document.write( "Question 955773: The perimeter of a triangle is 30 units.The length of the sides form an arithmetic sequence.if each length is a whole number,determine all possible sets of the lengths of the sides of the triangle. \n" ); document.write( "

Algebra.Com's Answer #583878 by MathLover1(20850) $\"\"$ $\"About$

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given:
\n" ); document.write( " $\"P=30\"$ units
\n" ); document.write( "The length of the sides form an arithmetic sequence.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " let $\"a\"$ be the shortest side therefore the first term
\n" ); document.write( "let $\"r\"$ be the common difference\r
\n" ); document.write( "\n" ); document.write( "the series is therefore:
\n" ); document.write( " $\"a\"$ , $\"a%2Br\"$ , $\"a%2B2r\"$ \r
\n" ); document.write( "\n" ); document.write( "then $\"P=a%2Ba%2Br%2Ba%2B2r\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"30=3a%2B3r\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"30=3%28a%2Br%29\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"30%2F3=a%2Br\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"10=a%2Br\"$ \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "all possible sets of the lengths of the sides of the triangle:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "if $\"a=9\"$ and $\"r=1\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=9\"$
\n" ); document.write( " $\"a%2Br=a%2B1=10\"$
\n" ); document.write( " $\"a%2B2r=a%2B2=11\"$ .........first solution: possible sides are $\"highlight%289%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2811%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=8\"$ and $\"r=2\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=8\"$
\n" ); document.write( " $\"a%2Br=8%2B2=10\"$
\n" ); document.write( " $\"a%2B2r=8%2B4=12\"$ .........2nd solution: possible sides are $\"highlight%288%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2812%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=7\"$ and $\"r=3\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=7\"$
\n" ); document.write( " $\"a%2Br=7%2B3=10\"$
\n" ); document.write( " $\"a%2B2r=7%2B6=13\"$ .........3rd solution: possible sides are $\"highlight%287%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2813%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=6\"$ and $\"r=4\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=6\"$
\n" ); document.write( " $\"a%2Br=6%2B4=10\"$
\n" ); document.write( " $\"a%2B2r=6%2B8=14\"$ .........4th solution: possible sides are $\"highlight%286%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2814%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=5\"$ and $\"r=5\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=5\"$
\n" ); document.write( " $\"a%2Br=5%2B5=10\"$
\n" ); document.write( " $\"a%2B2r=5%2B10=15\"$ .........5th solution: possible sides are $\"highlight%285%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2815%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=4\"$ and $\"r=6\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=4\"$
\n" ); document.write( " $\"a%2Br=4%2B6=10\"$
\n" ); document.write( " $\"a%2B2r=4%2B12=16\"$ .........6th solution: possible sides are $\"highlight%284%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2816%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=3\"$ and $\"r=7\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=3\"$
\n" ); document.write( " $\"a%2Br=3%2B7=10\"$
\n" ); document.write( " $\"a%2B2r=3%2B14=17\"$ .........7th solution: possible sides are $\"highlight%283%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2817%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=2\"$ and $\"r=8\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=2\"$
\n" ); document.write( " $\"a%2Br=2%2B8=10\"$
\n" ); document.write( " $\"a%2B2r=2%2B16=18\"$ .........8th solution: possible sides are $\"highlight%282%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2818%29\"$ \r
\n" ); document.write( "\n" ); document.write( "if $\"a=1\"$ and $\"r=9\"$ , then the lengths of the sides of the triangle are\r
\n" ); document.write( "\n" ); document.write( " $\"a=1\"$
\n" ); document.write( " $\"a%2Br=1%2B9=10\"$
\n" ); document.write( " $\"a%2B2r=1%2B18=19\"$ .........9th solution: possible sides are $\"highlight%281%29\"$ , $\"highlight%2810%29\"$ ,and $\"highlight%2819%29\"$ \r
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