I had made a practice of learning mathematics in English with my friend, Rininda Ulfa Arizka on Friday, January 9th, 2009 in Karangmalang. I took the topic of “Points, Lines, and Planes Position in Space”.
I chose this topic because it was the base topic of studying mathematics especially geometry. We will meet the other topics of mathematics that related with the points, lines, or planes.
Firstly, I explained her about the definition of Points, Lines, and Planes in Space; and then the position of points, lines, and planes in space. I also gave her the example and some exercises.
A.Definition of Point, Line, and Plane
The point is a position in space. Point doesn’t have length and width thickness. A point is represented by a dot and named with a capital alphabet.
The line is a set of points lined up very long and infinite, but doesn’t have the width. A line is represented by the ray with arrows in both ends. The arrows show that the line is not finish.
The plane is a set of points lined up and in rows closely and infinitely, but doesn’t have thickness. A plane is represented by figure of parallelogram., and named by using capital alphabet or Greece alphabet.
B.The Position of Points, Lines, and Planes in Space
Position of two points
Definition1 :
Two points coincide is two points that are same.
Position of point and line
Definition2 :
Points in a line (collinear) are the points that located in one line (the points are not placed in one line are non-collinear).
Position of point and plane
Axiom : Any three points placed minimum in one plane
Any three non-collinear points placed in one pane
Definition : coplanar
The points are called coplanar if and if only there is a plane consisting all that points.
Position of two lines
Definition : Parallelism and crossing lines
Two different lines are parallel if and if only the lines are coplanar and don’t intersect each other.
Two different lines are crossing if and if only the lines are non-coplanar.
If two different lines intersect each other, both are placed in one plane exactly.
Position of lines and plane
If there are a line and a plane, there will be some occurrences, those are the line break through its plane, or the line is placed in the plane.
Position of two planes
If there are two planes, there will be some occurrences, those are both of planes intersect each other or both of planes are parallel.
Position of three planes
If there are three different planes, there are some occurrences also, the planes intersect each other, or the planes are parallel. If the lines intersect each other, there is a point of intersect, a line of intersect, or pairs of them intersect each other and forming three lines that are parallel.
C.Examples
I drew a figure of parallelogram, some points inside and lines connected the points. From the figure I told Ninda the names of points, lines and plane of parallelogram. The name the parallelogram is plane-α. Plane- α consist of points A,B, C, D, E, F, G. BE and GC are on plane-α and intersect in point F. HD breakthrough plane-α in point D.
We could write the points and lines in plane- α as below :
A Є α = it means point A in plane-α
B Є BE = it means point B in BE
BE Є α = it means BE in plane-α
F = the point intersect of BE and GC
D = the point intersect of HD and plane-α
α = plane(BE,GC) = it means α is plane consisting BE and GC
D.Exercises
1.Explain about the position of points, lines and planes in the three dimension space!
2.What’s the meaning of points that are collinear?
3.What’s the meaning of points that are coplanar?
4.I drew a cube of ABCD.EFGH and the lines pass through the edges. I had Ninda to say name of lines and the position of a point of edge and the edge, the position of the point and the side, the position of the edge and the side.
E.Conclusion
From the explanation the topic above and after I saw the Ninda’s answers I could conclude that she had understood the topic of “Points, Lines, and Planes Position in Space”.
I also asked her how about my method of explaining the topic above. And she said that I had explained the topic clearly enough so she could be understanding the topic easily.
Thank you Ninda….
Reference : Suplemen Bahan Kuliah Geometri by Murdanu, M.Pd
Kamis, 15 Januari 2009
Rabu, 07 Januari 2009
PRE-CALCULUS
Let’s begin discussing the graphs of a rational function which can have discontinuities. A rational function has a polynomial in the denominator which means you’re dividing something that’s very big quantity. So you can be sure with the bottom of the fraction it has.
It’s impossible that some values of x is divided by zero. If it shows like that, the value is off limits.
Let f(x)=(x+2)/(x-1) and when x = 1 the function value becomes f(1)=(1+2)/(1-1)=3/0. In this case 0 is the denominator.
From this function, choosing x=1 is a bad idea. It makes a break in function graph.
For example : Suppose we have a graph of f(x)=(x+2)/(x-1). Insert 0 for x, so f(0)=(0+2)/(0-1)=-2. So the point of function in graph is (0,-2).
