GEOMETRY

GEOMETRY

The standard Competences:

Determine the position, distance, and angle that include point, line and plane in the three dimension space.

The Basic Competence:

> Determine the position of point, line and plane in the 3 dimension space

> Determine the distance from point to line and from point to plane

> Determine the size of angle, between line and plane and between two plane

1. POSITION OF POINT, LINE AND PLANE IN SPACE

Point, line and space are parts of the three dimension space. These part are named as space elements. In this chapter, we will study about position of point, line and plane in space. The beginning of this study is discussed about the meaning of point, line and plane, along with the axioms and theorems that relevant with them.

1.1 The Meaning of Point, Line and Plane

Space elements (point, line ,plane) are base term in geometry. As we had known that base term is terminology that only can be described in the following explain.

1. Point

Point only can be identified by the place, but it doesn’t have size(has no dimension). Point is shown by dot sign, then follow with the name of that point. The name of that point usually use capital letter, like A,B,C,P,Q and R.

2. Line

We can lengthen the line(means straight line ) as long as we want. But remember limit of the place that we use to draw the line, a line is drawn by just the part of line. This part of the line is called representative line. Line just has the size of long, and doesn’t have the size of wide. Name of a line can be identified by the name of representative line with use small letter, like g,h,k or say the name of segment line from starting point to the end of point.

3. Plane

Plane can be expanded. Generally a plane just drawn by the representative plane. The representative plane has two size, long and wide. Figure from plane has shape like square or rhombus, rectangle or parallelogram. The name of representative plane is written in the corner of the plane and use character a,b,c or H,U,W or with mention angle points from that plane.

4. Axioms of line and plane

Beside of the base terms, study geometry need axioms (usually called postulate). Axiom is statement that consider truth in a system and that truth receive without proof. In space of geometry, there are three important axioms. Those axioms are introduced by Euclides (300BC), he is mathematician from Alexandria.

These are the Euclides’s axioms :

Axiom 1 : Across two any kind of points, it can be made a straight line.

Axiom 2 : If a line and a plane has two alliance points, then that line is on plane.

Axiom 3 : across three any kind of points we can make a plane.

Notes:

1. In axiom 1, two kind of points are that two points are not same

2. In axiom 3, three kind of points are that points are not in one line.

According to the three axioms, then it can be 4 theorems to determine a plane.

Theorem 1: a plane is determined by three any kind of points

Theorem 2: a plane is determined by a line and a point (the point outside the line).

Theorem 3: a plane is determined by two intersection lines

Theorem 4: a plane is determined by two parallel lines.

Base term about point, line and plane, then axioms and theorems about line and plane are already known. Then we will discuss about position of point, line and plane in three dimension space.

1.2 Position of Point to Line and Point to Plane

1. Position of point to line

> The point is in the line

If line g across point A, then point A is in line g

> The point is in outside the line

If line h does not across point B, then point B is not on line h (outside the line g)

Example, look at cube ABCD-EFGH. Segment AB as a represent of line g.

a) The vertect of cube that on line g are point A and B

b) The vertect of cube that in outside line g are point C,D,E,F,G, and H.

2. Position of Point to Plane

> The point is on plane

If point A can be acrossed by plane a, then point A is on plane a.

> the point is in outside plane

If point B cannot be acrossed by plane b, then point B is in outside plane b

Example, look at cube ABCD-EFGH. Plane DCGH as the representative of plane U.

a) The vertect of cube that on plane U are point C,D,G and H.

b) The vertect of cube that in outside of plane U are point A,B,F and E.

Exercise 1.

1. Use the top of index finger as point A and pencil as line g.

a) Display the position of point A , if point A is on line g!

b) Display the position of point A , if point A in outside line g!

c) Is there any position of point A to another line g?

2. There is a cube ABCD-EFGH, BC represent line k, DE represent line l and AG represent line m. mention the angles of that cube that:

a) on line k

b) in the outside of line k

c) on line l

d) in the outside of line l

e) online m

f) in the outside of line m

3. Use the top of index finger as point A and notebook as plane U.

a) Display the position of point A , if point A is on plane U!

b) Display the position of point A , if point A in outside of plane U!

c) Is there any position of point A to another plane U?

