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Geometry and Geometry and Measurement Measurement Brad Fewins Stephen Hummel
| Date post: | 12-May-2015 |
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Geometry and Geometry and MeasurementMeasurement
Brad FewinsStephen Hummel
Table of Contents:Table of Contents:Pythagorean TheoremPythagorean Theorem
•Pythagoras of Samos•History•More on History•Pythagoras Quotes•References to the Pythagorean Theorem•More References•Proving the Theorem•Real-World Application•Works Cited
Table of Contents: ShapesTable of Contents: Shapes
Circle Triangle Square
Rectangle Rhombus
Additional Help Works Cited
Pythagoras of SamosPythagoras of Samos
• Pythagoras was an extremely important mathematician in history.
• He is called the first pure mathematician by many.
• Unfortunately, we know relatively little about his mathematical achievements.
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HistoryThere is a lot of debate whether
the theorem was discovered once or many times.
Many believe that the theorem was known to the Babylonians
1000 years previous to Pythagoras but he may have
been the first to prove it.
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More on history
• Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples.
There is a legend that Pythagoras sacrificed 100 oxen in light of the discovery.
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Pythagoras Quotes
• Number is the ruler of forms and ideas, and the cause of gods and demons.
• Every man has been made by God in order to acquire knowledge and contemplate.
• Geometry is knowledge of the eternally existent.
• Number is the within of all things. • There is geometry in the humming of the
strings. • Time is the soul of this world.
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References to the Pythagorean Theorem
• ~In the Wizard of Oz when the scarecrow gets his diploma from the wizard he immediately shows off his knowledge by exclaiming an incorrect version of the formula, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh, joy, oh, rapture. I've got a brain!"
• ~In an episode of the Simpson's, Homer quotes the scarecrow’s version of the theorem A man nearby then yells out, "That's a right triangle, you idiot!" (although that still doesn’t completely correct the scarecrows version)
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More References
• ~The speech software on the MacBook also references the previous incorrect statement of the theorem. It is a sample speech, Ralph is the voice setting.
• ~Also, Uganda released a coin with the shape of a right triangle inscribed on it. The coin has a picture of Pythagoras and the Pythagorean theorem on it.
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Proving the TheoremProving the Theorem
• This website includes an interactive java applet that allows the audience to follow along well enough to understand the geometry involved.
• http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html
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The answer to this real world application
• a=90 Since the distance• b=90 between home plate• c^2=a^2+b^2 and second base is • c^2=90^2+90^2 the same as the• c^2=8100+8100 distance between • c^2=16200 first base and third• c= base, the answer for • c=127.279 both distances will be
Back to the problem the same.
16200
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CircleCircle
Area of a Circle:Area of a Circle: A=A=∏(3.14)·r²∏(3.14)·r²
Or Or ∏∙r∙r∏∙r∙r Example:Example:
R= 3 inches, what is R= 3 inches, what is the area?the area?
∏∙ ∏∙3 inches·3 inches = 28.26in²3 inches·3 inches = 28.26in²
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Triangle
Area=Area=½· base · height½· base · height
Base=12cmBase=12cm Height=9cmHeight=9cm ½·12·9=½·12·9=
Click image to Click image to reveal answer!reveal answer!
Answer:Answer:
A=A=½·108 in²=½·108 in²=
A=54 inches²A=54 inches²
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SquareSquare Area= width Area= width · height· height X= 6 meters, what isX= 6 meters, what is the area?the area?
6m·6m=6m·6m=36m²36m²
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RectangleRectangle
Area=Area= Width Width · Height· Height If s=4, what is the Area?If s=4, what is the Area?
Click image for answerClick image for answer
Solution:Solution:
If s= 4cmIf s= 4cm Area= 9cmArea= 9cm · 4cm · 4cm
Answer= Answer= 36cm²36cm²
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RhombusRhombus
Area for base times Area for base times height method:height method:
Click image for solution!Click image for solution!
Area=Area=
base base · altitude or · altitude or heightheight
Example:Example: If base= 129cmIf base= 129cm Height= 34cmHeight= 34cm Area= ?Area= ?
Answer: RhombusAnswer: Rhombus
Area= 129cm Area= 129cm · 34cm=· 34cm= 4386 cm²4386 cm²
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Additional Help
Area of a Rectangle
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Works cited
• http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html
• http://en.wikipedia.org/wiki/Pythagorean_theorem• http://www.geom.uiuc.edu/~demo5337/Group3/hist.html• http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBC
Examples/Pythagoras/pythagoras.html• http://www.youtube.com/watch?v=1ZReTq9V2RI• http://www.youtube.com/watch?v=ECJfSyg_Obo
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