Guest Posted October 21, 2011 Report Share Posted October 21, 2011 (edited) A triangle has integer legs where one leg is equal to x and another that is three times as long. The other leg is 15. What is the maximum perimeter that the triangle can have? Edited October 21, 2011 by Fizzicks Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 70 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 (edited) Case 1: three sides are x, 3x, and 15. So, x+3x>15 and 15+x>3x which means 3.75<x<7.5 and for maximum integer value, x = 7. So perimeter = 7+21+15 = 43. Case 2: 15 is the other side having value 3x. so x=5. third side can be maximum 19. Perimeter = 5+15+19 = 39. Case 3: 15 is the value of x. Other side is 45. third side can be maximum as 59. Perimeter = 15+45+59 = 119. Wordings in the question was not clear to me. The phrase "other side" looks a bit confusing. Assuming, 15 is the third side actually, maximum perimeter would be 43. Edited October 21, 2011 by swapnil Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 (edited) I get 43 We know the three legs are x, 3x and 15 where x is an integer. For any triangle, the sum of the lengths of any two legs must be greater than the length of the third leg. Therefore 15 + x > 3x => 15 > 2x => 7.5 > x. So x is less than 7.5 and to maximize perimeter it should be the greatest integer less than 7.5. x is 7, 3x is 21 and the 3rd leg is 15 for a perimeter of 43. edit - SWAPNIL!! Edited October 21, 2011 by maurice Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 .For integer value 43 otherwise 44.99........ Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 Just a fraction below 45. Since the longest side of any triangle MUST be less than the sum of the remaining two and supposing 3x is the longest side: 3x < x + 15 this will give you the set of possible values for x. Maximizing this value is just below 7.5. Hence, the perimeter is: less_than(7.5)+15+less_than(3*7.5)=a_little_bit_less_than(45) 14.swapnil.14 is correct! Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 Just a fraction below 45. Since the longest side of any triangle MUST be less than the sum of the remaining two and supposing 3x is the longest side: 3x < x + 15 this will give you the set of possible values for x. Maximizing this value is just below 7.5. Hence, the perimeter is: less_than(7.5)+15+less_than(3*7.5)=a_little_bit_less_than(45) 14.swapnil.14 is correct! Sides are integer lengths... Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 (edited) Sides are integer lengths... Damn true! I missed that point... well, a few adjustments are required: Since the result of the inequality is x < 7.5, the greatest integer that satisfiess such solution is x = 7 Therefore, the perimeter is: 7 + 21 + 15 = 43 Nevertheless, is is a specific case of my previous answer. Anyway, I think the Space Cowboy got it first! Edited October 21, 2011 by jagdmc Quote Link to comment Share on other sites More sharing options...
0 Guest Posted October 21, 2011 Report Share Posted October 21, 2011 Damn true! I missed that point... well, a few adjustments are required: Since the result of the inequality is x < 7.5, the greatest integer that satisfiess such solution is x = 7 Therefore, the perimeter is: 7 + 21 + 15 = 43 Nevertheless, is is a specific case of my previous answer. Anyway, I think the Space Cowboy got it first! I wish...the other swapnil (what's up with that btw) just beat me to it. Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted October 22, 2011 Report Share Posted October 22, 2011 If you don't mind the word "degenerate" you get a larger result. Quote Link to comment Share on other sites More sharing options...
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A triangle has integer legs where one leg is equal to x and another that is three times as long. The other leg is 15. What is the maximum perimeter that the triangle can have?
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