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# The probability of finding an isosceles triangle.

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I agree with Prime provided the OP is worded,

What fraction of integer-sided triangles whose perimeter is 12 are isosceles?

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Depending on random selection process, there is another possibility.

If sides are randomly selected, rather than entire triangles, the answer is different.

For example, in a triangle of (2,4,6), the sides may be assigned in 6 different ways, whereas the sides of (2,5,5) may be assigned only in 3 different ways, and (4,4,4) -- only one combination.
Again that takes into account all triangles appearing as a segment: (0,6,6); (1,5,6); (2,4,6); and (3,3,6).
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I think the triangle inequality will rule out many of the triangles mentioned

What's triangle inequality?

If "triangle inequality" means that one side cannot exceed the sum of the other two, then we didn't forget that. Both 4/7 and 10/28 answers take that into account.

All triangles I see are these:

(0,6,6) => 3 variations for side assignment

(1,5,6) => 6 variations

(2,4,6) => 6

(2,5,5) => 3

(3,3,6) => 3

(3,4,5) => 6

(4,4,4) => 1

In all 7 different triangles. Total of 28 variatios with random side assignment.

If you pick triangles at random, the probability is 4/7. If you pick sides at random -- 10/28.

If you harbor a prejudice against triangles with one or two angles equal to zero, then the answers are 2/3 and 4/10 correspondingly.

Edited by Prime
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I think the triangle inequality will rule out many of the triangles mentioned

What's triangle inequality?

If "triangle inequality" means that one side cannot exceed the sum of the other two, then we didn't forget that. Both 4/7 and 10/28 answers take that into account.

All triangles I see are these:

(0,6,6) => 3 variations for side assignment

(1,5,6) => 6 variations

(2,4,6) => 6

(2,5,5) => 3

(3,3,6) => 3

(3,4,5) => 6

(4,4,4) => 1

In all 7 different triangles. Total of 28 variatios with random side assignment.

If you pick triangles at random, the probability is 4/7. If you pick sides at random -- 10/28.

If you harbor a prejudice against triangles with one or two angles equal to zero, then the answers are 2/3 and 4/10 correspondingly.

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. Note: This rule must be satisfied for all 3 conditions of the sides.

In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a
.

so triangles like 0,6,6 fail the triangle inequality theorem since 0+6 is not greater than 6

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I think the triangle inequality will rule out many of the triangles mentioned

What's triangle inequality?

If "triangle inequality" means that one side cannot exceed the sum of the other two, then we didn't forget that. Both 4/7 and 10/28 answers take that into account.

All triangles I see are these:

(0,6,6) => 3 variations for side assignment

(1,5,6) => 6 variations

(2,4,6) => 6

(2,5,5) => 3

(3,3,6) => 3

(3,4,5) => 6

(4,4,4) => 1

In all 7 different triangles. Total of 28 variatios with random side assignment.

If you pick triangles at random, the probability is 4/7. If you pick sides at random -- 10/28.

If you harbor a prejudice against triangles with one or two angles equal to zero, then the answers are 2/3 and 4/10 correspondingly.

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. Note: This rule must be satisfied for all 3 conditions of the sides.

In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a triangle .

so triangles like 0,6,6 fail the triangle inequality theorem since 0+6 is not greater than 6

So, 0-angle triangles are being discriminated. That's an injustice.

Still, my last post gives two different answers even for that case.

Edited by Prime
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I think the triangle inequality will rule out many of the triangles mentioned

What's triangle inequality?

If "triangle inequality" means that one side cannot exceed the sum of the other two, then we didn't forget that. Both 4/7 and 10/28 answers take that into account.

All triangles I see are these:

(0,6,6) => 3 variations for side assignment

(1,5,6) => 6 variations

(2,4,6) => 6

(2,5,5) => 3

(3,3,6) => 3

(3,4,5) => 6

(4,4,4) => 1

In all 7 different triangles. Total of 28 variatios with random side assignment.

If you pick triangles at random, the probability is 4/7. If you pick sides at random -- 10/28.

If you harbor a prejudice against triangles with one or two angles equal to zero, then the answers are 2/3 and 4/10 correspondingly.

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. Note: This rule must be satisfied for all 3 conditions of the sides.

<

blockquote>

In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a triangle&

nbsp;.

so triangles like 0,6,6 fail the triangle inequality theorem since 0+6 is not greater than 6

So, 0-angle triangles are being discriminated agianst. That's an injustice.

Still, my last post gives two different answers even for that case.

2/3 or 4/10

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I think the triangle inequality will rule out many of the triangles mentioned

What's triangle inequality?

If "triangle inequality" means that one side cannot exceed the sum of the other two, then we didn't forget that. Both 4/7 and 10/28 answers take that into account.

All triangles I see are these:

(0,6,6) => 3 variations for side assignment

(1,5,6) => 6 variations

(2,4,6) => 6

(2,5,5) => 3

(3,3,6) => 3

(3,4,5) => 6

(4,4,4) => 1

In all 7 different triangles. Total of 28 variatios with random side assignment.

If you pick triangles at random, the probability is 4/7. If you pick sides at random -- 10/28.

If you harbor a prejudice against triangles with one or two angles equal to zero, then the answers are 2/3 and 4/10 correspondingly.

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side. Note: This rule must be satisfied for all 3 conditions of the sides.

<

blockquote>

In other words, as soon as you know that the sum of 2 sides is less than (or equal to ) the measure of a third side, then you know that the sides do not make up a triangle&

nbsp;.

so triangles like 0,6,6 fail the triangle inequality theorem since 0+6 is not greater than 6

So, 0-angle triangles are being discriminated agianst. That's an injustice.

Still, my last post gives two different answers even for that case.

2/3 or 4/10

That depends on how "probability" was meant in the OP. The way OP was worded ("probability of forming"), it looks more like 4/10. In this context, I assume, we form a triangle by randomly picking sides for it. If OP said: "What is the probability that the triangle is isosceles," then I would be inclined to answer: 2/3.

Edited by Prime
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I think the triangle inequality will rule out many of the triangles mentioned

What's triangle inequality?

If "triangle inequality" means that one side cannot exceed the sum of the other two, then we didn't forget that. Both 4/7 and 10/28 answers take that into account.

All triangles I see are these:

(0,6,6) => 3 variations for side assignment

(1,5,6) => 6 variations

(2,4,6) => 6

(2,5,5) => 3

(3,3,6) => 3

(3,4,5) => 6

(4,4,4) => 1

In all 7 different triangles. Total of 28 variatios with random side assignment.

If you pick triangles at random, the probability is 4/7. If you pick sides at random -- 10/28.

If you harbor a prejudice against triangles with one or two angles equal to zero, then the answers are 2/3 and 4/10 correspondingly.

2:3 iso.JPG

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