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