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Can an isosceles right triangle ever have an integer as its perimeter?
- Thread starter Jedster
- Start date
Jedster
Flying through space
Hmm..that's a fail!!!!!
SpeaksForTheTrees
Well-Known Member
Can a right isos tri have its perimeter as whole number ?
3 ,4 ,5 right isos tri = 12 perimeter ?
Can all the lengths of the sides of an isos right tri add up to a whole number.?
Simple rule , is such a thing as a 3,4,5 right angled triangle
Where 3 , 4 and 5 are the length of each side , exact .
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Corthos
Great Old One
I don't see why not...
Nearly any problem in math can have any answer, though I don't know how you'd do it if the side length of the triangle was an integer (seems like there would have to be decimals). I've got a pretty annoying head cold right now, so I'll let someone else pull out the specific equations needed. XD
Nearly any problem in math can have any answer, though I don't know how you'd do it if the side length of the triangle was an integer (seems like there would have to be decimals). I've got a pretty annoying head cold right now, so I'll let someone else pull out the specific equations needed. XD
SpeaksForTheTrees
Well-Known Member
Integer is a number with no fractional parts ie 1,2,3
Easy one.
If one leg length = an integer multiple of the reciprocal of (2 + square root of 2)
If one leg length = an integer multiple of the reciprocal of (2 + square root of 2)
Corthos
Great Old One
I know. Note that I said side length, and not perimeter. =/
SpeaksForTheTrees
Well-Known Member
Therefore the "going" of one leg length = 3.4142236
Jedster
Flying through space
3,4,5 is not isosceles.
Ouroboros
Coincidentia oppositorum
"Can an isosceles right triangle ever have an integer as its perimeter?"
(Note to other posters, perimeter is the sum of the sides. The "circumference" or boundary.)
Yes. Infinitely many.
The simplest one would have the sides 1, 1, 1. Adds up to the perimeter 3.
Start with any integer number, lets say 11. Divide by three. Each side is 11/3 in length. The sum of the sides then is 11.
Wait... Doh!!!
"... right triangle..." Missed the right angle part. Let me think...
...
(Note to other posters, perimeter is the sum of the sides. The "circumference" or boundary.)
Yes. Infinitely many.
The simplest one would have the sides 1, 1, 1. Adds up to the perimeter 3.
Start with any integer number, lets say 11. Divide by three. Each side is 11/3 in length. The sum of the sides then is 11.
Wait... Doh!!!
"... right triangle..." Missed the right angle part. Let me think...
...
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Jedster
Flying through space
Well done...you get the gold star.
Each of the smaller angles is 45°.
Suppose the equal sides are of length x, the using sin45°= opposite/hypotenuse= x/h, then
x/h=1/√2, so h=x*√2 , so
perimeter = x+x+x*√2
= 2x+x*√2
=x(2+√2)
So if x = 5/(2+√2), then the perimeter = 5.
SO, generally, in an isosceles right triangle, the perimeter is (2 + √2) times the length of a side.
If the side length is for example, 5/(2 + √2), then the perimeter is the integer 5.
However, the side length and perimeter cannot both be integers.
SpeaksForTheTrees
Well-Known Member
Jedster
Flying through space
Perhaps I should have written
Can an isosceles right-angled triangle ever have an integer as its perimeter?
Ouroboros
Coincidentia oppositorum
No. Absolutely not. It was a typical question that would be on a test. When you read, you have to pay attention. That's the cool part of it.
rosends
Well-Known Member
so far I have checked all the ones with legs between 1 and 679 and the answer is "no." 268 is closest. Should I keep checking?
It seems that the sum of two identical squares is never a perfect square. All I found on line is that the sum of two squared odd integers can't equal a perfect square. Nothing about identical integers, or I just haven't found it yet.
It seems that the sum of two identical squares is never a perfect square. All I found on line is that the sum of two squared odd integers can't equal a perfect square. Nothing about identical integers, or I just haven't found it yet.
SpeaksForTheTrees
Well-Known Member
Two equal sides , has been over 30 years )
Jedster
Flying through space
The question only asks if the perimeter can be an integer See post #12 above which says in the last line
However, the side length and perimeter cannot both be integers.
rosends
Well-Known Member
Oh, I was starting with the presumption that the side legs were integers. If the legs don't have to be integers then that's a different kettle of fish.
Jedster
Flying through space
Me too , when I first tried the question.