The sides of a triangle are in the ratio of $\frac{1}{4}:\frac{1}{5}:\frac{1}{6}$. If the perimeter of the triangle is 37 cm, then find the length of the greatest side of the triangle?

15 cm

The correct answer is Option (2) → 15 cm

To find the length of the greatest side of the triangle, we first need to simplify the given ratio of the sides and then use the perimeter to find the actual lengths.

1. Simplify the Ratio

The given ratio of the sides is:

$\frac{1}{4} : \frac{1}{5} : \frac{1}{6}$

To convert this into a whole number ratio, find the Least Common Multiple (LCM) of the denominators (4, 5, and 6).

• LCM(4, 5, 6) = 60

Now, multiply each term of the ratio by 60:

• $\frac{1}{4} \times 60 = 15$
• $\frac{1}{5} \times 60 = 12$
• $\frac{1}{6} \times 60 = 10$

The simplified ratio of the sides is $15 : 12 : 10$.

2. Set up the Equation for Perimeter

Let the lengths of the three sides be $15x$, $12x$, and $10x$.

The perimeter is the sum of all sides:

$\text{Perimeter} = 15x + 12x + 10x = 37 \text{ cm}$

$37x = 37$

$x = 1$

3. Calculate the Side Lengths

Substitute $x = 1$ back into the expressions for the side lengths:

• Side 1 = $15(1) = 15$ cm
• Side 2 = $12(1) = 12$ cm
• Side 3 = $10(1) = 10$ cm

4. Identify the Greatest Side

Comparing the three lengths ($15$ cm, $12$ cm, and $10$ cm), the greatest side is 15 cm.

Final Answer:

The length of the greatest side of the triangle is 15 cm.