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Pythagorean Theorem Proof: Four Right Triangles


Date: 10/7/96 at 17:15:49
From: Tess Bethune
Subject: Pythagorean Theorem

I don't really understand the Pythagorean Theorem.  


Date: 10/22/96 at 18:44:14
From: Doctor Yiu
Subject: Re: Pythagorean Theorem

Dear Tess,

Let's say you have never heard of the Pythagorean theorem, but you
have a rectangular cardboard of sides 5 inches and 12 inches. 
You want to figure out how long the DIAGONAL is.

With pictures, things will be easier.

     

Make a copy of the two pictures and have them in front of you before
you proceed.


Take two identical cardboards of sides 5 inches and 12 inches, and
cut each one along a diagonal. In this way you will have four right 
triangles.

With the four right triangles, form a large square as in the 
picture on the lefthand side.   Each side of the square has length 
5 + 12 = 17 inches.

        

Let's calculate areas.

The area of the large square is 17^2 = 289.

The four triangles together make up two rectangular cardboards and
together have area 2 times 5 times 12 = 120.

Therefore, the 'hollow' square in the middle must have area 
289 - 120 = 169.

Each side of this middle square is the square root of 169, 
which is 13.

Conclusion: if the legs of a right triangle are 5 and 12, then the
hypotenuse is 13.  (It is the same to say that if the two sides of 
a rectangle are 5 and 12, then each diagonal is 13).

..................................................................

This method actually applies to ALL right triangles. 

You might want to try to find the hypotenuse of a right triangle with 
legs 8 and 15.

Following the above method, you should get 17 for the answer.

..................................................................

People in ancient times pondered on this simple method of calculation 
and discovered a general law, which we usually call the Pythagorean 
theorem. It tells of a simple relation among the three sides of a 
right triangle.

Instead of working with specific numbers, we shall now use symbols 
a and b to stand for the legs, and c for the hypotenuse of a right 
triangle.

The first picture on the righthand side above is a rearrangement of 
the same square that you saw on the lefthand side.

Here, you saw the same four right triangles, and two squares in the
lower lefthand and upper righthand corners

A comparison of the two pictures clearly tells you that (by 
removing the four right triangles from each picture) the total area 
of the two squares in the second picture is the same as the square in 
the middle of the first picture.  

        

In other words,

the sum of the squares on the legs of a right triangle is equal to the
square on its hypotenuse.

This is the famous Pythagorean Theorem:  a^2 + b^2 = c^2.


Now,  you no longer need to repeat the method of subtraction of areas 
in the beginning of this message. For example, from a = 8 and b = 15,
you would easily get  c^2 = 8^2 + 15^2 = 64 + 225 = 289, and by taking
square root, c = 17.

I hope this makes the Pythagorean theorem understandable.

Many thanks to Doctor Jodi for valuable comments, and Doctor Ken and 
Doctor Sarah for putting up the pictures.

Doctor Yiu,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Basic Algebra
High School Geometry
High School Triangles and Other Polygons
Middle School Algebra
Middle School Geometry
Middle School Triangles and Other Polygons

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