The Sides Of A Rectangle Are In The Ratio Of 5:6. If Its Longer Side Is 7.2 In., Find The Shorter Side, is a classic problem in geometry that combines understanding ratios, proportions, and basic algebra. Such problems are common in math exams and serve as excellent exercises for developing problem-solving skills. In this article, we will explore how to approach this problem systematically, understand the concepts behind ratios, and apply algebraic methods to find the missing measurement.
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Understanding Ratios and Their Significance in Geometry
Before delving into the specific problem, it is essential to understand what ratios are and how they are used in geometry.
What Is a Ratio?
A ratio is a comparison of two quantities expressing how many times one value contains or is contained within the other. It is generally written in the form of a:b, where a and b are numbers representing the quantities being compared.Example:
If the sides of a rectangle are in the ratio 5:6, it means for every 5 units of one side, the other side measures 6 units.
Ratios in Geometric Figures
Ratios help describe the proportions of geometric figures, such as rectangles, triangles, and circles. Understanding ratios allows us to determine unknown lengths when the proportional relationship is known.---
Problem Breakdown and Approach
Let's restate the problem for clarity:
> The sides of a rectangle are in the ratio 5:6. If its longer side measures 7.2 inches, find the length of the shorter side.
Step-by-step approach:
- Identify the known and unknown quantities.
- Set up the ratio as a proportion.
- Create an algebraic equation based on the ratio and the known measurement.
- Solve for the unknown side length.
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Step 1: Assign Variables to the Sides
Let the shorter side be represented by \( x \).
Since the ratio of the sides is 5:6, the longer side can be expressed as:
\[ \text{Longer side} = \frac{6}{5} \times x \]
Alternatively, if we denote:
- Shorter side = \( 5k \)
- Longer side = \( 6k \)
where \( k \) is a common multiplier.
Given that the longer side is 7.2 inches, we set:
\[ 6k = 7.2 \]
and solve for \( k \).
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Step 2: Setting Up the Equation
Using the ratio, the shorter side:
\[ \text{Shorter side} = 5k \]
and from the given longer side:
\[ 6k = 7.2 \]
Solve for \( k \):
\[ k = \frac{7.2}{6} \]
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Step 3: Calculating the Shorter Side
Perform the division:
\[ k = \frac{7.2}{6} = 1.2 \]
Now, find the shorter side:
\[ \text{Shorter side} = 5k = 5 \times 1.2 = 6 \text{ inches} \]
Thus, the shorter side measures 6 inches.
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Summary of the Solution
- The ratio of sides is 5:6.
- The longer side (6 parts) measures 7.2 inches.
- Find the value of one part, \( k \), by dividing 7.2 by 6.
- Multiply \( k \) by 5 to find the shorter side.
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Additional Concepts and Variations
Understanding this problem opens up avenues to solve similar questions involving ratios and proportions in various geometric figures.
1. Ratios in Different Shapes
Ratios are not limited to rectangles. They are widely used in:- Triangles: ratios of sides or angles.
- Circles: ratios of circumference to diameter (pi).
- Similar figures: ratios of corresponding sides.
2. Proportional Reasoning
Proportional reasoning involves setting up equations where two ratios are equal, enabling the solving of unknown quantities.Example:
If in a triangle, sides are proportional to 3:4:5, and one side is known, the others can be calculated using ratios.
3. Applying Ratios in Real-Life Contexts
Ratios are used in:- Scale drawings and maps.
- Recipes, to maintain ingredient proportions.
- Construction, to ensure proper dimensions.
Practice Problems for Mastery
To reinforce your understanding, try solving these similar problems:- In a rectangle, the ratio of length to width is 3:4. If the width is 8 inches, what is the length?
- A triangle has sides in the ratio 7:24:25. If the shortest side is 7 inches, what are the lengths of the other sides?
- The ratio of the circumference to diameter of a circle is pi. If the diameter measures 10 meters, what is the circumference?
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Conclusion
Understanding ratios and their application in geometry allows for quick and effective problem solving. In the case of the rectangle with sides in the ratio 5:6, knowing one side's measurement enables us to find the other using simple algebra. Mastery of such concepts enhances mathematical reasoning and prepares students for more complex geometric problems.Whether you're preparing for exams or solving real-world measurement challenges, grasping the concept of ratios is fundamental. Remember, always identify the ratio, assign variables, set up proportions, and solve systematically.
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In summary:
- The sides of the rectangle are in the ratio 5:6.
- The longer side (6 parts) is 7.2 inches.
- Calculated the value of one part, \( k \), as 1.2 inches.
- Found the shorter side (5 parts) as 6 inches.
This approach exemplifies the power of ratios and algebra in solving geometric problems efficiently and accurately.
