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I want to understand right triangles a little better.

So if you have a Triangle and the given facts are:

1) Right triangle (so one side is 90*) 2) The hypotunse is equal to 10

Can we conclude the triangle is a 6-8-10 ( a variant of the 3-4-5) triangle? If not, can we calculate the permiter of the triangle?

How about the introduction of a new fact:

1) Right triangle (so one side is 90*) 2) The hypotunse is equal to 10 3) All the sides of the triangle are integer values.

Now can we concludes it is a 6-8-10 triangle?

With the exception of having a 90* degree angle, what conditions do we need to ensure a triangle is a Pythagorean triangle?

There are various ways in which you can have a right triangle with hypotenuse 10. The sides could be 1, \(\sqrt{99}\), 10 or \(\sqrt{2}\), \(\sqrt{98}\), 10 or 6, 8, 10 etc

If sides are integral, it does add constraint to the values that the sides can take. Pythagorean triplets give sides of right triangles that have all integral sides. But that doesn't ensure that the hypotenuse will imply a single value for the other two sides in all cases (but it does in most). If hypotenuse is 10, the sides will be 6 and 8 but if the hypotenuse is 65, the other two sides could be (16, 63) or (33, 56)

1) Right triangle (so one side is 90*) 2) The hypotunse is equal to 10 3) All the sides of the triangle are integer values.

Does the third condition gaurentee that we have a unique triplet triangle? Because as you mentioned we can have a right triangle with hypotunse of 10 with three possible side orders:

But notice only one of choices has all of its sides as integer values!! (1) and (2) do not have integer values as sides.

Can I therefore state the condition :

IF the triangle is: 1) A right triangle 2) One of the sides is triplet value (3,4,5) (6,8,10) (5,12,13) (8,15,17) etc 3) All of the sides are integer value

1) Right triangle (so one side is 90*) 2) The hypotunse is equal to 10 3) All the sides of the triangle are integer values.

Does the third condition gaurentee that we have a unique triplet triangle? Because as you mentioned we can have a right triangle with hypotunse of 10 with three possible side orders: (actually, there are innumerable ways in which you can have the hypotenuse 10. I just gave 3 examples. But yes, only one of them has all integral sides)

But notice only one of choices has all of its sides as integer values!! (1) and (2) do not have integer values as sides.

Can I therefore state the condition :

IF the triangle is: 1) A right triangle 2) One of the sides is triplet value (3,4,5) (6,8,10) (5,12,13) (8,15,17) etc 3) All of the sides are integer value

Then the triangle is a triplet and unique?

Read the above post again: "If sides are integral, it does add constraint to the values that the sides can take. Pythagorean triplets give sides of right triangles that have all integral sides. But that doesn't ensure that the hypotenuse will imply a single value for the other two sides in all cases (but it does in most). If hypotenuse is 10, the sides will be 6 and 8 but if the hypotenuse is 65, the other two sides could be (16, 63) or (33, 56)"

Let me re-phrase it:

If hypotenuse is 10, the sides must be 6, 8, 10 - unique If hypotenuse is 65, the sides are not unique. There are two pythagorean triplets with hypotenuse 65. They are (16, 63, 65) and (33, 56, 65).

Often, with a given hypotenuse, only one pythagorean triplet is formed, but it is not necessary.
_________________

There are various ways in which you can have a right triangle with hypotenuse 10. The sides could be 1, \(\sqrt{99}\), 10 or \(\sqrt{2}\), \(\sqrt{98}\), 10 or 6, 8, 10 etc

More than various, there is actually an infinite amount of right triangles with hypotenuse = 10!!

1) Right triangle (so one side is 90*) 2) The hypotunse is equal to 10

Q. Can we conclude the triangle is a 6-8-10 ( a variant of the 3-4-5) triangle? A. No you CANNOT conclude that its a 6-8-10 triangle as you have only the following things known:-- Suppose base is b height is d and hypotenous is h then what you know is h=10 and a^2 + b^2 = h^2 = 10^2 So there can be any number of values for a and b! So you CANNOT conclude that it is a 6-8-10 traingel

Q. If not, can we calculate the permiter of the triangle? A. NO you cannot find the perimeter of teh trainagle as you don't know the individual sides of the traingle/(as explained above)

How about the introduction of a new fact:

1) Right triangle (so one side is 90*) 2) The hypotunse is equal to 10 3) All the sides of the triangle are integer values.

Q. Now can we concludes it is a 6-8-10 triangle? A. Suppose base is b height is d and hypotenous is h then what you know is h=10 and a^2 + b^2 = h^2 = 10^2 Since a and b can only be positive integers so, a can be 6 or 8 and b can be 8 0r 6 respectively. YES you can conclude that it is a 6-8-10 triangle

Q. With the exception of having a 90* degree angle, what conditions do we need to ensure a triangle is a Pythagorean triangle? A. Pythagorean Triangle. I doubt if there is a term like that! But i know what you mean. You mean what all do we need to apply Pythagorean Theorm. Pythagorean Theorm is defined only for right angled triangles. Any triangle which is a right angled triangle => You can use Pythagorean Theorm. Any triangle which is NOT a right andled triangle => You CANNOT use Pythagorean Theorm.

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