Understanding the Triangle Sides Ratio Problem

The problem provides the ratio of the sides of a triangle as fractions and the semi-perimeter. We need to find the length of the longest side of the triangle. First, we need to convert the fractional ratio into a simple ratio of whole numbers. Then, using the semi-perimeter, we can determine the actual lengths of the sides and identify the longest one.

Converting Fractional Ratio to Whole Number Ratio

The given ratio of the sides is \( \frac{1}{3} : \frac{1}{4} : \frac{1}{5} \). To convert this into a ratio of whole numbers, we find the least common multiple (LCM) of the denominators, which are 3, 4, and 5.

The LCM of 3, 4, and 5 is 60.

Now, multiply each part of the ratio by the LCM (60):

• \( \frac{1}{3} \times 60 = \frac{60}{3} = 20 \)
• \( \frac{1}{4} \times 60 = \frac{60}{4} = 15 \)
• \( \frac{1}{5} \times 60 = \frac{60}{5} = 12 \)

So, the ratio of the sides of the triangle is \( 20 : 15 : 12 \).

Representing the Sides and Using Semi-perimeter

Let the sides of the triangle be \( 20x \), \( 15x \), and \( 12x \), where \( x \) is a common multiplier.

The perimeter of the triangle is the sum of its sides:

Perimeter \( = 20x + 15x + 12x = (20 + 15 + 12)x = 47x \) cm.

The semi-perimeter (s) is half of the perimeter.

Semi-perimeter \( s = \frac{\text{Perimeter}}{2} = \frac{47x}{2} \) cm.

We are given that the semi-perimeter of the triangle is 47 cm. So, we can set up the equation:

\( \frac{47x}{2} = 47 \)

Solving for the Multiplier 'x'

To find the value of \( x \), we solve the equation \( \frac{47x}{2} = 47 \).

Multiply both sides by 2:

\( 47x = 47 \times 2 \)

\( 47x = 94 \)

Divide both sides by 47:

\( x = \frac{94}{47} \)

\( x = 2 \)

The common multiplier \( x \) is 2.

Calculating the Lengths of the Sides

Now that we have the value of \( x \), we can find the actual lengths of the sides of the triangle:

• Side 1 \( = 20x = 20 \times 2 = 40 \) cm
• Side 2 \( = 15x = 15 \times 2 = 30 \) cm
• Side 3 \( = 12x = 12 \times 2 = 24 \) cm

The lengths of the sides of the triangle are 40 cm, 30 cm, and 24 cm.

Identifying the Longest Side

Comparing the lengths of the three sides (40 cm, 30 cm, and 24 cm), the longest side is 40 cm.

Revision Table: Triangle Sides and Semi-perimeter Concepts

Additional Information: Triangle Properties and Calculations

Understanding ratios and how to use the semi-perimeter are key skills for solving geometry problems involving triangles. Here are a few related points:

• Types of Triangles by Side: Triangles can be classified based on their side lengths: Equilateral (all sides equal), Isosceles (two sides equal), and Scalene (all sides different). The triangle in this problem is a scalene triangle as its sides (40 cm, 30 cm, 24 cm) are all different.
• Area of a Triangle using Semi-perimeter: The semi-perimeter is often used in Heron's formula to calculate the area of a triangle when only the side lengths are known: Area \( = \sqrt{s(s-a)(s-b)(s-c)} \), where a, b, and c are the side lengths and s is the semi-perimeter.
• Ratios in Geometry: Ratios are fundamental in geometry for comparing lengths, areas, and volumes, and for working with similar figures. Converting fractional ratios to whole number ratios, as done in this problem, is a common technique.

By using the ratio of the sides and the given semi-perimeter, we successfully calculated the actual side lengths and identified the longest side.