Now, next to try insert x=1. So we get f(1)=(1+2)/(1-1)=3/0. We know that is impossible. It also means that the graph of function we make has no any point of y when x=1. We can also call the graph is break or discontinuity.
Rational functions don’t always work this way. Take a graph of f(x)=1/(x+1), so f(1)=1/(1+1)=1/2.
Not all of rational functions will give zero in denominator, so the graph is smooth or not broken.
But don’t forget! The denominator of rational function can be zero!
It’s impossible that some values of x is divided by zero. If it shows like that, the value is off limits.
Let f(x)=(x+2)/(x-1) and when x = 1 the function value becomes f(1)=(1+2)/(1-1)=3/0. In this case 0 is the denominator.
From this function, choosing x=1 is a bad idea. It makes a break in function graph.
For example : Suppose we have a graph of f(x)=(x+2)/(x-1). Insert 0 for x, so f(0)=(0+2)/(0-1)=-2. So the point of function in graph is (0,-2).
Now, next to try insert x=1. So we get f(1)=(1+2)/(1-1)=3/0. We know that is impossible. It also means that the graph of function we make has no any point of y when x=1. We can also call the graph is break or discontinuity.
Rational functions don’t always work this way. Take a graph of f(x)=1/(x+1), so f(1)=1/(1+1)=1/2.
Not all of rational functions will give zero in denominator, so the graph is smooth or not broken.
But don’t forget! The denominator of rational function can be zero!
RIGHT TRIANGLE
The right triangle is a triangle where is one of its interior angles is a right angle (90 degrees).
Right triangles figure prominently in various branches of mathematics. For example, trigonometry concerns itself almost exclusively with the properties of right triangles, and the famous theorem by Pythagoras defines the relationship between the three sides of a right triangle:
a^2 + b^2 = h^2
where h is the length of the hypotenuse
a, b are the lengths of the other two sides
Attributes
Hypotenuse : The side opposite the right angle. This will always be the longest side of a right triangle.
Sides : The two sides that are not the hypotenuse. They are the two sides making up the right angle itself.
Properties
• A right triangle can also be isosceles if the two sides that include the right angle are equal in length (AB and BC in the figure above)
• A right triangle can never be equilateral, since the hypotenuse (the side opposite the right angle) is always longer than the other two sides.
Things to Ponder
1. Can a right triangle also be isosceles?
Yes, if its sides have the same length. In the diagram above, try to create such a triangle
2. Can a right triangle also be a scalene triangle?
Yes, if the two sides have different length. The hypotenuse is always the longest side, so it must be a different length to the sides
3. One angle of a right triangle is 41°. What are the other two?
49° and 90°. The interior angles of any triangle always add up to 180°
4. What is the area of a right triangle with sides of length 12 and 15cm?
Somewhere between 89 and 94 square cm
5. A right triangle has a hypotenuse 10 meters long, and one leg 6 meters long. What is the length of the other leg?
Somewhere between 7.1 and 8.9 meters
Right triangles figure prominently in various branches of mathematics. For example, trigonometry concerns itself almost exclusively with the properties of right triangles, and the famous theorem by Pythagoras defines the relationship between the three sides of a right triangle:
a^2 + b^2 = h^2
where h is the length of the hypotenuse
a, b are the lengths of the other two sides
Attributes
Hypotenuse : The side opposite the right angle. This will always be the longest side of a right triangle.
Sides : The two sides that are not the hypotenuse. They are the two sides making up the right angle itself.
Properties
• A right triangle can also be isosceles if the two sides that include the right angle are equal in length (AB and BC in the figure above)
• A right triangle can never be equilateral, since the hypotenuse (the side opposite the right angle) is always longer than the other two sides.
Things to Ponder
1. Can a right triangle also be isosceles?
Yes, if its sides have the same length. In the diagram above, try to create such a triangle
2. Can a right triangle also be a scalene triangle?
Yes, if the two sides have different length. The hypotenuse is always the longest side, so it must be a different length to the sides
3. One angle of a right triangle is 41°. What are the other two?
49° and 90°. The interior angles of any triangle always add up to 180°
4. What is the area of a right triangle with sides of length 12 and 15cm?
Somewhere between 89 and 94 square cm
5. A right triangle has a hypotenuse 10 meters long, and one leg 6 meters long. What is the length of the other leg?
Somewhere between 7.1 and 8.9 meters
Minggu, 21 Desember 2008
THE PROBLEM SOLVING
1. Make the graph of y = g(x). If the function h is defined by h(x) = g(2x) + 2, what is the value of h(1)?
Solution:
We are looking for h(1). We can get the first information from the graph of y = g(x) and the function of h(x) = g(2x) + 2. So, what’s the value of h(x) when x=1?
