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 chapter explores various geometric constructions using traditional tools like the compass and straightedge. It begins with basic constructions—such as drawing perpendiculars, parallels, and identifying concurrent lines—to more complex figures including circles, segments, and the Nine-Point Circle. The section then introduces the concept of locus, which involves finding sets of points meeting specific conditions. Mastery of these concepts is fundamental not only in theoretical geometry but also in practical applications like design, architecture, and technology-enabled explorations.

Learning Objectives

1

Explain the fundamental ideas behind geometric constructions, including constructing perpendiculars, parallels, and concurrent lines.

2

Demonstrate the ability to perform more complex constructions involving circles, segments, and special figures.

3

Understand the concept of locus, including its meaning and how to solve locus problems through construction.

4

Apply construction techniques in technological explorations and real-world problem solving.

5

Review and prepare for cumulative assessments and college entrance exam questions through practical exercises and mixed reviews.

Key Concepts

CONCEPT

DEFINITION

Construction

A methodical process for drawing geometric figures using only a compass and straightedge as tools.

Perpendiculars

Lines or segments that intersect at a 90-degree angle, essential in establishing right angles in geometric constructions.

Parallels

Lines in a plane that do not meet; they remain equidistant and have the same orientation.

Concurrent Lines

Three or more lines that intersect at a single point.

Circles and Segments

Curved constructions used to define arcs, radii, chords, and central angles within geometric figures.

Special Constructions

Unique geometric configurations such as the Nine-Point Circle that showcase important properties in triangle geometry.

Locus

A set of points that satisfy a particular condition or a rule, often used to define curves or regions in geometric spaces.

Nine-Point Circle

A circle that can be constructed for any given triangle, which passes through nine significant concurrent points including the midpoint of each side.

Example Problems

Example 1

On your paper, draw two segments roughly like those shown. Use these segments in Exercises $1-4$ to construct a segment having the indicated length. (Segment can't copy) $$a+b$$

Example 2

On your paper, draw two segments roughly like those shown. Use these segments in Exercises $1-4$ to construct a segment having the indicated length. (Segment can't copy) $$b-a$$

Example 3

On your paper, draw two segments roughly like those shown. Use these segments in Exercises $1-4$ to construct a segment having the indicated length. (Segment can't copy) $$3 a-b$$

Example 4

On your paper, draw two segments roughly like those shown. Use these segments in Exercises $1-4$ to construct a segment having the indicated length. (Segment can't copy) $$a+2 b$$

Example 5

Using any convenient length for a side, construct an equilateral triangle.
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Step-by-Step Explanations

QUESTION

How can you construct a perpendicular line from a given point on a line?

STEP-BY-STEP ANSWER:

Step 1: Place the compass point on the given point on the line.
Step 2: Draw an arc that intersects the line at two distinct points.
Step 3: Without changing the compass width, draw arcs from these two intersection points above (or below) the line.
Step 4: Mark the intersection point of these arcs.
Step 5: Draw a straight line connecting the given point and the intersection point. This line is perpendicular to the original line.
Final Answer: A line drawn from the given point perpendicular to the initial line using intersecting arcs.

Constructing a Perpendicular Line

QUESTION

How do you determine the locus of points equidistant from a fixed point and a fixed line?

STEP-BY-STEP ANSWER:

Step 1: Identify the fixed point and the fixed line in the plane.
Step 2: Recognize that the locus forms a parabola with the fixed point as the focus.
Step 3: Use geometric methods or a technology tool to plot points that are equidistant from the fixed point and the line.
Step 4: Draw the curve that passes through these points, forming the parabola.
Final Answer: The curve representing all points equidistant from a fixed point and a fixed line, known as a parabola.

Determining a Locus of Points

QUESTION

What are the steps to construct a parallel line through a given point not on the original line?

STEP-BY-STEP ANSWER:

Step 1: Identify the original line and the external point.
Step 2: Place the compass point on the external point and draw an arc that intersects the original line.
Step 3: With the same compass width, replicate the arc's intersection on the other side of the external point.
Step 4: Use angle-copying techniques to construct congruent angles between the original line and the new line through the given point.
Step 5: Draw the line through the given point that maintains equal angles, ensuring it is parallel to the original line.
Final Answer: A methodical construction of a line parallel to the given line through the external point using arcs and congruent angles.

Constructing a Parallel Line

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Common Mistakes

  • Confusing the steps of constructing perpendicular lines with those used for constructing parallel lines.
  • Forgetting to maintain the same compass width during the construction process, leading to inaccurate figures.
  • Misidentifying the locus as a single point or line instead of the full set of points satisfying the given condition.
  • Overlooking the importance of error checking through intersections in constructions, which can result in misaligned figures.