The sides of the triangle are in the ratio 9:12:18. The longest side of the triangle measures 12 cm. What are the lengths of the other sides?

Hint: If the ratio of the sides of any figure is $a:b:c$ , then taking any common factor say, $x$, we may take the sides to be $ax,{\text{ }}bx{\text{ and }}cx$. Then, to find the value of this common factor $x$, use the given conditions. Once you have calculated the value of $x$, rest of the values can be easily found.

Complete step-by-step solution
Let us consider the ratio of the sides given in the question,
9:12:18
Let the common factor between the sides of the triangle be $x$.
Then we get the sides of the triangle as: $9x,12x,{\text{ and }}18x$. You must also observe that among these sides of the triangle, $9x$ is the smallest side, and $18x$ is the largest one.

It is also given in the question that the dimension of the longest side is 12 cm. Because the largest side is $18x$, we get;
$
  18x = 12 \\
   \Rightarrow x = \dfrac{{12}}{{18}} \\
   \Rightarrow x = \dfrac{2}{3} \\
 $

Now we have obtained the value of $x$. Substituting this value into the sides $9x,12x,{\text{ and }}18x$, we can easily find their lengths.

Let us make the substitution to find the lengths now.

The smallest side is given by $9x$, find its length.

$ 9x = 9\left( {\dfrac{2}{3}} \right) \\
   \Rightarrow 9x = 6{\text{ cm}} \\
$

Thus, we get the length of the smallest side as 6 cm.

The middle side is given by $12x$, find its length.

$
  12x = 12\left( {\dfrac{2}{3}} \right) \\
   \Rightarrow 12x = 8{\text{ cm}} \\
 $

Thus, we get the length of the middle side as 8 cm.

The longest side is given by $18x$, find its length.

$
  18x = 18\left( {\dfrac{2}{3}} \right) \\
   \Rightarrow 18x = 12{\text{ cm}} \\
$

Thus, we get the length of the longest side as 12 cm, which verifies the given condition.

Thus, the lengths of the other two sides of the triangle with the longest side as 12 cm, are 6 cm and 8 cm.

Note: While determining the sides of the triangle, remember that you will be taking only one number to multiply each number in the given ratio. If more than one number is taken, say x, y and z, then when the numbers given in the ratio are multiplied by x, y and z respectively, the ratio of the sides obtained after this multiplication, might not be the ratio that was provided to you in the question.