We substitute 1 into that situation
h(1) = g(2) + 2
Based on the graph above we find when x is equal to 2, y is equal to 1. Or we can say when x=2, g(x)=1, so h(1) = 1 + 2 = 3
Thus we get h(1) = 3
2. Let the function f be defined by f(x) = x + 1.
If 2f(p) = 20, what’s the value of f(3p)?
Solution:
We are looking for f(3p), it means that we are looking for the value of f(x) when x= 3p.
The first information is the function of f(x) = x + 1. And the next information is 2f(p) = 20. The question is “what’s f(x) when x = 3p”.
By using the function of 2f(p) = 20, we divide both side by 2, so we get f(p) = 10.
f(p) is just the implication of f(x) where x = p. We write
f(p) = 10
p + 1 = 10
p = 9
We are looking for the value of f(x) when x = 3p, so we can take p = 9 into x, thus x = 27.
We back to the first equation f(x) = x + 1. We substitute x = 27 into the equation, so that f(27) = 27 + 1 = 28.
So the value of f(3p) is 28.
Solution:
We are looking for h(1). We can get the first information from the graph of y = g(x) and the function of h(x) = g(2x) + 2. So, what’s the value of h(x) when x=1?
We substitute 1 into that situation
h(1) = g(2) + 2
Based on the graph above we find when x is equal to 2, y is equal to 1. Or we can say when x=2, g(x)=1, so h(1) = 1 + 2 = 3
Thus we get h(1) = 3
2. Let the function f be defined by f(x) = x + 1.
If 2f(p) = 20, what’s the value of f(3p)?
Solution:
We are looking for f(3p), it means that we are looking for the value of f(x) when x= 3p.
The first information is the function of f(x) = x + 1. And the next information is 2f(p) = 20. The question is “what’s f(x) when x = 3p”.
By using the function of 2f(p) = 20, we divide both side by 2, so we get f(p) = 10.
f(p) is just the implication of f(x) where x = p. We write
f(p) = 10
p + 1 = 10
p = 9
We are looking for the value of f(x) when x = 3p, so we can take p = 9 into x, thus x = 27.
We back to the first equation f(x) = x + 1. We substitute x = 27 into the equation, so that f(27) = 27 + 1 = 28.
So the value of f(3p) is 28.
INVERSE FUNCTION
Before we know about the definition of inverse function, suppose that we have relation of a function F(x,y)=0 where function y=f(x) as the Vertical Line T. and also x=g(y) as the Horizontal Line T.
We can solve the function by using function y=f(x) where x=g(x). If we looking for the function of y=x2 and then x=g(y) is a horizontal line which intersects the graph in two points.
We can graph the function of x=g(y) with the domain x is bigger than or equal to zero (0 ≤ x) and then we have the square roots of y=x2
Let’s write y=2x-1 and make the graph of that function. There is a straight line with the point of (0,-1), slope 2
(0,-1)
So, in the x-intersect we find point of (½,0).
And let’s look at the line of y=x and for reason we can substitute y=x into the function y = 2x - 1, so : x = 2x - 1
1 + x = 2x
1 = x
So, we have intersection point between line y = 2x – 1 and y = x, of course in point of (1,1).
Now, from these relation we want to solve the equation
2x – 1 = y
2x = y + 1
x = ½(y + 1)
x = ½y + ½
Then this equation is also equal to : y = ½x + ½
We looking back on the graph then we get the other line. Let we graph a line containing of point (1,1) and (0,-1).
We have f(x) = 2x – 1 and g(x) = ½x + ½, so
f(g(x)) = 2(g(x)) – 1
= 2 (½x + ½) – 1
= x + 1 – 1
= x
On the other hand
g(f(x)) = ½(f(x)) + ½
= ½(2x – 1) + ½
= x - ½ + ½
= x
So the important problem of two functions is g = f-1
f(g(x)) = f(f-1(x))
= x
g(f(x)) = f-1(f(x))
= x
Let’s do one more example
Write the function of y=(x-1)/(x+2)
The x-intercept is gonna be equal to 1, and y-intercept is gonna be -1.
