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 key aspects of right triangles including the Pythagorean Theorem, its converse, the properties of special right triangles, and the concept of similarity in right triangles. Mastery of these topics allows students to solve for missing sides, verify the nature of triangles, and apply geometric principles to real-world problems.

Learning Objectives

1

Explain and apply the Pythagorean Theorem to solve for missing sides in right triangles.

2

Demonstrate understanding of the Converse Pythagorean Theorem to determine whether a given triangle is right.

3

Analyze and recognize the properties of special right triangles, including 45-45-90 and 30-60-90 triangles.

4

Apply the concept of similarity in right triangles to solve geometric problems.

Key Concepts

CONCEPT

DEFINITION

Pythagorean Theorem

A fundamental relation in Euclidean geometry among the three sides of a right triangle, expressed as a² + b² = c², where c is the hypotenuse.

Converse Pythagorean Theorem

A theorem stating that if the square of one side of a triangle equals the sum of the squares of the other two sides, then the triangle is a right triangle.

Special Right Triangles

Right triangles with specific angle measures (commonly 45°-45°-90° and 30°-60°-90°) that have predictable relationships between their side lengths.

Similarity in Right Triangles

The concept that two right triangles are similar if their corresponding angles are equal, leading to proportional relationships among their corresponding sides.

Example Problems

Example 1

Simplify. $$\sqrt{12}$$

Example 2

Simplify. $$\sqrt{72}$$

Example 3

Simplify. $$\sqrt{45}$$

Example 4

Simplify. $$\sqrt{75}$$

Example 5

Simplify. $$\sqrt{800}$$
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Step-by-Step Explanations

QUESTION

How do you use the Pythagorean Theorem to find the missing side of a right triangle?

STEP-BY-STEP ANSWER:

Step 1: Identify the sides. Label the two legs as a and b, and the hypotenuse as c.
Step 2: Write the equation a² + b² = c².
Step 3: Substitute the known values into the equation.
Step 4: Solve the equation for the unknown side by isolating the variable.
Step 5: Take the square root of both sides if solving for the hypotenuse, or rearrange accordingly for a leg.
Final Answer: The unknown side is determined by the computed value from the equation.

Pythagorean Theorem

QUESTION

How can you determine if a triangle is right using the Converse Pythagorean Theorem?

STEP-BY-STEP ANSWER:

Step 1: Identify the sides of the triangle and assume the longest side is the hypotenuse candidate.
Step 2: Square the lengths of all three sides.
Step 3: Check if the sum of the squares of the two shorter sides equals the square of the longest side.
Step 4: If the equality holds, the triangle is a right triangle.
Final Answer: The triangle is right if a² + b² = c² holds, where c is the longest side.

Converse Pythagorean Theorem

QUESTION

What are the side ratios for common special right triangles and how are they applied?

STEP-BY-STEP ANSWER:

Step 1: For a 45°-45°-90° triangle, recognize that the legs are congruent and the hypotenuse is the leg length multiplied by √2.
Step 2: For a 30°-60°-90° triangle, note that the shortest side (opposite the 30° angle) is half the hypotenuse, and the longer leg (opposite the 60° angle) is the shorter side multiplied by √3.
Step 3: Use these ratios to solve for unknown sides when one side is given.
Final Answer: The side lengths can be quickly found using the ratios 1:1:√2 for 45°-45°-90° triangles and 1:√3:2 for 30°-60°-90° triangles.

Special Right Triangles

QUESTION

How do you use similarity to solve for unknown side lengths in right triangles?

STEP-BY-STEP ANSWER:

Step 1: Confirm that the triangles are similar by ensuring corresponding angles are equal.
Step 2: Set up a proportion between the corresponding sides of the triangles.
Step 3: Substitute the known side lengths into the proportion.
Step 4: Solve the proportion to find the missing side length.
Final Answer: The unknown side is determined by solving the proportion between the sides of the similar triangles.

Similarity in Right Triangles

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Common Mistakes

  • Misidentifying the hypotenuse, especially in the converse theorem, leading to incorrect calculations.
  • Incorrectly applying the Pythagorean Theorem to non-right triangles.
  • Forgetting to take the square root when solving for the unknown side after summing the squares.
  • Mixing up the ratios for 45-45-90 and 30-60-90 triangles, resulting in erroneous side length determinations.