Question

A triangle has a perimeter of 351 meters. The lengths of the sides of the triangle are consecutive odd integers. What are the lengths of the sides?

Answer 1

See a solution process below:

Explanation:

The formula for the perimeter of a triangle is:

`p = a + b + c`

Where `a`, `b` and `c` are the three sides of the triangle.

Because we know `a`, `b` and `c` are three consecutive odd integers we can write our unknowns as:

`a = a`

`b = a + 2"m"`

`c = a + 4"m"`

We can no substitute these into our equation along with the perimeter which was giving in the problem and solve for `a`:

`p = a + b + c` becomes:

`351"m" = a + (a + 2"m") + (a + 4"m")`

`351"m" = a + a + 2"m" + a + 4"m"`

`351"m" = a + a + a + 2"m" + 4"m"`

`351"m" = 1a + 1a + 1a + 2"m" + 4"m"`

`351"m" = (1 + 1 + 1)a + (2"m" + 4"m")`

`351"m" = 3a + 6"m"`

`351"m" - color(red)(6"m") = 3a + 6"m" - color(red)(6"m")`

`345"m" = 3a + 0`

`345"m" = 3a`

`(345"m")/color(red)(3) = (3a)/color(red)(3)`

`115"m" = (color(red)(cancel(color(black)(3)))a)/cancel(color(red)(3))`

`115"m" = a`

`a = 115"m"`

We can then also find the lengths of sides `b` and `c` by substituting:

`b = 115"m" + 2"m" = 117"m"`

`c = 115"m" + 4"m" = 119"m"`

The lengths of the three sides of the triangle are:

`115"m" + 117"m" + 119"m" = 351"m"`

Answer 2

Let the lengths of three sides of the triangle are represented by three consecutive odd numbers as `(2 n+1);(2 n+3);(2n+5)` m.

By the problem it's perimeter is

`(2 n+1)+(2 n+3)+(2n+5)=351`

`=>6n+9=351`

`=>n=(351-9)/6=57`

So three sides are `(57*2+1)m,(59*2+1)mand(61*2+1)m`

Or,`115m,117mand119m`