# False Theorem

Hello
Below is the theorem tried proving geometrically in book
"Mathematics for Computer Science
Eric Lehman and Tom Leighton
2004"

False Theorem 11. 420 > 422

My question:
I would like to know the error in proof

Sham

Sham
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Author Commented:
Please let me know, if you have problem in accessing this book
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Commented:
The stray corners are just under 2 units high, and the large rectangle is
21.9047619 * 19
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Author Commented:
are you saying that
>2 mentioned for length of both new triangles is wrong?
It is <2

But I feel length of new traingle is more than 2 for both traiangles, Because length of rectangle is greater than breadth.
Sham
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Commented:
would you still feel that way if the upper piece had been slid until it had moved exactly 20 units leftward?
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Author Commented:
I feel that when upper piece of rectangle moved 2  units left, Breadth of 2 new triangle is 2 units.
If breadth of 2 new triangles is 2 units, then length of 2 new triangles will definetely be > 2 units.

Sham

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Commented:
What would you feel about the length if the breadth was 20 units?
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Author Commented:
Ya i already told you this point.
length of triangle will be > 20 if breadth of triangle is 20
Because length of any rectangle is obviously greater than breadth of rectangle.
Sham
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Commented:
what is the length and breadth of the original rectangle?
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Cut it in half diagonally to get two triangles   w= 21  and  h= 20.

Now cut the the little corner triangles with w= 2.

Don't slide the figures and estimate.
Calculate the heights of these figures using similar triangles.

Calculate the areas of all four figures (two trapezoids and two little triangles).
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Commented:
Do the calculations using fractions.  Don't use a calculator.
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Author Commented:
You can go thru False theorem in the book i mentioned

BTW  the length is 21 and breadth is 20
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Commented:
I did the calculations.  That's how I found the error.
You should do the calculations.  It's not that hard.

Why estimate a length as >2 when you can calculate it exactly???
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Author Commented:
sorry please find the dimension in the attached screenshot

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Commented:
The entire book is available here.  The problem is on page 23.

There is a simple similar triangle ratio calculation that will tell you the answer.
You can do it in your head.  But you do have to do it.
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Commented:
>>  The entire book is available here.  The problem is on page 23.

http://www.cs.princeton.edu/courses/archive/spr10/cos433/mathcs.pdf
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Commented:
>>  But I feel length of new traingle is more than 2 for both traiangles, Because length of rectangle is greater than breadth.

Why feel when you can calculate????
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Author Commented:
Hello glitch
The upper pience is moved left exact 2 units.
After shift, If i consider any one triangle,
what i understand is horizontal length is 2 units and vertical length is definitely  less than 2 units because upper piece was shifted horizontally and the horizontal length is 21 > vertical length 20
But How do i calculate second side of triangle if you do not know hypotenus ?
Sham

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Commented:
Similar triangles:     20/21 = X/2
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Author Commented:
i understand that they are similar triangles,
but
"20/21=x/2" is going above my head
i did not understand this
Sham
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Commented:
The concept of similar triangles is secondary school geometry, not college math.
You may need to do some reviewing.

http://www.mathopenref.com/similartriangles.html
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Commented:
The base of the big triangle is 21.  The height is 20.

The base of the small, similar triangle is 2.      The ratio of the heights to bases will be the same.
``````Height      20       X
------  =  ----  =  ---
Base       21       2

Cross multiply to find   21*X = 20*2   ==>   X = 40/21
``````
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Author Commented:
Hello glitch
I sincerely remembernow  that idid such exercise in school days 18 -20 years back.
you basically took big triangle and small triangle from any one piece.
But i strongly feel there is one theorem based on which you wrote this.
What was that theorem
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Author Commented:
Hello glitch
I feel you wrote
20/21 = x/2
based on one theroem
Because this idea is not intuitive
What is that theorem
Sham
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Commented:
From the link I posted earlier:

Properties of Similar Triangles

Corresponding angles are the same.
Corresponding sides are all in the same proportion.

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Author Commented:
Hello glitch
But before that, I would like to ask one question which is beyond this discussion,
I read this property in my school days.
How do you prove the following?
Properties of Similar Triangles

Corresponding angles are the same.
Corresponding sides are all in the same proportion.

Sham
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Author Commented:
perfect
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Commented:
If the height was >20 after being moved exactly 20 units leftward, it would be longer than the original rectangle.
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