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## .what is the length of three sides of the triangle

the perimeter of a triangle is 43ft. The shortest side is one-third of the length of the middle side. The longest side is 3ft more than four times the shortest side. find the lengths of the three sides.

• A triangle has 3 sides.
• The perimeter of a triangle equals to the sum of the  length of the 3 sides.

We will call those 3 sides A,B,C

A = shortest side,   B= middle side,  C= longest side.

Next, we will determine what we already know.

• We know that  B  =  3A (3 times as long as side A)
• We know C = 4A +3 (3ft more than 4 times of side A)

Therefore P= A+B+C

We can substitute for our known variables

43= A +  3A  (value for B) + 4A +3 (value for C)

43 = 8A +3

We will isolate our unknown variable 8A to one side.

To isolate 8A we will subtract 3 from both sides.  (Remeber whatever operations we perform on one side, we must perform the same identical operation on the other side.)

43-3 = 8A +3 -3

40 = 8A

We will divide both sides of our equation  by 8 to solve for A.

40/8 = 8A/8

5 = A

The shortest side is 5 ft.

The shortest side is 1/3 as long as the middle side or (the midde side is 3 times as long as the shortest side.)

Therefore side B =  3A  or 3(5) = 15 ft.

The  longest side is  3 ft more than  4 times the shortest side.

Therefore side C=  4A+ 3  or 4(5)+3 =   20+3 = 23 ft.

The final step in solving this equation is to substitute the values into the formula.

P= A+B+C

43= 5+15 +23

43=43

The  shortest side of the triangle is 5 ft.

The middle side is 15 feet.

The longest side is 23 ft.

We need to find the length of each side of a triangle based on given ratio between the sides and the total length (perimeter) of the triangle given in ft.

We can define a triangle ABC with 3 sides a b c.

Let side a be the shortest side. (a = shortest side)

Let side b be the middle side. (b = middle side)

Let c be the longest side. (c = longest side)

We are given the following information:

a = b/3   or:  b = 3a

c = 4a + 3ft

The given total length (perimeter) of triangle ABC is given as Perimeter=P=43ft

Lets put it together:

P= 43ft = a + b + c

providing b in and c in terms of a (b =3a, c = 4a + 3ft) ) into the p equation gives:

P = 43ft = a + 3a + (4a+3f) = 4a + 4a+3f = 8a + 3ft

So:   8a = 43ft-3ft = 40ft

Solving for a:   a = 40ft/8 = 5ft

Solving for b:  b = 3a = 3*5ft = 15ft

Solving for c:  c = 4a + 3ft = 4*5ft + 3ft = 20ft + 3ft = 23 ft.

Verify the answer:  Perimeter = P = 43ft = a + b + c = 5ft + 15ft + 23ft = 43ft

We will begin by naming our variables

x = shortest side

y = medium side

z = longest side

Next, we will write three equations from the information that is given

x + y + z = 43        Equation 1

x = y/3                  Equation 2

z = 3 + 4x              Equation 3

Now that we have 3 equations with 3 variables, I am giong to use substitution method to solve the problem

x = y/3                      Equation 2

3x = y                       Multiply both sides by 3

x + y + z = 43            Equation 1

x + 3x + 3 + 4x = 43   Substitution (value of y from equation 2 and z from equation 3)

8x + 3 = 43                Combine like terms

8x = 40                      Subtract 3 from each side

x = 5                         Divide each side by 8

z = 3 + 4x                   Equation 3

z = 3 + 4(5)               Substitute the value of x

z = 3 +20                   Simplify

z = 23                        Simplify

x = y/3                      Equation 2

5 = y/3                     Substitute the value of x

15 = y                       Multiply each side by 3

The sides of the triangle are 5, 15, and 23.