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.

To solve this equation let's start with a few basic concepts.

**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.**