The solution is :
y=(x-1)/(x+2)
y ( x + 2 ) = x - 1
yx + 2y = x – 1
yx – x = -1 – 2y
(y - 1) x = -1 – 2y
x=(-1-2y)/(y-1)
y^(-1)=(-1-2x)/(x-1)
Let x = 0, y = -1
y = 0, x = -½
There are vertical asymtot x = 1 and horizontal asymtot y = -2
We can see from the figure of function y=(x-1)/(x+2) and y^(-1)=(-1-2x)/(x-1) that the two functions are reflected each other. So, the favorite function to look for the inverses are two function.
We can solve the function by using function y=f(x) where x=g(x). If we looking for the function of y=x2 and then x=g(y) is a horizontal line which intersects the graph in two points.
We can graph the function of x=g(y) with the domain x is bigger than or equal to zero (0 ≤ x) and then we have the square roots of y=x2
Let’s write y=2x-1 and make the graph of that function. There is a straight line with the point of (0,-1), slope 2
(0,-1)
So, in the x-intersect we find point of (½,0).
And let’s look at the line of y=x and for reason we can substitute y=x into the function y = 2x - 1, so : x = 2x - 1
1 + x = 2x
1 = x
So, we have intersection point between line y = 2x – 1 and y = x, of course in point of (1,1).
Now, from these relation we want to solve the equation
2x – 1 = y
2x = y + 1
x = ½(y + 1)
x = ½y + ½
Then this equation is also equal to : y = ½x + ½
We looking back on the graph then we get the other line. Let we graph a line containing of point (1,1) and (0,-1).
We have f(x) = 2x – 1 and g(x) = ½x + ½, so
f(g(x)) = 2(g(x)) – 1
= 2 (½x + ½) – 1
= x + 1 – 1
= x
On the other hand
g(f(x)) = ½(f(x)) + ½
= ½(2x – 1) + ½
= x - ½ + ½
= x
So the important problem of two functions is g = f-1
f(g(x)) = f(f-1(x))
= x
g(f(x)) = f-1(f(x))
= x
Let’s do one more example
Write the function of y=(x-1)/(x+2)
The x-intercept is gonna be equal to 1, and y-intercept is gonna be -1.
The solution is :
y=(x-1)/(x+2)
y ( x + 2 ) = x - 1
yx + 2y = x – 1
yx – x = -1 – 2y
(y - 1) x = -1 – 2y
x=(-1-2y)/(y-1)
y^(-1)=(-1-2x)/(x-1)
Let x = 0, y = -1
y = 0, x = -½
There are vertical asymtot x = 1 and horizontal asymtot y = -2
We can see from the figure of function y=(x-1)/(x+2) and y^(-1)=(-1-2x)/(x-1) that the two functions are reflected each other. So, the favorite function to look for the inverses are two function.
Minggu, 23 November 2008
THE DEFINITION AND THE SENTENCE OF WORDS
A. dispatch
Definition : dispose of rapidly and without delay and efficiently
Sentence : He dispatched the task he was assigned.
B. engaging
Definition : carrying out / participating in an activity; be involved in
Sentence : The students are engaging the National Seminar of Mathematics.
C. pita = ribbon
Definition : any long object resembling a thin line
Sentence : We describe a geometric interpretation of k-ribbon Fibonacci tableaux and use this interpretation to describe a notion of P equivalence for k-ribbon Fibonacci tableaux.
D. kurs = kurs; the exchange rate
Definition : the price of one country’s currency expressed in another country’s currency
Sentence : The higher the exchange rate for one dollar in terms of rupiah, the lower the relative value of rupiah.
Definition : dispose of rapidly and without delay and efficiently
Sentence : He dispatched the task he was assigned.
B. engaging
Definition : carrying out / participating in an activity; be involved in
Sentence : The students are engaging the National Seminar of Mathematics.
C. pita = ribbon
Definition : any long object resembling a thin line
Sentence : We describe a geometric interpretation of k-ribbon Fibonacci tableaux and use this interpretation to describe a notion of P equivalence for k-ribbon Fibonacci tableaux.
D. kurs = kurs; the exchange rate
Definition : the price of one country’s currency expressed in another country’s currency
Sentence : The higher the exchange rate for one dollar in terms of rupiah, the lower the relative value of rupiah.
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