4. There is a regular pyramid T.ABCD

a) Mention the vertect of pyramid that are located in the flanks of side.

b) Mention the vertect of pyramid that are located in the flanks of base.

c) Mention the vertect of pyramid that are located in the planes of side.

d) Mention the vertect of pyramid that are located in the base of plane.

e) Is there any vertect of the pyramid that in the outside of flanks

f) Mention the agle of pyramid that in the outside of the base flank

g) Mention the agle of pyramid that in the outside of base plane

5. There is a cube KLMN-PQRS, plane KLMN represent plane a, plane KLQP represent plane b and plane KMRP represent plane c. Mention the vertect of that cube that:

a) are located on plane a

b) are located in the outside of plane a

c) are located on plane b

d) are located in the outside of plane b

e) are located in plane c

f) are located in the outside of plane c

1.3 Position of Line to Line and Line to Plane

A. Position of Line to Another Line

The position possibility of a line to another line in three dimension space are intersect, parallel or cross.

> Two intersect lines

Line g intersect to line h if line g and line h are located in a plane (remember theorem 3) and have alliance point. In geometry of plane, the alliance point is called the intersection point between two lines.

Notes:

If two lines are intersect in more than one point, so that two line are same.

> Two parallel lines

Line g and line h are parallel lines, if line g and line h are located in a plane (remember theorem 4) and have not any alliance point

> Two across lines

Line g across (not intersect and not parallel) to line h, if line g and line h are not located in a same plane.

Example :

Look at the cube ABCD-EFGH

Side AB represent line g

a) The sides of that cube that intersect to line g are AD,AE,BC and BF.

b) The sides of that cube that parallel to line g are BC, EF and HG.

c) The sides of that cube that across to line g are CG ,DH, EH and FG.

d) Is there any side of cube that same with line g?

> Axiom of two parallel lines :

Axiom 4 :

Across a point that is located in the outside a line , it only can be made a line that parallel with that line.

> Theorems of two parallel lines

According to the position of two parallel lines, it can be pulled same conclusion. The conclusions are described in these theorems:

- Theorem 5 :

If line k parallel to line l, and line l parallel to line m, so line k parallel to line m.

- Theorem 6 :

If line k parallel to line h and intersect line g, line l parallel to line h and intersect line g too, so line k, l and g are located in a plane.

- Theorem 7 :

If line k parallel to line l, and line l pierce plane a, so line k pierce plane a.

THE VIDEOS

Exercise 4:

THE VIDEOS

A. Grammar

> Tipe dasar dari kalimat sederhana (simple sentence).

Kalimat sederhana adalah semua elemen yang paling dasar pada kalimat yang terdiri dari subjek atau predikat.

> Subjek adalah yang melakukan tindakan dari kata kerja pokok.

Subjek sederhana adalah kata benda khusus yang menunjukkan sebuah kegiatan.

Contoh :

The happy little child kicked the gnome over fence.

Subjek : child

Simple subjek : the happy little child

Predikat : kicked

> Predikat dari sebuah kalimat terdiri dari : kata kerja utama dan yang mengikutinya.

Contoh:

The happy little child kicked.

The gnome over fence.

> kalimat sederhana dapat diperoleh tanpa subjek dan predikat.

Kalimat perintah -> memerintah seseorang untuk melakukan sesuatu.

Contoh:

Kick that gnome over the fence.

Kalimat di atas tidak ada subjeknya.

B. Verb ( Kata Kerja)

Kata kerja adalah untuk menjelaskan suatu kejadian.

Kata kerja adalah kata terpenting dalam suatu kalimat.

Contoh :

Dave runs.

Runs adalah kata kerja, karena menjelaskan kejadianyangdilakukan Dave.

Dalam tata bahasa inggris, perubahan bentuk kata kerja menunjukkan ssiapa yang sedamg melakukan kegiatan tersebut.

They, we, I,you -> verb (kk)

She, he and it -> verb (kata kerja +s)

Kata kerja – to be :

- I am , you are, he is, she is, it is -> kata ganti tunggal

- We are, they are -> kata ganti jamak.

D. Basic Trigonometry

Trigonometry is from Greek, that is from trigon and metron.

To remember the function of trigonometry we can use

Soh cah toa

Soh = sine is opposite over hypotenuse

Cah = cosine is adjacent over hypotenuse

Toa = tangent is opposite over adjacent

Sine a = opp/hyp

Cos a = adj/hyp

Tan a = opp/adj.

E. Compound Sentence (Kalimat Majemuk)

Contoh :

It is the end of the world as we know it and I feel fine.

