The problem in the image states:

• The sides of a triangle are in the ratio $\frac{1}{3} : \frac{1}{4} : \frac{1}{6}$, and the perimeter is given as 108 cm. We are asked to find the length of the largest side of the triangle.

Solution:

1. The ratio of the sides is given as $\frac{1}{3} : \frac{1}{4} : \frac{1}{6}$. To simplify, we find a common denominator for all the fractions.

   • $\frac{1}{3} = \frac{4}{12}$
   • $\frac{1}{4} = \frac{3}{12}$
   • $\frac{1}{6} = \frac{2}{12}$

   So, the simplified ratio of the sides becomes $4 : 3 : 2$.

2. Let the sides of the triangle be $4x$, $3x$, and $2x$, where $x$ is a common factor.

3. The perimeter of the triangle is the sum of its sides, which is given as 108 cm. So: $4x + 3x + 2x = 108$ Simplifying: $9x = 108$ $x = \frac{108}{9} = 12$

4. Now, the largest side is $4x$, so: $4x = 4 \times 12 = 48 \, \text{cm}$

Answer:

The length of the largest side of the triangle is 48 cm (Option D).

Would you like further details or have any questions?

5 Related Questions:

1. How can the perimeter of a triangle be used to find individual side lengths?
2. What are some other ways to express ratios of sides in triangles?
3. How would the solution change if the perimeter was different?
4. Can you apply this method to non-right triangles, and why?
5. How does the triangle inequality apply to the sides in this problem?

Tip:

When given a ratio of sides in fractions, always look for a common denominator to simplify the calculations and work with whole numbers.