# The perimeter of a triangle is 29 mm. The length of the first side is twice the length of the second side. The length of the third side is 5 more than the length of the second side. How do you find the side lengths of the triangle?

Feb 29, 2016

${s}_{1} = 12$
${s}_{2} = 6$
${s}_{3} = 11$

#### Explanation:

The perimeter of a triangle is the sum of the lengths of all its sides. In this case, it is given that the perimeter is 29mm. So for this case:

${s}_{1} + {s}_{2} + {s}_{3} = 29$

So solving for the length of the sides, we translate statements in the given into equation form.

"The length of the 1st side is twice the length of the 2nd side"

In order to solve this, we assign a random variable to either ${s}_{1}$ or ${s}_{2}$. For this example, I would let $x$ be the length of the 2nd side to avoid having fractions in my equation.

so we know that:
${s}_{1} = 2 {s}_{2}$

but since we let ${s}_{2}$ be $x$, we now know that:
${s}_{1} = 2 x$
${s}_{2} = x$

"The length of the 3rd Side is 5 more than the length of the 2nd Side."

Translating the statement above to equation form...
${s}_{3} = {s}_{2} + 5$

once again since we let ${s}_{2} = x$
${s}_{3} = x + 5$

Knowing the values (in terms of $x$) of each side, we would now be able to compute for $x$ and ultimately compute for the length of each side.

[Solution]
${s}_{1} = 2 x$
${s}_{2} = x$
${s}_{3} = {s}_{2} + 5$

${s}_{1} + {s}_{2} + {s}_{3} = 29$

$2 x + x + x + 5 = 29$
$4 x + 5 = 29$
$4 x = 29 - 5$
$4 x = 24$
$x = \frac{24}{4}$
$x = 6$

Using the computed value of $x$, we would be able to compute for the values of ${s}_{1}$, ${s}_{2}$, and ${s}_{3}$

${s}_{1} = 2 x$
${s}_{1} = 2 \left(6\right)$
${s}_{1} = 12$

${s}_{2} = x$
${s}_{2} = 6$

${s}_{3} = x + 5$
${s}_{3} = 6 + 5$
${s}_{3} = 11$

[Checking]
${s}_{1} + {s}_{2} + {s}_{3} = 29$
$12 + 6 + 11 = 29$
$29 = 29$