Question 618648
Hi, there--
.
Recall that a parallelogram has two pairs of parallel sides. We will use two variables to solve this problem.
.
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.
.
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
.
{{{48=x+x+y+y}}}
{{{2x+2y=48}}}
.
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
.
{{{y/x=3.5}}}
.
NOTE: We set up the variable so that y is longer than x. The ratio is greater than 1 so we know that we are comparing the longer side to the shorter one.
.
Now we have two variables and two equations. We can solve the system to find the values of x and y.
.
Rewrite the first equation in terms of x.
.
{{{2x+2y=48}}}
{{{2y=48-2x}}}
{{{y=24-x}}}
.
Substitute 24-x for y in the second equation.
.
{{{y/x=3.5}}}
{{{(24-x)/x=3.5}}}
.
Solve for x.
.
{{{24-x=3.5x}}}
.
I'm not a fan of decimal coefficients, so I multiply every term by 2.
.
{{{48-2x=7x}}}
{{{48=9x}}}
{{{x=48/9=16/3}}}
.
We now know that the shorter sides have a length of 16/3 cm. Substitute 16/3 for x in the first equation.
.
{{{2x+2y=48}}}
{{{2(16/3)+2y=48}}}
{{{32/3+2y=48}}}
{{{2y=48-32/3}}}
{{{y=24-32/6=56/3}}}
.
According to this equation, the length of the longer side is 56/3 cm.
.
.
Now we side to check our work. The perimeter is 48 cm.
.
{{{16/3+16/3+56/3+56/3=144/3}}}
{{{144/3=48}}}
.
The perimeter checks out. Now we need to check that the ratio of the two sides reduces to 3.5.
.
{{{(56/3)/(16/3)=56/16}}}
{{{56/16=3.5}}}
.
Check! So the lengths of the sides of the parallelogram are 16/3 cm and 56/3 cm.
.
Hope this helps. Feel free to email me if you have questions about this.
.
Ms.Figgy
math.in.the.vortex@gmail.com