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 July 2nd, 2017, 06:29 PM #1 Banned Camp   Joined: Jul 2010 Posts: 118 Thanks: 0 Special triangles A new method that produces right triangles , with angles close 45 deg' Select a number close to ( 1 + root of 2 ) for example a = 2.4141 b = 0.5 ( a^2 - 1 ) = 2.41394 c = b +1 = 3.41394 a^2 + b^2 = c^2
 July 2nd, 2017, 06:57 PM #2 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,591 Thanks: 546 Math Focus: Yet to find out. No it doesn't. And what does "Select a number close to (1 + root of 2)" have to do with anything? Why not just say, select a number a and b and the result is another arbitrary constant. Thanks from topsquark and JeffM1
 July 2nd, 2017, 07:17 PM #3 Banned Camp   Joined: Jul 2010 Posts: 118 Thanks: 0 It's so simple I choose a = 2.41 and that's all. b = 0.5 ( a^2 - 1) = 2.40405 c = b + 1 = 3.40405 a^2 + b^2 = c^2
July 2nd, 2017, 07:24 PM   #4
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Quote:
 Originally Posted by aetzbar A new method that produces right triangles , with angles close 45 deg' Select a number close to ( 1 + root of 2 ) for example a = 2.4141 b = 0.5 ( a^2 - 1 ) = 2.41394 THIS IS NOT CORRECT c = b +1 = 3.41394 SO THIS IS NOT CORRECT a^2 + b^2 = c^2 SO THIS ALSO IS NOT CORRECT
I am not sure why we are supposed to pay attention to a post where the arithmetic is not correct. But even if the arithmetic were correct, what would be the significance of finding a right triangle where the other two angles are not equal to 45 degrees? There is nothing untoward with a right triangle having angles of 44, 46, and 90 degrees.

I went to Columbia in the 60's, when you could drink in New York at 18. I heard far more cogent discussions in the freshman dorm after the bars closed.

 July 2nd, 2017, 07:24 PM #5 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 12,396 Thanks: 828 We don't need your method. Quit being silly... Thanks from topsquark and JeffM1
July 2nd, 2017, 07:38 PM   #6
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The perfect triangle

Quote:
 Originally Posted by JeffM1 I am not sure why we are supposed to pay attention to a post where the arithmetic is not correct. But even if the arithmetic were correct, what would be the significance of finding a right triangle where the other two angles are not equal to 45 degrees? There is nothing untoward with a right triangle having angles of 44, 46, and 90 degrees. I went to Columbia in the 60's, when you could drink in New York at 18. I heard far more cogent discussions in the freshman dorm after the bars closed.
a = 1 + root of 2
b = 0.5 ( a^2 - 1 ) = 1 + root of 2
c = b + 1 = 2 + root of 2

a^2 + b^2 = c^2

July 2nd, 2017, 08:04 PM   #7
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 Originally Posted by aetzbar I choose a = 2.41 and that's all. b = 0.5 ( a^2 - 1) = 2.40405 c = b + 1 = 3.40405 a^2 + b^2 = c^2
$a = \sqrt{2} + 1 \implies a^2 = 2\sqrt{2} + 3.$

$b = 0.5(a^2 - 1) = 0.5(2\sqrt{2} + 3 - 1) = \sqrt{2} + 1 \implies \\ b^2 = 2\sqrt{2} + 3.$

$c = b + 1 = \sqrt{2} + 2 \implies c^2 = 4\sqrt{2} + 6.$

$a^2 + b^2 = 2\sqrt{2} + 3 + 2\sqrt{2} + 3 = 4\sqrt{2} + 6.$

$a^2 + b^2 = c^2.$

And we deduce from this what exactly?

July 2nd, 2017, 09:16 PM   #8
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Quote:
 Originally Posted by aetzbar A new method that produces right triangles , with angles close 45 deg
One of the basic lessons in school is that the isosceles triangle with legs $\sqrt2$ and hypotenuse $1$ has angles of exactly $45^\circ$.

We also have methods for generating every single right-angled triangle with integer sides. Do you really think the world cares about a right-angled triangle with angles "close" to $45^\circ$?

If you are going to use irrational numbers, you really ought to get comfortable with them.

 July 2nd, 2017, 09:36 PM #9 Banned Camp   Joined: Jul 2010 Posts: 118 Thanks: 0 The number a (that selected,between zero and infinite) Determines all the triangle data. b = 0.5 ( a^2 - 1 ) c = 0.5 ( a^2 - 1 ) + 1 circumference = a^2 + a aria = 0.25 ( a^3 - a) height = ( a^3 - a) : (a^2 +1) angle tg against a = 2a : ( a^2 - 1) half angle tg against a = 1 : a thanks
 July 2nd, 2017, 10:26 PM #10 Global Moderator   Joined: Dec 2006 Posts: 18,954 Thanks: 1601 If b = (a² - 1)/2, a² = 2b + 1, and so a² + b² = b² + 2b + 1 = (b + 1)² = c², satisfying Pythagoras. This isn't new. As a = 1 + √2 satisfies (a² - 1)/2 = a, choosing a to be "close to" 1 + √2 ensures that b is also "close to" 1 + √2, and so the acute angles of the triangle are close to 45°. In particular, a = 2.4141 implies b = 2.413939405 and c = 3.413939405. To ensure the triangle has an angle of exactly 90°, the values of b and c shouldn't be rounded to fewer decimal places.

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