M2 Geometry Journal 1:Chapter 1 Geometry

José Antonio Weymann

POINT,LINE AND PLANE Point: A single dot in space, used to describe location.

It is described with a dot and a capital letter. P Line: A straight connection of points that goes on

forever in both directions. XY

YX Plane: A flat surface that goes on forever, it has no

thickness and it contains 3 points. T

COLLINEAR AND COPLANAR POINTS The difference between collinear and

coplanar points is collinear points coexist in in the same line and coplanar points coexist in the same plane.

Collinear Coplanar:

P N A B M

COLLINEAR AND COPLANAR POINTS Collinear Counterexapmple Coplanar Counterexample

LINE, SEGMENT, AND RAY Line: A straight connection of points that goes on forever in

both directions. XY

YX Segment: Any straight collection of dots that has a beginning

and an end (endpoints) XY Ray: A straight collection of points that has one end point

and goes on forever. XY *This are related because they all are ways to use points and

lines, also we will be using them for the rest of the school year. Besides their everyday location and distance applications.

INTERSECTION OF LINES AND PLANES It is when two lines intersect through the

same line in any situation. Like in Postulates 1-1-4 (If two lines intersect, then they intersect in exactly one point) and 1-1-5 (If two planes intersect, then they intersect in exactly one line)

POSTULATE, AXIOM AND THEOREM

The Difference between Postulate, Axiom and Theorem is between Postulate and Axiom is nothing they are interchangeable terms (this are accepted truths as fact with out proof), but between this and theorem is that a theorem is a theoretical proposition, later to be proved by other propositions and formulas.

RULER POSTULATE (POSTULATE 1-2-1) To measure segments use a ruler and just

subtract the values at the end points. A Field is 120 ft long and a player starts running from 60 ft , what

is the distance when he runs to point 120 ft? Answer: 60 ft, The road is 130km long and a car starts its trip from south park

which is 90km up the road, when he reaches the 130 km line how many km does the car have traveled? Answer: 40km

The sidewalk is 850mt long and an old lady starts walking from 500mt, when she reaches the end of the sidewalk, what’s her current distance

Answer: 350mt.

SEGMENT ADDITION POSTULATE (1-2-2) If A,B,C (our three collinear points) and B is

between A & C, then AB+BC=AC 24 46 C A B 24+46=

70, so CB = 70 so CA+AB= CB 72 20 A B C 72+20=

92, so AC = 92 so AB+BC=AC 75 55 E F G 75+55= 130,

so EG = 130 so EF+FG=EG

DISTANCE BETWEEN TWO POINTS ON A COORDINATE PLANE To find the distance between two points on a coordinate

plane you have to take the X1 and X2 coordinates and square them then the Y1 & Y2 coordinates and do the same, add and then square the answer. d = √ (X₁-X₂)₂+(Y₁ -Y₂)₂

AB= √(5-0)₂+(1-3)₂ * CD= √(-3 –(-1)₂+(-4 - 1)₂ √5₂ + (-2)₂ √(-2)₂+(-5)₂ √25+ 4 √ 4+25 This are congruent √29

√29 d= √(4-1)₂+(2-6)₂ √3₂+(-4)₂ √9+16 √25 = 5 this is a random problem

CONGRUENCE AND EQUALITY Congruence means and equal measure, not

necessarily a value, we are comparing names; this contrasts with equality because that means 2 things with same value, therefore it is comparing values. They are similar because both are comparing two things. They are different because congruent is two objects exactly the same regardless of orientation, and equality is shape size and angles, like two squares they are congruent but don’t have measurements meaning we don’t know if they are equal.

≈ =

PYTHAGOREAN THEOREM (THEOREM 1-6-1)

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a₂+b₂=c₂

a=4 and b=5 a=3 and b=4 find the value of c c₂= a₂+b₂ c₂= a₂+b₂ a₂+b₂=c₂ a=4     b=5 =4₂+5₂ = 3₂+4₂ 4₂+5₂= c₂ =16+25 =9+16 16+25= c₂ =41 =25 41= c₂ c=√41 c=√25 c≈ 6.4 c≈ 5

ANGLES Angles are 2 rays that share a common

end point. There are acute (smaller than 90°), right (90°), obtuse (bigger than 90°) and finally straight angles (180°)

Right Acute Obtuse Straight

Exterior Interior

ANGLE ADDITION POSTULATE (1-3-2) 2 small angles add up to the big angle. 1.angle ABD=150 2.angle EFI=180 3.angle

VXY=110 angle ABC=45 angle FGI=90 angle

YXZ=60 150-45= 180-90=

110+60= CBD=105 EFG=90

VXY=170

MIDPOINT (CONSTRUCTION AND MIDPOINT FORMULA)

Midpoint is what we call the middle of the segment, equidistant from the endpoints; cuts into two equal smaller segments. To construct a midpoint first draw a segment, second draw a line a little bit passed the approximate middle, third draw two arcs from both endpoints in the two sides of the segment, fourth draw a straight line through the crossings and you have your constructed midpoint.

