Understanding the Quadrilateral Problem

This problem involves a quadrilateral, which is a four-sided polygon. We are given the ratio of the lengths of its sides (often called arms in this context) and its total perimeter. Our goal is to find the length of the shortest side.

Setting up the Ratio and Sides

The ratio of the arms of the quadrilateral is given as 2 ∶ 3 ∶ 4 ∶ 5.

We can represent the lengths of the four arms using a common multiplier, let's call it \(x\). So, the lengths of the arms are:

• Arm 1: \(2x\)
• Arm 2: \(3x\)
• Arm 3: \(4x\)
• Arm 4: \(5x\)

The smallest arm corresponds to the smallest ratio part, which is \(2x\).

Using the Perimeter to Find the Multiplier

The perimeter of any polygon is the sum of the lengths of all its sides. For this quadrilateral, the perimeter is the sum of the four arm lengths.

Perimeter = Arm 1 + Arm 2 + Arm 3 + Arm 4

We are given that the perimeter is 616 cm.

So, we can write the equation:

\(2x + 3x + 4x + 5x = 616\)

Solving for the Multiplier \(x\)

Now, let's solve the equation for \(x\):

Combine the terms on the left side:

\((2 + 3 + 4 + 5)x = 616\)

\(14x = 616\)

To find \(x\), divide both sides by 14:

\(x = \frac{616}{14}\)

Let's perform the division:

So, the value of \(x\) is 44.

Finding the Length of the Smallest Arm

The lengths of the arms are \(2x\), \(3x\), \(4x\), and \(5x\). The smallest arm is \(2x\).

Substitute the value of \(x = 44\) into the expression for the smallest arm:

Smallest arm = \(2x = 2 \times 44\)

Calculate the product:

\(2 \times 44 = 88\)

So, the length of the smallest arm is 88 cm.

Verification (Optional but Recommended)

Let's find the lengths of all arms and check if their sum equals the perimeter:

• Arm 1: \(2x = 2 \times 44 = 88\) cm
• Arm 2: \(3x = 3 \times 44 = 132\) cm
• Arm 3: \(4x = 4 \times 44 = 176\) cm
• Arm 4: \(5x = 5 \times 44 = 220\) cm

Sum of arms = \(88 + 132 + 176 + 220\)

Sum = \(220 + 176 + 132 + 88\)

Sum = \(396 + 132 + 88\)

Sum = \(528 + 88\)

Sum = \(616\) cm

The sum matches the given perimeter (616 cm), confirming our calculation for \(x\) and the arm lengths is correct.

Conclusion

The smallest arm of the quadrilateral has a length of 88 cm.

Revision Table: Quadrilateral Arms Ratio

Additional Information: Working with Ratios in Geometry

When a problem gives the ratio of side lengths of a shape, it means the sides are proportional. Using a multiplier (\(x\)) is a standard way to represent the actual lengths based on the ratio. For example, if a ratio is 2:3, the lengths are \(2x\) and \(3x\).

The multiplier \(x\) is a constant value that scales the ratio up or down to the actual size of the shape. To find \(x\), you usually use another piece of information given in the problem, such as the perimeter or area.

In this problem, the perimeter is the sum of the actual lengths. By setting the sum of the represented lengths (\(2x+3x+4x+5x\)) equal to the given perimeter (616 cm), we create an equation that allows us to solve for \(x\). Once \(x\) is known, the actual length of any side can be calculated by multiplying the corresponding ratio number by \(x\).

Understanding ratios is crucial for solving many geometry problems involving proportional dimensions.