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

Similar polygons are geometric figures that maintain the same shape due to equal corresponding angles and proportional corresponding sides. Understanding these properties allows one to calculate scale factors and determine unknown side lengths, which is essential in solving practical geometry problems, designing models, and applying proportional reasoning in real-world contexts.

Learning Objectives

1

Define similar polygons and explain the conditions for two polygons to be similar.

2

Identify and calculate the scale factor (ratio) between corresponding sides of similar polygons.

3

Apply the properties of similar polygons to find missing side lengths and solve geometric problems.

4

Develop problem-solving strategies that use proportional reasoning in similar polygon contexts.

Key Concepts

CONCEPT

DEFINITION

Similar Polygons

Polygons that have all corresponding angles equal and all corresponding sides in proportion.

Corresponding Angles

The angles in similar positions in two similar polygons; these angles are congruent.

Corresponding Sides

The sides of similar polygons that occupy the same relative positions; the ratios of their lengths are equal.

Scale Factor

The constant ratio between the lengths of corresponding sides of similar polygons.

Proportionality

A relationship that states two ratios are equal; used to find unknown side lengths in similar polygons.

Example Problems

Example 1

$A B C D$ is a parallelogram. Find the value of each ratio. $$A B: B C$$

Example 2

$A B C D$ is a parallelogram. Find the value of each ratio. $$A B: C D$$

Example 3

$A B C D$ is a parallelogram. Find the value of each ratio. $$m \angle C: m \angle D$$

Example 4

$A B C D$ is a parallelogram. Find the value of each ratio. $$m \angle B: m \angle C$$

Example 5

$A B C D$ is a parallelogram. Find the value of each ratio. $$AD: \text{perimeter of}\quad A B C D$$
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Step-by-Step Explanations

QUESTION

Given two similar polygons where one has a side length of 4 and the corresponding side in the other polygon is 6, how do you calculate the scale factor and use it to find an unknown side length?

STEP-BY-STEP ANSWER:

Step 1: Identify the corresponding sides. Here, 4 corresponds to 6.
Step 2: Calculate the scale factor (k) by dividing the length of a side in the second polygon by the corresponding side length in the first polygon: k = 6 / 4 = 1.5.
Step 3: Use the scale factor to find unknown side lengths. For example, if a side of the first polygon is 8, then the corresponding side in the second polygon will be 8 * 1.5 = 12.
Final Answer: The scale factor is 1.5, and an unknown side of length 8 in the first polygon corresponds to a side of length 12 in the second polygon.

Finding the Scale Factor

QUESTION

How can you verify that two polygons are similar?

STEP-BY-STEP ANSWER:

Step 1: Compare all corresponding angles to ensure they are equal.
Step 2: Check that all corresponding side lengths are in the same ratio (scale factor).
Step 3: Confirm that both conditions (equal angles and proportional sides) hold true.
Final Answer: Two polygons are similar if all their corresponding angles are equal and the ratios of the lengths of their corresponding sides are constant.

Checking Similarity

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Common Mistakes

  • Confusing similar polygons with congruent polygons; recall that congruent polygons are identical in size, while similar polygons can have different sizes.
  • Failing to correctly match corresponding vertices, which can lead to incorrect ratios and calculations.
  • Ignoring the necessity to check all corresponding angles and sides for proportionality.
  • Arithmetic errors while calculating the scale factor, leading to incorrect solutions.