Question 618657
Hi, there--
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Recall that a parallelogram has two pairs of parallel sides. We will use two variables to solve this problem.
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Let x be the length of each of the shorter pair of parallel sides.
Let y be the length of each of the longer pair of parallel sides.
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We will use the information given in the problem to write two equations that model the problem. The perimeter (48 cm.) is the sum of the four sides, so
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{{{48=x+x+y+y}}}
{{{2x+2y=48}}}
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The ratio of two sides of the parallelogram is 3:5. Since the parallel sides of a parallelogram have equal length, we know that we are comparing the lengths x and y here, so
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{{{x/y=3/5}}}
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NOTE: We set up the variables so that x is shorter than y. So "x is to y as 3 is to 5."
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Now we have two variables and two equations. We can solve the system to find the values of x and y.
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Rewrite the first equation in terms of x.
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{{{2x+2y=48}}}
{{{2y=48-2x}}}
{{{y=24-x}}}
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Substitute 24-x for y in the second equation.
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{{{x/y=3/5}}}
{{{x/(24-x)=3/5}}}
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Solve for x. I will skip a bunch of algebra steps and cross-multiply.
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{{{5x=3(24-x)}}}
{{{5x=72-3x}}}
{{{8x=72}}}
{{{x=9}}}
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We now know that the shorter sides have a length of 9 cm. Substitute 9 for x in the first equation.
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{{{2x+2y=48}}}
{{{2(9)+2y=48}}}
{{{18+2y=48}}}
{{{2y=48-18=30}}}
{{{y=15}}}
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According to this equation, the length of the longer side is 15 cm.
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Now we side to check our work. The perimeter is 48 cm.
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{{{9+9+15+15=48}}}
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The perimeter checks out. Now we need to check that the ratio of the two sides reduces to 3:5. The ratio 9:15 is equivalent to 3:5 because we can divide both parts by their common factor,3. That is, 9/3=3 sand 15/3=5.
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Check! So the lengths of the sides of the parallelogram are 9 cm and 15 cm.
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Hope this helps. Feel free to email me if you have questions about this.
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Ms.Figgy
math.in.the.vortex@gmail.com