MIDPOINT FORMULA: (X₁+X₂/2,Y₁+Y₂/2) 1.(6+4)(4+1)/2 m=1.5 2.(-3+0)(-1,1)/2 m=-1.5 3. (-3+-1.5)(4+1)/2 m= -.75,2.5

ANGLE BISECTOR (CONSTRUCTION) An angle bisector is a line which cuts and

angle into two equal parts, therefore the word bisect means to cut in have. First lock your compass, make an arc in each side put the point in each arc and make an arc in the interior; then connect the vertex to the intersection point.

ADJACENT, VERTICAL AND LINEAR PAIRS Adjacent: are 2 angles that have the same vertex

and a same side. Linear Pairs of Angles: 2 adjacent straight angles

that will form a straight line. Meaning all Linear Pairs of Angles are supplementary (Linear Pair Postulate, L.P.P)

Vertical: Non adjacent angles formed when 2 lines intersect; vertical angles are always congruent.

Adjacent: Linear Pairs: Vertical:

COMPLEMENTARY AND SUPPLEMENTARY Complementary angles are 2 angles that add

up to 90º and supplementary add up to 180º; supplementary are always linear pairs but complementary aren`t, they are always adjacent.

Complementary: 90º Supplementary: 180º

PERIMETER AND AREA Square: to take the perimeter of a square add all 4

sides (P=4s) and area just square one side (A=s₂) Rectangle: to take the perimeter of a rectangle add

the 2 lengths and 2 widths (P=2l+2w) and area multiply length time width (A=lw)

Triangle: to take the perimeter of a triangle (a+b+c) and to take the area (bh/2)

Square: P=4(4), P=16ft. Rectangle: 5ft. 4ft. A=4₂, A= 16ft₂ 4ft.

P=5+5+4+4, P=18ft.

A=5*4, A=20ft.₂

PERIMETER AND AREATriangle: P= 2x+3x+5+10 A=1/2(3x+5)(2x) = 5x+15 =3x₂+5x

2nd round of examples: Square: Rectangle: 3cm. P=6(4) P=24 5cm. A= 6₂ A= 36 P=3+3+5+5 P=

16cm. A=3*5 A=15cm. ₂

Triangle: P=4x+5x+3+9 A=1/2(5x+9)(4x) = 9x+12 =5x₂+5x

CIRCUMFERENCE OF A CIRCLE The circumference of a circle is the distance

around the circle. Circumference (C) is given by C=(Pi)d or C=2(Pi)r.

C=2(Pi)r C=2(Pi)r =2(Pi)(3)=6(Pi) =2(Pi)(11) ≈ 18.8cm =22(Pi) ≈69.1cm.

3cm. 11cm.

AREA OF A CIRCLE A = (Pi)r₂ A= (Pi)r₂ = (Pi)(3)₂ = 9(Pi) 3cm. ≈ 28.3cm₂ A= (Pi)r₂ = (Pi)(11)₂ = 121(Pi) ≈380.1cm₂ 11cm.

FIVE STEP PROBLEM SOLVING PROCESS

1.READ IT CAREFULLY2.WRITE DOWN ALL IMPORTANT INFORMATION3.DRAW A PICTURE4.WRITE AND SOLVE THE EQUATION5.ANSWER THE QUESTION

FIVE STEP PROBLEM SOLVING PROCESS 1. Read it Carefully:The quilt pattern includes 32 small triangles. Each has a

base of 3 inches and a height of 1.5 in. Find the amount of fabric used to make the 32 triangles.

2. Write down all important information: 3 inches, 32 small triangles, 1.5 inches 3. Draw a Picture h:1.5in. b:

FIVE STEP PROBLEM SOLVING PROCESS4. Write and Solve the equation b: 3in. h:1.5in. bh/2 (3)(1.5)/2 *32 = 72

5. Answer the Question Answer: The amount of fabric used to make the 32

triangles in the quilt is 72in₂

INTRODUCTION TO TRANSFORMATIONS Things to know before entering

transformations Pre-image: Image:

A

ABC A’B’C’

A

TRANSFORMATIONS A Transformation is a change in the position

of the original object. Translation : The slide of an object in any

direction (usually is the most common) (X,Y) (X+a,Y+b)

TRANSFORMATIONS Rotation: to rotate a figure around a

point.

TRANSFORMATIONS Reflection: When you mirror your Pre-image

across any line. If across Y-axis (X,Y)(-X,Y). If across X-axis (X,Y)(X,-Y)

JOURNAL HAS BEEN CONCLUDED This geometry chapter 1 journal has

ended, good luck on the quiz and I hope to have covered the topics In a satisfactory matter.