To find the length of the smallest side of a triangle with sides that are consecutive even integers and a perimeter of 54, let me guide you step-by-step through the solution:
1. Define the sides of the triangle:
Let's denote the consecutive even integers as [tex]\( x \)[/tex], [tex]\( x+2 \)[/tex], and [tex]\( x+4 \)[/tex].
2. Set up the equation for the perimeter:
The perimeter of a triangle is the sum of its three sides. So, we have the following equation for the perimeter:
[tex]\[
x + (x + 2) + (x + 4) = 54
\][/tex]
3. Combine like terms:
Simplify the equation by adding the [tex]\( x \)[/tex] terms together and the constant terms together:
[tex]\[
3x + 6 = 54
\][/tex]
4. Isolate the variable [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], first subtract 6 from both sides of the equation:
[tex]\[
3x = 48
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = 16
\][/tex]
6. Identify the smallest side:
Since [tex]\( x \)[/tex] represents the smallest side of the triangle, and we've determined [tex]\( x = 16 \)[/tex], the smallest side is 16.
Therefore, the length of the smallest side of the triangle is:
(d) 16
