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 detailed the five key quadrilaterals: square, rectangle, parallelogram, rhombus, and trapezoid. Each shape has unique properties that define its side lengths, angles, and symmetry, which help in determining formulas for area and perimeter. By understanding and applying these formulas, students can solve practical problems involving these shapes and appreciate their roles in both geometry and real-world applications.

Learning Objectives

1

Identify and describe the five common quadrilaterals: square, rectangle, parallelogram, rhombus, and trapezoid.

2

Compare and contrast the properties such as side lengths, angles, and symmetry of each quadrilateral.

3

Apply formulas to calculate the perimeter and area of each quadrilateral.

4

Solve problems involving quadrilaterals by recognizing their characteristics and using appropriate strategies.

Key Concepts

CONCEPT

DEFINITION

Quadrilateral

A four-sided polygon with four vertices and four angles.

Square

A quadrilateral with all sides equal in length and all interior angles equal to 90 degrees.

Rectangle

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

Parallelogram

A quadrilateral where opposite sides are parallel and equal in length, but angles are not necessarily 90 degrees.

Rhombus

A quadrilateral with all sides equal in length, opposite sides parallel, and diagonals that intersect at right angles.

Trapezoid

A quadrilateral with at least one pair of parallel sides.

Example Problems

Example 1

Refer to $\square C R E W.$ If $O E=4$ and $W E=8 .$ name two segments congruent to $\overline{W E}$.

Example 2

Refer to $\square C R E W.$ If $\overline{W R} \perp \overline{C E}$, name all angles congruent to $\angle R C E.$

Example 3

Refer to $\square C R E W.$ If $\overline{W R} \perp \overline{C E}$. name all segments congruent to $\overline{W E}$.

Example 4

Refer to $\square C R E W.$ If $R E=E W$, name all angles congruent to $\angle E R W.$

Example 5

$P Q R S$ is a parallelogram. Find the values of $a, b$ $x,$ and $y.$ FIGURE CAN'T COPY.
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Step-by-Step Explanations

QUESTION

Given a square with side length 5 units, find its perimeter and area.

STEP-BY-STEP ANSWER:

Step 1: Calculate the perimeter by multiplying the side length by 4. Perimeter = 4 × 5 = 20 units.
Step 2: Calculate the area by squaring the side length. Area = 5 × 5 = 25 square units.
Final Answer: The square’s perimeter is 20 units and its area is 25 square units.

Square Area and Perimeter

QUESTION

Given a rectangle with a length of 8 units and a width of 3 units, determine its area and perimeter.

STEP-BY-STEP ANSWER:

Step 1: Calculate the perimeter using the formula: Perimeter = 2(length + width). Thus, 2(8 + 3) = 2 × 11 = 22 units.
Step 2: Calculate the area using the formula: Area = length × width. Thus, 8 × 3 = 24 square units.
Final Answer: The rectangle’s perimeter is 22 units and its area is 24 square units.

Rectangle Area and Perimeter

QUESTION

If a parallelogram has a base of 10 units and a height of 4 units, find its area.

STEP-BY-STEP ANSWER:

Step 1: Use the area formula for a parallelogram: Area = base × height.
Step 2: Multiply the base and the height: 10 × 4 = 40 square units.
Final Answer: The area of the parallelogram is 40 square units.

Parallelogram Area

QUESTION

A rhombus has diagonals measuring 12 units and 8 units. Compute its area.

STEP-BY-STEP ANSWER:

Step 1: Recall the area formula for a rhombus: Area = 1/2 × (diagonal1 × diagonal2).
Step 2: Multiply the diagonals and then multiply by 1/2: 1/2 × (12 × 8) = 1/2 × 96 = 48 square units.
Final Answer: The area of the rhombus is 48 square units.

Rhombus Area

QUESTION

A trapezoid has bases of 6 units and 10 units with a height of 5 units. What is its area?

STEP-BY-STEP ANSWER:

Step 1: Use the trapezoid area formula: Area = 1/2 × (sum of parallel sides) × height.
Step 2: Calculate the sum of the parallel sides: 6 + 10 = 16 units.
Step 3: Multiply by the height and then take half: 1/2 × 16 × 5 = 8 × 5 = 40 square units.
Final Answer: The area of the trapezoid is 40 square units.

Trapezoid Area

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Common Mistakes

  • Confusing the properties of squares and rectangles, particularly regarding side lengths and angles.
  • Mixing up the area formulas for different quadrilaterals, such as using the base and height incorrectly in parallelograms.
  • Overlooking the fact that trapezoids require the average of the two bases for area calculations.
  • Assuming all quadrilaterals have right angles, which is not true for shapes like parallelograms or rhombuses.