Understanding the Quadrilateral Problem

The question gives us information about a quadrilateral. A quadrilateral is a polygon with four sides. We are given the ratio of the lengths of its sides and the total length of its perimeter. The perimeter of any polygon is the sum of the lengths of all its sides.

Setting up the Side Lengths using the Ratio

The sides of the quadrilateral are in the ratio 3 ∶ 4 ∶ 6 ∶ 8. This means that the actual lengths of the sides are proportional to these numbers. We can represent the lengths of the four sides using a common multiple, let's call it 'x'.

• Side 1 length = \(3x\)
• Side 2 length = \(4x\)
• Side 3 length = \(6x\)
• Side 4 length = \(8x\)

Here, 'x' is a positive value that represents the unit length for the ratio.

Calculating the Perimeter

The perimeter of the quadrilateral is the sum of these four side lengths. We are given that the perimeter is 84 cm.

Perimeter = Side 1 + Side 2 + Side 3 + Side 4

So, we can write the equation:

\(3x + 4x + 6x + 8x = 84\)

Solving for the Unit Length 'x'

Now, we need to solve this equation to find the value of 'x'. First, sum the terms with 'x':

\((3 + 4 + 6 + 8)x = 84\)

\(21x = 84\)

To find 'x', divide both sides of the equation by 21:

\(x = \frac{84}{21}\)

\(x = 4\)

So, the unit length 'x' is 4 cm.

Finding the Length of Each Side

Now that we know the value of 'x', we can calculate the actual length of each side of the quadrilateral:

• Side 1 length = \(3x = 3 \times 4 = 12\) cm
• Side 2 length = \(4x = 4 \times 4 = 16\) cm
• Side 3 length = \(6x = 6 \times 4 = 24\) cm
• Side 4 length = \(8x = 8 \times 4 = 32\) cm

Identifying the Longest Side

The question asks for the longest side of the quadrilateral. Comparing the lengths we calculated (12 cm, 16 cm, 24 cm, and 32 cm), the longest side is the largest value among these.

The lengths are 12 cm, 16 cm, 24 cm, and 32 cm.

The largest length is 32 cm.

Summary of Side Lengths

The longest side of the quadrilateral measures 32 cm.

Revision Table: Quadrilateral Side Ratio

Additional Information: Ratios and Proportions

Understanding ratios and proportions is key to solving problems like this. A ratio \(a : b : c : d\) means the quantities are in the proportion \(\frac{\text{quantity}_1}{a} = \frac{\text{quantity}_2}{b} = \frac{\text{quantity}_3}{c} = \frac{\text{quantity}_4}{d} = x\), where 'x' is the constant of proportionality. In this problem, the quantities are the side lengths, and their ratio is 3:4:6:8. By setting the sides as \(3x, 4x, 6x, 8x\), we are using this concept of a constant of proportionality 'x'. Once 'x' is found using the given perimeter, we can determine the exact value of each side length.