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584 Chapter 9

Extending Perimeter, Circumference, and Area

9A Developing Geometric Formulas

9-1 Developing Formulas for Triangles and Quadrilaterals

Lab Develop π

9-2 Developing Formulas for Circles and Regular Polygons

9-3 Composite Figures

Lab Develop Pick’s Theorem for Area of Lattice Polygons

9B Applying Geometric Formulas

9-4 Perimeter and Area in the Coordinate Plane

9-5 Effects of Changing Dimensions Proportionally

9-6 Geometric Probability

Lab Use Geometric Probability to Estimate π

584

KEYWORD: MG7 ChProj

This map of Texas counties shows the elevations of different regions.

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Extending Perimeter, Circumference, and Area 585

Vocabulary Match each term on the left with a definition on the right.

1. area

2. kite

3. perimeter

4. regular polygon

A. a polygon that is both equilateral and equiangular

B. a quadrilateral with exactly one pair of parallel sides

C. the number of nonoverlapping unit squares of a given size that exactly cover the interior of a figure

D. a quadrilateral with exactly two pairs of adjacent congruent sides

E. the distance around a closed plane figure

Convert Units Use multiplication or division to change from one unit of measure to another.

5. 12 mi = yd Length

Metric Customary

1 kilometer = 1000 meters 1 meter = 100 centimeters

1 centimeter = 10 millimeters

1 mile = 1760 yards 1 mile = 5280 feet

1 yard = 3 feet 1 foot = 12 inches

6. 7.3 km = m

7. 6 in. = ft

8. 15 m = mm

Pythagorean Theorem Find x in each right triangle. Round to the nearest tenth, if necessary.

9. 10. 11.

Measure with Customary and Metric Units Measure each segment to the nearest eighth of an inch and to the nearest half of a centimeter.

12. 13. 14.

Solve for a Variable Solve each equation for the indicated variable.

15. A = 1_ 2

bh for b 16. P = 2b + 2h for h

17. A = 1_ 2(b 1 + b 2)h for b 1 18. A =

1_ 2

d 1d 2 for d 1

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586 Chapter 9586

Geometry TEKS Les. 9-1

9-2 Geo. Lab

Les. 9-2

Les. 9-3

9-3 Geo. Lab

Les. 9-4

Les. 9-5

Les. 9-6

9-6 Geo. Lab

G.3.C Geometric structure* use logical reasoning to prove statements are true ...

�

G.3.E Geometric structure* use deductive reasoning to prove a statement

�

G.5.A Geometric patterns* use ... geometric patterns to develop algebraic expressions representing geometric properties

� � � �

G.5.B Geometric patterns* use ... geometric patterns to make generalizations about geometric properties, including ... ratios in similar figures ...

�

G.7.A Dimensionality and the geometry of location* use ... two- dimensional coordinate systems to represent ... figures

�

G.7.B Dimensionality and the geometry of location* use slopes ... to investigate geometric relationships ...

�

G.8.A Congruence and the geometry of size* find areas of ... circles, and composite figures

� � � � � �

G.8.C Congruence and the geometry of size* ... use the Pythagorean Theorem

� �

G.11.D Similarity and the geometry of shape* describe the effect on perimeter, area, ... when one or more dimensions of a figure are changed ...

�

* Knowledge and skills are written out completely on pages TX28–TX35.

Key Vocabulary/Vocabulario apothem apotema

center of a circle centro de un circulo

center of a regular polygon

centro de un poligono regular

central angle of a regular polygon

ángulo central de un poligono

circle circulo

composite figure figuras compuestas

geometric probability

probabilidad geométrica

Vocabulary Connections

To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like.

1. How can you use the everyday meaning of the word center to understand the term center of a circle ?

2. The word composite means “of separate parts.” What do you think the term composite figure means?

3. What does the word probability mean? How do you think geometric probability differs from theoretical probability?

4. The word apothem begins with the root apo-, which means “away from.” The apothem of a regular polygon is measured “away from” the center to the midpoint of a side. What do you think is true about the apothem and the side of the polygon?

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Extending Perimeter, Circumference, and Area 587

Study Strategy: Memorize Formulas Throughout a geometry course, you will learn many formulas, theorems, postulates, and corollaries. You may be required to memorize some of these. In order not to become overwhelmed by the amount of information, it helps to use flash cards.

Try This

1. Choose a lesson from this book that you have already studied, and make flash cards of the formulas or theorems from the lesson.

2. Review your flash cards by looking at the front of each card and trying to recall the information on the back of the card.

To create a flash card, write the name of the formula or theorem on the front of the card. Then clearly write the appropriate information on the back of the card. Be sure to include a labeled diagram.

Front Back

In a right triangle, the two sides that form the right angle are the legs . The side across from the right angle that stretches from one leg to the other is the hypotenuse . In the diagram, a and b are the lengths of the shorter sides, or legs, of the right triangle. The longest side is called the hypotenuse and has length c.

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a 2 + b 2 = c 2

Theorem 1-6-1 Pythagorean Theorem

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588 Chapter 9 Extending Perimeter, Circumference, and Area

Literal Equations

A literal equation contains two or more variables. Formulas you have used to find perimeter, circumference, area, and side relationships of right triangles are examples of literal equations.

If you want to evaluate a formula for several different values of a given variable, it is helpful to solve for the variable first.

Example

Danielle plans to use 50 feet of fencing to build a dog run. Use the formula P = 2� + 2w to find the length � when the width w is 4, 5, 6, and 10 feet.

Solve the equation for �.

First solve the formula for the variable.

P = 2� + 2w Write the original equation.

Subtract 2w from both sides.

Divide both sides by 2.

P - 2w = 2�

P - 2w _ 2

= �

Use your result to find � for each value of w.

� = P - 2w _ 2

= 50 - 2 (4)

_ 2

= 21 ft Substitute 50 for P and 4 for w.

Substitute 50 for P and 5 for w.

Substitute 50 for P and 6 for w.

Substitute 50 for P and 10 for w.

� = P - 2w _ 2

= 50 - 2 (5)

_ 2

= 20 ft

� = P - 2w _ 2

= 50 - 2 (6)

_ 2

= 19 ft

� = P - 2w _ 2

= 50 - 2 (10)

_ 2

= 15 ft

Try This

1. A rectangle has a perimeter of 24 cm. Use the formula P = 2� + 2w to find the width when the length is 2, 3, 4, 6, and 8 cm.

2. A right triangle has a hypotenuse of length c = 65 ft. Use the Pythagorean Theorem to find the length of leg a when the length of leg b is 16, 25, 33, and 39 feet.

3. The perimeter of �ABC is 112 in. Write an expression for a in terms of b and c, and use it to complete the following table.

a b c

48 35

36 36

14 50

588

See Skills Bank page S59

Algebra

TAKS Grades 9–11 Obj. 1

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9- 1 Developing Formulas for Triangles and Quadrilaterals 589

Why learn this? You can use formulas for area to help solve puzzles such as the tangram.

A tangram is an ancient Chinese puzzle made from a square. The pieces can be rearranged to form many different shapes. The area of a figure made with all the pieces is the sum of the areas of the pieces.

The area of a region is equal to the sum of the areas of its nonoverlapping parts.

Postulate 9-1-1 Area Addition Postulate

Recall that a rectangle with base b and height h has an area of A = bh. You can use the Area Addition Postulate to see that a parallelogram has the same area as a rectangle with the same base and height.

A triangle is cut off one side and translated to the other side.

The area of a parallelogram with base b and height h is A = bh.

Area Parallelogram

Remember that rectangles and squares are also parallelograms. The area of a