Transcript
CSO: M.O.8.4.3
Students will solve right triangle problems where the existence of triangles is not
obvious using the Pythagorean Theorem.
Objective:
Legs – The sides that form the Right (90⁰) angle.
Hypotenuse – The side opposite the right angle, it is the longest side of the triangle.
Converse – reversing the parts.
Helpful Vocabulary
Pythagorean TheoremDescribes the relationship between the lengths of the legs and the hypotenuse for any right triangle
HypotenuseLeg 1
Leg 2
IN WORDS AND SYMBOLS
• In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the length
•c2 = a2 + b2
Who is Pythagoras?
Born in Samos (Island in Aegean Sea)Around 570 - 495 BC
Greek Philosopher, mathematician, and Mystic
known as an expert on the fate of our soul after death
Believed to be first Western Vegetarian.
founder of the religious movement called Pythagoreanism in Southern Italy
Proofs
First Video Proof
Second Video Proof
Third Video Proof
Historical Note
While we call it Pythagoras‘ Theorem, it was alsoknown by Indian, Greek, Chinese and babylonian mathematicians well before he lived !
Using a Centimeter Grid to find area
Area = 1 cm
Area = 16 cm squared
Area = 48 cm squared
3-4-5 Rule
This rule is used to check for the existence of a Right corner.Simply Stated:The measure of any side of 3 units, plus the next side of 4 units has to have a diagonal side of 5 units.
3-4-5 Rule Expanded
This is the 3-4-5 Rule3 squared is 94 squared is 169+16 = 25Square Root of 25 is 5
Make a ConjectureIf the length of one side is 6 andLength of the next side is 8,What would be the length of the longest side if this was a Right Triangle and 6 and 8 were the two shorter sides?
10
The answer is 15 since we will not have a negative side to the triangle
Now let’s try a problem together
Side a =12 ft
Side b = 18 ft
c
Find the length of the hypotenuse of the above Right Triangle?
Start withc2 = a2 + b2
Fill in with knowns
c2 = a2 + b2
c2 = (12)2 + (18)2
Square the sides
c2 = 144 + 324
Add
c2 = 468
Find the Square Root of Both Sides
√c2 = √468
Round
c = 21.63
If you reverse the parts of the pythagorean theorem,
you have formed itsConverse, and it is also
true
Funny Break
a2 b2 c2
3 4 5
5 12 13
7 24 25
8 15 17
9 40 41
11 60 61
12 35 37
Pythagorean Triples
Irrational numbers and Pythagoras
An irrational number is a number that cannot be expressed as the quotient a/b where a and b are integers and b ≠ 0
Every square root of an imperfect square is an irrational number.Example:
√10 = 3.1622776……..This number continues indefinitely with no repetition
Problems to try
c2 = a2 + b2
c2 = 24yds2 + 18yds2
c2 = 576 + 324c2 = 900c = 30
b2 = c2 - a2
b2 = 82 - 32
b2 = 64 – 9b2 = 55b = 7.42
a2 = c2 - b2
a2 = 20cm2 - 17cm2
a2 = 400 - 289a2 = 111a = 10.54
Answer to this problem using Pythagoras is 8ft
22 ft
14 ft
How tall does the ladder need to be to reach the coconuts?
Hope you learned something about Pythagoras and his theorem.
References
Who2 Biography. Copyright © 1998-2010 by Who2, LLC. All rights reserved. See the Pythagoras biography from Who2.
Pierce, Rod. "Math is Fun - Maths Resources" Math Is Fun. Ed.
Rod Pierce. 19 Apr 2010. 1 Oct 2010 http://www.mathsisfun.com/
http://www.glencoe.com/ose/showbook.php
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