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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

9-2 Developing Formulas for Circles and Regular Polygons

9-3 Composite Figures

Lab Develop Picks 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

KEYWORD: MG7 ChProj

It measures up!How would you find the area of a field or the floor of an irregularly shaped building? You can use the ideas in this chapter to find out!

Extending Perimeter, Circumference, and Area 585

VocabularyMatch 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 UnitsUse multiplication or division to change from one unit of measure to another.

5. 12 mi = yd Length

Metric Customary

1 kilometer = 1000 meters1 meter = 100 centimeters

1 centimeter = 10 millimeters

1 mile = 1760 yards1 mile = 5280 feet

1 yard = 3 feet1 foot = 12 inches

6. 7.3 km = m

7. 6 in. = ft

8. 15 m = mm

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

9.

10.

11.

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

12. 13. 14.

Solve for a VariableSolve 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

586 Chapter 9

You will study areas and perimeters of

figures whose vertices are given by ordered pairs.

areas and perimeters of figures whose dimensions are found by using the Pythagorean Theorem.

areas and perimeters of figures in customary and metric units.

proofs of formulas for area and perimeter.

Previously, you graphed ordered pairs. developed and used the

Pythagorean Theorem.

measured with customary and metric units.

used formulas for area and perimeter.

You can use the skills learned in this chapter in your future math classes,

such as Calculus, to find the area under a curve.

in other classes, such as in Geography to find lengths of borders and areas of countries.

outside of school to plan a garden, analyze data in the newspaper, and solve puzzles.

KeyVocabulary/Vocabularioapothem 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

ngulo central de un poligono

circle circulo

composite figure figuras compuestas

geometric probability

probabilidad geomtrica

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?

Extending Perimeter, Circumference, and Area 587

Study Strategy: Memorize FormulasThroughout 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

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

See Skills Bank page S59

Algebra

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 square with side s is A = s 2 , and the perimeter is P = 4s.

1E X A M P L E Finding Measurements of ParallelogramsFind each measurement.

A the area of the parallelogramStep 1 Use the Pythagorean Theorem to find the height h. 3 2 + h 2 = 5 2

h = 4

Step 2 Use h to find the area of the parallelogram. A = bh Area of a parallelogram

Substitute 6 for b and 4 for h.

Simplify.

A = 6 (4) A = 24 in 2

ObjectivesDevelop and apply the formulas for the areas of triangles and special quadrilaterals.

Solve problems involving perimeters and areas of triangles and special quadrilaterals.

9-1 Developing Formulas for Triangles and Quadrilaterals

The height of a parallelogram is measured along a segment perpendicular to a line containing the base.

590 Chapter 9 Extending Perimeter, Circumference, and Area

Find each measurement.

B the height of a rectangle in which b = 5 cm and A = (5 x 2 - 5x) cm 2 A = bh Area of a rectangle

Substitute 5 x 2 - 5x for A and 5 for b.

Factor 5 out of the expression for A.

Divide both sides by 5.

Sym. Prop. of =

5 x 2 - 5x = 5h5 ( x 2 - x) = 5h x 2 - x = h h = ( x 2 - x) cm

C the perimeter of the rectangle, in which A = 12 x ft 2

Step 1 Use the area and the height to find the base. A = bh Area of a rectangle