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 lays the groundwork for geometry by defining the basic figures: points, lines, planes, segments, rays, and angles. Students learn fundamental postulates and theorems that establish relationships between these figures, which serve as the building blocks for more advanced geometric reasoning. Mastery of these concepts is crucial for problem-solving and real-world applications such as plotting points and measuring distances.

Learning Objectives

1

Define and differentiate between points, lines, planes, segments, rays, and angles.

2

Understand and apply basic geometry postulates and theorems related to points, lines, and planes.

3

Solve problems involving distance and location by using basic geometric principles.

4

Interpret and analyze geometric figures through real-world applications and problem-solving exercises.

Key Concepts

CONCEPT

DEFINITION

Point

A location in space with no dimensions; it has no length, width, or height.

Line

A straight one-dimensional figure having no thickness and extending infinitely in both directions.

Plane

A flat, two-dimensional surface that extends infinitely in all directions.

Segment

A part of a line that is bounded by two distinct endpoints.

Ray

A part of a line that starts at one endpoint and extends infinitely in one direction.

Angle

The figure formed by two rays or line segments sharing a common endpoint.

Postulate

A statement accepted without proof that serves as a starting point for further reasoning and arguments.

Example Problems

Example 1

Copy and complete the table. Refer to the diagrams. (FIGURES AND TABLE CAN'T COPY)

Example 2

Use a centimeter ruler. Copy the points $F . T .$ and $P$ from the diagram. a. Draw a line to indicate all points equidistant from $F$ and $T$. b. Draw a circle to indicate points $6 \mathrm{cm}$ from $P .$ If you don't have a compass, draw as well as you can freehand. c. How many points are equidistant from $F$ and $T$, and are also $6 \mathrm{cm}$ from $P ?$ (FIGURE CAN'T COPY)

Example 3

Use a centimeter ruler. Repeat Exercise 2 , but use $2 \mathrm{cm}$ instead of $6 \mathrm{cm}$.

Example 4

Use a centimeter ruler. There is a distance you could use in parts (b) and (c) of Exercise 2 that would lead to the answer one point in part (c). Estimate that distance.

Example 5

Which is greater, the distance from $R$ to $S$ or the distance from $T$ to $U ?$ (FIGURE CAN'T COPY)
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Step-by-Step Explanations

QUESTION

How do you calculate the distance between two points on a line?

STEP-BY-STEP ANSWER:

Step 1: Identify the coordinates of the two points on the line.
Step 2: Subtract the smaller coordinate value from the larger one.
Step 3: Use the absolute value if necessary to ensure the distance is non-negative.
Final Answer: The absolute difference between the two coordinates is the distance between the points.

Distance Between Points

QUESTION

How do you locate a point on a plane using a coordinate system?

STEP-BY-STEP ANSWER:

Step 1: Set up a coordinate system with a horizontal (x-axis) and a vertical (y-axis).
Step 2: Identify the x-coordinate (horizontal position) and y-coordinate (vertical position) of the point.
Step 3: Plot the point at the intersection corresponding to these coordinates.
Final Answer: The point is uniquely located on the plane at the coordinates (x, y).

Locating a Point on a Plane

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Common Mistakes

  • Confusing points with lines, not realizing that points have no size while lines extend infinitely.
  • Mistaking the directionality of a ray for the bounded nature of a segment.
  • Overlooking the infinite nature of lines and planes, leading to limited or incorrect interpretations.
  • Incorrectly plotting points due to misunderstanding the coordinate system layout.