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The perimeter of a triangle is 47 miles. The first side is 5 miles less than twice the second and the third is 2 miles more than the first.

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3 Answers

This word problem is about a triangle whose perimeter is 47 miles.  Since it's asking to find the length of each side, then there are 3 unknowns.  So here's what we do:

Let the variables S1 be the length of side 1 in miles,

S2 be the length of side 2 in miles,

and S3 be the length of side 3 in miles.

 

Now, the next thing to do with these variables is to set up equations that follow the word problem.  Here they are as follows:

S1+S2+S3=47         the perimeter of the triangle by adding the lengths of all three sides

S1=(2*S2)-5             1st side (represented by S1) is (represented by =) 5 less (represented by -5) than twice the 2nd side (represented by 2*S2).

S3=S1+2                  3rd side (represented by S3) is 2 more (represented by +2) than the 1st side. 

 

Next, since we have more than one unknown, we should express one equation in terms of one variable.  Since only one variable has a coefficient which is S2 in the term, 2*S2, I suggest that we express the other variable equations of S1 and S3 in terms of S2.  Here's how it goes:

S1=(2*S2)-5           as S1 is already expressed in terms of S2

S3=S1+2=[(2*S2)-5]+2        I plugged in the S1 equation to express S3 in terms of S2.

               =(2*S2)-5+2          I combined like terms.

               =(2*S2)-3.

Now that I have expressed both S1 and S3 in terms of S2, I can plug those equations into that for the triangle perimeter:

[(2*S2)-5]+S2+[(2*S2)-3]=47

(2*S2)-5+S2+(2*S2)-3=47             Again, I combined like terms.

(5*S2)-8=47

          +8=+8                        I added 8 on both sides of the equation to eliminate -8 on the left side.

5*S2=55

5*S2/5=55/5

S2=11

S1=(2*11)-5=22-5=17      Now that I have found the value of S2, I can plug it into the S1 equation.

S3=17+2=19                    Now that I have found the value of S1, I can plug it into the S3 equation.

So now here are the lengths of each side of the triangle.  Side 1 is 17 miles, side 2 is 11 miles, and side 3 is 19 miles.

 

Recall that the perimeter of a triangle is equal to the sum of the measures of its three sides. That is,

     Perimeter = measure of side 1 + measure of side 2 + measure of side 3

The problem states that the measure of side 1 is equal to 5 miles less than twice the measure of side 2 and the measure of side 3 is 2 miles more than the measure of side 3. Since the measure of side 3 is dependent on the measure of side 1 and the measure of side 1 is dependent on the measure of side 2, then both the measures of sides 1 and side 3 are dependent on the measure of side 2. With this, we can assign a constant variable to the measure of side 2 and determine the measures of the other sides using this variable.

     measure of side 2 = x

     measure of side 1 = 2(x) - 5 = 2x - 5

     measure of side 3 = (2x - 5) + 2 = 2x - 3

We are given that the perimeter of the triangle in question is 47 miles. Using this and the equations we've found for the measures of each side, we can solve for x and use this value to determine the measure of each side.

     Perimeter = measure of side 1 + measure of side 2 + measure of side 3

               47 = (2x - 5) + (x) + (2x - 3)

After combining like terms, we arrive at the following:

          47 = 5x - 8

Add 8 to both sides of the equation then divide both sides by 5 to solve for x:

         47 + 8 = 5x - 8 + 8

         55 = 5x

         55/5 = 5x/5

         11 = x

Given that  x = 11 , the measure of each side is as follows:

     measure of side 1:   2x - 5 = 2·11 - 5 = 22 - 5 = 17 miles

     measure of side 2:   x = 11 miles

     measure of side 3:   2x - 3 = 2·11 - 3 = 22 - 3 = 19 miles

First we know that the perimeter of a triangle is found by the addition of it's sides. So, 47 = to all the sides of the triangle added together. 

The information given is wordy and may appear tricky, so let's break it down.

Draw a triangle and assign the value of x to the 2nd side.

We are told that the first side is 5 miles less than 2x the second. In variables that would mean:

Side 1 = 2x -5 (note that x = to the value of your second side) 

The next bit of information says that

Side 3 = 2 + (2x - 5) (where 2x - 5 = side 1)

Now we have to solve: 47 = [2 + 2x - 5] + [2x - 5] + [x] (I put brackets to signify that each part of this equation represents sides i.e. 47 = [side 3] + [side 1] + [side 2])

Then it's as easy as solving:

47 = 5x - 8

Solve for x by adding 8 = 8 + 47 = 55 and divide by 5 so that x = 11.

But you aren't done yet! Remember the question asks for each side, so we must plug our value of x = 11 into each side equation

Side 1 = 2x - 5 = 2(11) - 5 = 17

Side 2 = x = 11

Side 3 = 2 + (2*(11) - 5) = 19