> kalimat di atas terdiri dari 2 klausa dan dihubungkan dengan kata penghubung(and).

> Independent clause adlah klausa yang dapat berdiri dalam kalimat.

> jika dalam kalimat terdiri dari 2 klausa, maka kalimat tersebut disebut kalimat majemuk.

> untuk menggabungkan dua klausa digunakan :

1.titik dua (:)

Contoh : I love my two sisters : they bake me pie.

2.titik koma (;)

Contoh : It’s the end of the world as we know it; I feel fine.

3. garis hubung (-)

Terdiri dari beberapa elemen yang mengejutkan.

> Klausa dependent adalah klausa yang tidak bida berdiri sendiri dan bukan kalimat lengkap.

> Kalimat kompleks adalah kalimat yang terdiri dari klausa independent dan klausa dependent.

F. Limit by Inspection

Limit by inspection can be two condition :

1. x goes to positive or negative infinity.
2. limit involves a polynomial divided by a polynomial

We must see the power of x in the numerator and in the denominator.

1. If the highest power of x is greater in numerator, limit is positive or negative infinity.

Example: limit x -> inf [(x^3+4)/(x^3+x+1)] = inf

2. If the highest power of x in numerator less than in the denominator, the limit is zero.

Example : limit x-> inf [(x^3+3)/(x^3+1)] = 0

3. If the highest power of x in numerator is same as highest power of x in denominator, the quotient of the coefficient of the two highest powers.

Example : limit x -> inf [(4x^3+x^2+1)/(3x^3+4)] = 4/3

G. Pre Calculus Graph

> Graph of a rational function which can have dis continuities because has polynomial in the denominator.

Example :

f(x) = (x+2)/(x-1)

f(1) = the value became (1+2)/(1-1)=3/0 -> bad idea, it will break in the graph

f(0) = (0+2)/(0-1)=-2

> Rational function do not always work this way.

Take graph f(x) = 1/(x^2) not all rational function will give zero in denominator.

> Rational function denominator can be zero.

> example:

y = (x^2-x-6)/(x-3) , if x = 3

y = 0/0

when the result 0/0, we can simplify that equation,

y = (x^2-x-6)/(x-3)

= [(x+2)(x-3)]/(x-3)

y = x+2

H. Trigonometry Function

Trigonometry is study about right triangle. To remember the function of trigonometry we can use soh cah toa.

Trigonometry function is ratio of different sides of triangles.

Trigonometry function :

1. sine theta = opp/hyp

2. cosine theta = adj/hyp

3. tangent theta = opp/adj

4. cosecant theta = hyp/opp

5. secant theta = hyp/adj

6. cotangent theta = adj/opp

notes:

opp = side opposite theta

adj = side adjacent to theta

hyp = hypotenuse

Do You Believe in Me

Exercise 2:

Do You Believe in Me

There is a student, a boy that has no fear to talk in front of a lot of people. He is from Dallas. He stand on stage, talk to all audience to affect them to believe what did he says. He studies in Dallas ISD and joins Charles Rice Learning Centre.

He says that he can do anything, become anything. He says that every single of us can realize our dreams. He says that we must not give up to learn in Dallas. He always says ‘do you believe in me?’.

He tells that each of Dallas people can be graduate, ready for college and get a work place. He ask to the audience to believe that we can reach a highest potential.

He says that people in Dallas must believe that they can be anything what they want. He believes that all of people in Dallas can be a success people with their ability.

Exercise 6:

The definition, explanation and the example of the term questioning by :

1. Tiada yang lain selain dirimu yang akan selalu aku tunggu

-There is no body beside of you that I will always waiting for.

2. Garis singgung lingkaran dalam segitiga.

-tangent of a circle in triangel.

3. Juring lingkaran : sector of circle

Ex : The sector of that circle is not known.

4. berboncengan disaat gerimis dating.

-Give aride whwn drizzle is coming.

5. Garis yang saling bersilangan

-Line that across each other

6. Tali busur : khord

Ex : AB is the khord of circle O

7. Proyeksi orthogonal : orthogonal projection.

Ex : Find the orthogonal projection of x^2+y^2=0

8. Sudut yang mengapit diameter lingkaran.

-Angle that imposed tha diameter of the circle.

9. Menghitung peluang dengan pendekatan frekuensi relative.

- Find the vacant with approximation of relative frequency.

10. Jangkauan interquartil : interquartil range.

Ex : Find the interquartil range from these data.