6-4 Practice Rectangles Answer Key With Work

6-4 Practice Rectangles: Answer Key with Detailed Work

Finding the perfect answer key for geometry problems can be challenging. This comprehensive guide provides detailed solutions and explanations for 6-4 practice problems focused on rectangles, ensuring a thorough understanding of the concepts. Whether you're a student looking for help with homework or a teacher seeking supplementary materials, this resource will empower you to master rectangle properties and calculations.

Understanding Rectangles: A Foundation for Problem-Solving

Before diving into the practice problems, let's solidify our understanding of rectangles. A rectangle is a quadrilateral (a four-sided polygon) with four right angles. This fundamental property leads to several crucial characteristics:

• Opposite sides are equal and parallel: This means the lengths of opposite sides are identical, and those sides never intersect, even if extended infinitely.
• Adjacent sides are perpendicular: This reinforces the right angle characteristic; where two sides meet, they form a 90-degree angle.
• Diagonals bisect each other: The diagonals of a rectangle (lines connecting opposite corners) intersect at their midpoints, creating four congruent triangles.

Key Formulas for Rectangle Calculations

Mastering the following formulas is essential for solving rectangle problems:

• Perimeter: P = 2(l + w), where 'l' is the length and 'w' is the width. The perimeter is the total distance around the rectangle.
• Area: A = l * w. The area represents the space enclosed within the rectangle.
• Diagonal: d = √(l² + w²) (using the Pythagorean theorem). This formula calculates the length of a diagonal.

6-4 Practice Rectangles: Problem Breakdown and Solutions

Let's tackle several example problems, providing step-by-step solutions and explanations. These problems will cover a range of difficulty levels, focusing on perimeter, area, and diagonal calculations.

Problem 1: Finding the Perimeter

A rectangle has a length of 12 cm and a width of 5 cm. Calculate its perimeter.

Solution:

1. Identify the knowns: l = 12 cm, w = 5 cm
2. Apply the perimeter formula: P = 2(l + w) = 2(12 cm + 5 cm) = 2(17 cm) = 34 cm

Answer: The perimeter of the rectangle is 34 cm.

Problem 2: Calculating the Area

A rectangular garden measures 8 meters in length and 6 meters in width. What is its area?

Solution:

1. Identify the knowns: l = 8 m, w = 6 m
2. Apply the area formula: A = l * w = 8 m * 6 m = 48 m²

Answer: The area of the garden is 48 square meters.

Problem 3: Determining the Diagonal Length

A rectangle has a length of 15 inches and a width of 8 inches. Find the length of its diagonal.

Solution:

1. Identify the knowns: l = 15 inches, w = 8 inches
2. Apply the diagonal formula: d = √(l² + w²) = √(15² + 8²) = √(225 + 64) = √289 = 17 inches

Answer: The length of the diagonal is 17 inches.

Problem 4: Finding the Width given the Area and Length

A rectangle has an area of 72 square feet and a length of 9 feet. What is its width?

Solution:

1. Identify the knowns: A = 72 ft², l = 9 ft
2. Rearrange the area formula to solve for width: w = A / l = 72 ft² / 9 ft = 8 ft

Answer: The width of the rectangle is 8 feet.

Problem 5: Finding the Length given the Perimeter and Width

The perimeter of a rectangle is 46 meters, and its width is 8 meters. What is its length?

Solution:

1. Identify the knowns: P = 46 m, w = 8 m
2. Rearrange the perimeter formula to solve for length: P = 2(l + w) => 46 m = 2(l + 8 m) => 23 m = l + 8 m => l = 23 m - 8 m = 15 m

Answer: The length of the rectangle is 15 meters.

Problem 6: Word Problem involving Area and Perimeter

A farmer wants to fence a rectangular field. The field has an area of 100 square yards and a perimeter of 40 yards. What are the dimensions (length and width) of the field?

Solution: This problem requires a bit more algebraic manipulation.

1. Use the area formula: A = l * w = 100 square yards
2. Use the perimeter formula: P = 2(l + w) = 40 yards, which simplifies to l + w = 20 yards.
3. Solve the system of equations: We can solve for 'l' in the perimeter equation: l = 20 - w.
4. Substitute into the area equation: (20 - w) * w = 100
5. Expand and solve the quadratic equation: 20w - w² = 100 => w² - 20w + 100 = 0 This factors to (w - 10)(w - 10) = 0.
6. Therefore, w = 10 yards. Substituting back into the equation l = 20 - w, we get l = 10 yards.

Answer: The field is a square (a special type of rectangle) with dimensions of 10 yards by 10 yards.

Problem 7: Problem involving Similar Rectangles

Two rectangles are similar. The smaller rectangle has a length of 6 cm and a width of 4 cm. The larger rectangle has a length of 9 cm. What is the width of the larger rectangle?

Solution: Similar rectangles have proportional sides.

1. Set up a proportion: 6/4 = 9/x (where x is the width of the larger rectangle)
2. Cross-multiply and solve for x: 6x = 36 => x = 6 cm

Answer: The width of the larger rectangle is 6 cm.

Problem 8: Area of a Composite Figure

A figure is composed of two rectangles. Rectangle A has a length of 10 cm and a width of 5 cm. Rectangle B has a length of 7 cm and a width of 3 cm. What is the total area of the figure?

Solution: Find the area of each rectangle and add them together.

1. Area of Rectangle A: 10 cm * 5 cm = 50 cm²
2. Area of Rectangle B: 7 cm * 3 cm = 21 cm²
3. Total Area: 50 cm² + 21 cm² = 71 cm²

Answer: The total area of the figure is 71 square centimeters.

Advanced Rectangle Problems and Concepts

The problems above covered basic concepts. Let's explore more complex scenarios:

• Rectangles inscribed in circles: These problems involve finding the dimensions of a rectangle that fits perfectly inside a circle. This often requires using the Pythagorean theorem and the properties of right-angled triangles.

• Optimization problems: These often involve finding the maximum area of a rectangle given a fixed perimeter or vice-versa. Calculus or algebraic techniques are often necessary for solving these optimization problems.

• Problems involving three-dimensional rectangles (rectangular prisms): This expands the concept to include volume calculations (V = l * w * h) and surface area calculations.

Tips for Mastering Rectangle Problems

• Draw diagrams: Visualizing the problem through a well-labeled diagram significantly simplifies the solution process.

• Organize your work: A structured approach, clearly showing each step, minimizes errors and improves understanding.

• Check your units: Ensure consistent units throughout the calculations (e.g., all measurements in centimeters or all in meters).

• Practice regularly: Consistent practice with a variety of problems strengthens your understanding and problem-solving skills.

• Seek help when needed: Don't hesitate to ask for clarification or assistance if you encounter difficulties.

This comprehensive guide provides a robust foundation for tackling 6-4 practice problems on rectangles. Remember that consistent practice and a thorough understanding of the underlying concepts are key to mastering these geometrical calculations. By applying the formulas and techniques explained here, you'll confidently solve a wide range of rectangle problems.