15: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 16: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 17: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Cone

V = 1/3 πr2h

Sphere

V = 4/3 πr3

S = 4πr2

Cylinder

V = πr2h

S = 2πr2 + 2πrh

Pyramid

V = (s2h)/3

Page 18: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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)

Page 19: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 20: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Right Triangles

- Pythagorean Theorem: c2 = a2 + b2

Special Right Triangles

30°

60°90°

45°

45°

90°

x√3

x

2x √2 x

x

x

Page 21: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Similarity

- if the corresponding angles are equal, and the corresponding sides are proportional

- AAA, SSS, and SAS Similarity Theorems

Page 22: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 23: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 24: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

- 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

Page 25: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

Congruence

- triangles are congruent if the corresponding sides and angles are congruent

- SAS, ASA, SSS, SAA postulates

- Hypotenuse-Leg theorem

Page 26: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 27: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

- 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

Page 28: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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)

Page 29: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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

Page 30: Given two points (x 1 ,y 1 ) and (x 2 ,y 2 )

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