Finding the Smallest Side of a Quadrilateral Using Ratio and Perimeter

This problem involves finding the length of the smallest side of a quadrilateral when the ratio of its sides and its total perimeter are given. We can use the given ratio to represent the side lengths with a variable, and then use the perimeter information to solve for that variable.

Understanding Quadrilateral Side Ratios and Perimeter

A quadrilateral is a four-sided polygon. The ratio of sides tells us the proportional relationship between the lengths of the four sides. The perimeter of any polygon is the total distance around it, which is the sum of the lengths of all its sides.

• The ratio of sides is given as 2 ∶ 3 ∶ 4 ∶ 5.
• The perimeter is given as 560 cm.

Setting up the Equation

Let the common ratio factor be 'x'. Then the lengths of the four sides of the quadrilateral can be represented as:

• Side 1: 2x
• Side 2: 3x
• Side 3: 4x
• Side 4: 5x

The perimeter is the sum of these side lengths. So, we can write the equation:

\( \text{Perimeter} = \text{Side 1} + \text{Side 2} + \text{Side 3} + \text{Side 4} \)

\( 560 = 2x + 3x + 4x + 5x \)

Solving for the Common Ratio Factor (x)

Combine the terms on the right side of the equation:

\( 560 = (2 + 3 + 4 + 5)x \)

\( 560 = 14x \)

Now, isolate 'x' by dividing both sides by 14:

\( x = \frac{560}{14} \)

Let's perform the division:

\( x = 40 \)

So, the common ratio factor 'x' is 40.

Calculating the Side Lengths

Now that we know the value of x, we can find the length of each side:

• Side 1 = \( 2x = 2 \times 40 = 80 \) cm
• Side 2 = \( 3x = 3 \times 40 = 120 \) cm
• Side 3 = \( 4x = 4 \times 40 = 160 \) cm
• Side 4 = \( 5x = 5 \times 40 = 200 \) cm

Identifying the Smallest Side

Comparing the lengths of the four sides (80 cm, 120 cm, 160 cm, and 200 cm), the smallest side is the one corresponding to the smallest part of the ratio, which is 2.

The smallest side length is 80 cm.

Verification

Let's check if the sum of these sides equals the given perimeter:

\( 80 + 120 + 160 + 200 = 560 \)

The sum is indeed 560 cm, which matches the given perimeter. This confirms our calculations are correct.

Final Answer on Smallest Side

Based on the calculations, the smallest side of the quadrilateral is 80 cm.

Revision Table: Key Concepts for Quadrilateral Problems

Additional Information on Quadrilaterals

Quadrilaterals come in many forms, each with specific properties. Understanding these properties can be helpful in geometry problems.

• Types of Quadrilaterals: Some common types include squares, rectangles, parallelograms, rhombuses, trapezoids, and kites. Each has specific properties regarding side lengths, angles, and diagonals.
• Sum of Interior Angles: The sum of the interior angles of any quadrilateral is always 360 degrees.
• Applications: Quadrilaterals are fundamental shapes in geometry and are used in various applications, from architecture and design to physics and engineering.

Problems involving ratios and perimeters are common ways to test understanding of basic geometric concepts and algebraic problem-solving skills.