Book cover for McDougal Littell Jurgensen Geometry : Student Edition 2000

McDougal Littell Jurgensen Geometry : Student Edition 2000

Ray C. Jurgensen, Richard G. Brown

ISBN #9780395977279

1st Edition

2,712 Questions

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20,674 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the calculation of areas for various plane figures, including both common polygons (such as rectangles, parallelograms, trapezoids, and triangles) and curved figures (like circles and sectors). Understanding the formulas and processes for determining these areas not only facilitates problem solving in geometry but also enhances practical applications in real-world contexts such as design, architecture, and probability. Key takeaways include mastering standard area formulas, correct application of these formulas across different shapes, and the integration of geometric probability into problem-solving.

Learning Objectives

1

Calculate and compare the areas of various plane figures including rectangles, triangles, trapezoids, parallelograms, and rhombuses.

2

Apply formulas to determine the areas of circles, sectors, arcs, and regular polygons.

3

Utilize geometric ratios and concepts of similarity in solving area-related problems.

4

Integrate technology and calculator key-in explorations to validate area computations in complex figures.

Key Concepts

CONCEPT

DEFINITION

Polygon

A closed plane figure with at least three straight sides and angles.

Rectangle

A quadrilateral with opposite sides equal and all interior angles equal to 90°.

Rhombus

A quadrilateral with four equal-length sides and opposite sides parallel, but not necessarily with right angles.

Parallelogram

A quadrilateral with two pairs of parallel sides where opposite sides are equal in length.

Triangle

A polygon with three edges and three vertices.

Trapezoid

A quadrilateral with at least one pair of parallel sides; the parallel sides are referred to as the bases.

Regular Polygon

A polygon where all sides and interior angles are equal.

Circle

A set of all points in a plane that are at a given distance (radius) from a fixed point (center).

Circumference

The total distance around the circle; essentially, the perimeter of the circle.

Arc

A portion of the circumference of a circle.

Sector

A region bounded by two radii and the arc lying between them.

Geometric Probability

A probability measure where outcomes are determined using geometric measures such as length, area, or volume.

Example Problems

Example 1

Refer to rectangles. Complete the tables. $p$ is the perimeter. (TABLE CANNOT COPY)

Example 2

Refer to rectangles. Complete the tables. $p$ is the perimeter. (TABLE CANNOT COPY)

Example 3

Refer to rectangles. Complete the tables. $p$ is the perimeter. (TABLE CANNOT COPY)

Example 4

Refer to rectangles. Complete the tables. $p$ is the perimeter. (TABLE CANNOT COPY)

Example 5

Refer to rectangles. Complete the tables. $p$ is the perimeter. (TABLE CANNOT COPY)

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Step-by-Step Explanations

QUESTION

How do you calculate the area of a rectangle with a given length and width?

STEP-BY-STEP ANSWER:

Step 1: Identify the values for the length and width of the rectangle.
Step 2: Apply the formula Area = Length × Width.
Step 3: Multiply the length by the width to obtain the area.
Final Answer: The area is equal to the product of the length and width.

Area of a Rectangle

QUESTION

How do you compute the area of a trapezoid with bases b1 and b2, and height h?

STEP-BY-STEP ANSWER:

Step 1: Identify the lengths of the two parallel bases (b1 and b2) and the height h.
Step 2: Apply the formula Area = 1/2 × (b1 + b2) × h.
Step 3: Sum the lengths of the bases, divide by 2 and multiply by the height.
Final Answer: The area of the trapezoid is 1/2 × (b1 + b2) × h.

Area of a Trapezoid

QUESTION

How do you calculate the area of a circle and the area of a sector of that circle?

STEP-BY-STEP ANSWER:

Step 1: Identify the radius (r) of the circle.
Step 2: Use the circle area formula: Area = π × r².
Step 3: For a sector, determine the central angle (in degrees or radians).
Step 4: If using degrees, use the formula Sector Area = (θ/360) × π × r². If in radians, use Sector Area = 1/2 × r² × θ.
Final Answer: The area is computed using the appropriate formula based on whether you're finding the full circle's area or just a sector's.

Area of a Circle and Its Sectors

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Common Mistakes

  • Confusing the formulas for different polygons (e.g., using the rectangle's area formula for a parallelogram without considering height).
  • Forgetting to convert angles from degrees to radians when required by the sector area formula.
  • Misidentifying the bases in a trapezoid, which leads to incorrect multiplication and division in the formula.
  • Overlooking the use of technology or calculator key-ins that can simplify complex computations.