Posted March 23, 2013 Using integer side lengths, what is the probability of forming an isosceles triangle if the perimeter is 12? 0 Share this post Link to post Share on other sites
0 Posted March 23, 2013 4/7, counting funny cases of (3,3,6) and (6,6,0) 0 Share this post Link to post Share on other sites
0 Posted March 24, 2013 I agree with Prime provided the OP is worded, What fraction of integer-sided triangles whose perimeter is 12 are isosceles? 0 Share this post Link to post Share on other sites
0 Posted March 24, 2013 1110 129 138 147 156 228 237 246255 345444 : 2 vs 11 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 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. Then the answer is: 10/28. Again that takes into account all triangles appearing as a segment: (0,6,6); (1,5,6); (2,4,6); and (3,3,6). 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 I think the triangle inequality will rule out many of the triangles mentioned 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 (edited) 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 March 25, 2013 by Prime 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 (edited) 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 Edited March 25, 2013 by BMAD 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 (edited) 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 March 25, 2013 by Prime 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 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 . 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 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 (edited) 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 March 25, 2013 by Prime 0 Share this post Link to post Share on other sites
0 Posted March 25, 2013 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. 2:3iso.JPG 0 Share this post Link to post Share on other sites
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Using integer side lengths, what is the probability of forming an isosceles triangle if the perimeter is 12?
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