Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-02T20:41:14Z

AllisonMiller: /* What Can We Do With It? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem. This needs to be fleshed out with examples.

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 15/Homework 4

2010-10-20T05:44:14Z

AllisonMiller: /* Question 2 */

== Question 1 ==

== Question 2 ==

Bohao, Tim, Dylan, Chan and Stewart

5 players

3 of them are right handed
2 of them are left handed

3 of them are under 2m
2 of them are over 2m

We are looking for the centre player who is left handed and also over 2m.

-Tim or Chan must be right handed, because Dylan and Bohao are both right handed and Stewart is left handed.
-Bohao is over 2m tall so this means that either Dylan or Tim must be the same height as Chan and Stewart who are both under 2m.
-You can already see a common trend developing, the fact that we are trying to find both the variables for Tim's height as well as his handedness.
-This then narrows down to Dylan and TIm to be over 2m
-Dylan is right handed though

This leads us to the conclusion that Tim is the centre player because he is the only valid option, Dylan who is over 2m is right handed and therefor does not fit the description.

== Question 3 ==

== Question 4 ==

== Question 5 ==

User:AllisonMiller

2010-09-20T03:26:24Z

AllisonMiller: Created page with ' The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angl…'

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them.
One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.

The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.

Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2
c2= 62 + 82
c2 = 36 + 64
c2 = 100
c = 10

UBC Wiki - User contributions [en]

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

Course:MATH110/Archive/2010-2011/003/Groups/Group 15/Homework 4

User:AllisonMiller