common core state standards 10-2 g-mg.a.1 mp 1, mp 3, mp 4...

Problem 1

Lesson 10-2 Areas of Trapezoids, Rhombuses, and Kites 623

10-2

Objective To find the area of a trapezoid, rhombus, or kite

Areas of Trapezoids, Rhombuses, and Kites

Essential Understanding You can find the area of a trapezoid when you know its height and the lengths of its bases.

The height of a trapezoid is the perpendicular distance between the bases.

Rearranging figures into familiar shapes is an example of the Solve a Simpler Problem strategy.

b1

b2

Draw a trapezoid on a sheet of graph paper. Label the bases b1 and b2. Draw its midsegment. Cut out the trapezoid, and then cut it along the midsegment. Rotate the top part of the trapezoid 180° so that b1 and b2 now form one long base. How can you use this new figure to find the area of the trapezoid? Explain your reasoning.

Lesson Vocabulary

•height of a trapezoid

LessonVocabulary

Theorem 10-4 Area of a Trapezoid

The area of a trapezoid is half the product of the height and the sum of the bases.

A = 12 h(b1 + b2)

hsm11gmse_1002_t09235.ai

h

b1

b2

Area of a Trapezoid

Geography What is the approximate area of Nevada?

A = 12 h(b1 + b2) Use the formula for area of a

trapezoid.

= 12 (309)(205 + 511) Substitute 309 for h, 205 for

b1, and 511 for b2.

= 110,622 Simplify.

The area of Nevada is about 110,600 mi2.

1. What is the area of a trapezoid with height 7 cm and bases 12 cm and 15 cm?

LasVegas

RenoCarson City

309 mi

511 mi

205 mi

hsm11gmse_1002_a09884Nevada10p x 13p

1st proof12.15.08

Got It?

Which borders of Nevada can you use as the bases of a trapezoid?The two parallel sides of Nevada form the bases of a trapezoid.

G-MG.A.1 Use geometric shapes, their measures, and their properties to describe objects.

MP 1, MP 3, MP 4, MP 6

MATHEMATICAL PRACTICES

Common Core State Standards

Problem 2

Problem 3

624 Chapter 10 Area

Finding Area Using a Right Triangle

What is the area of trapezoid PQRS?

You can draw an altitude that divides the trapezoid into a rectangle and a 30° -60° -90° triangle. Since the opposite sides of a rectangle are congruent, the longer base of the trapezoid is divided into segments of lengths 2 m and 5 m.

h = 2 13 longer leg = shorter leg # 13

A = 12 h(b1 + b2) Use the trapezoid area formula.

= 12 (213 )(7 + 5) Substitute 213 for h, 7 for b1, and 5 for b2.

= 1213 Simplify.

The area of trapezoid PQRS is 1213 m2.

2. Reasoning In Problem 2, suppose h decreases so that m∠P = 45 while angles R and Q and the bases stay the same. What is the area of trapezoid PQRS?

Essential Understanding You can find the area of a rhombus or a kite when you know the lengths of its diagonals.

Theorem 10-5 Area of a Rhombus or a Kite

The area of a rhombus or a kite is half the product of the lengths of its diagonals.

A = 12 d1d2

Rhombus Kite

Rhombus Kitehsm11gmse_1002_t09243.ai

d1d2


d1

d2

Finding the Area of a Kite

What is the area of kite KLMN?

Find the lengths of the two diagonals: KM = 2 + 5 = 7 m and LN = 3 + 3 = 6 m.

A = 12 d1d2 Use the formula for area of a kite.

= 12 (7)(6) Substitute 7 for d1 and 6 for d2.

= 21 Simplify.

The area of kite KLMN is 21 m2.

3. What is the area of a kite with diagonals that are 12 in. and 9 in. long?


7 m

5 m

60�

S

Q

R

P


5 m2 m

5 m

60�

S

h

Q

R

P

Got It?


M

L

N

K

2 m3 m

3 m5 m

Got It?

Do you need to know the side lengths of the kite to find its area?No. You only need the lengths of the diagonals.

How are the sides related in a 30° -60° -90° triangle?The length of the hypotenuse is 2 times the length of the shorter leg, and the longer leg is 13 times the length of the shorter leg.

6.5 ft

A

BC

2.5 ft

Problem 4


Finding the Area of a Rhombus

Car Pooling The High Occupancy Vehicle (HOV) lane is marked by a series of “diamonds,” or rhombuses painted on the pavement. What is the area of the HOV lane diamond shown at the right?

△ABC is a right triangle. Using the Pythagorean

Theorem, AB = 26.52 - 2.52 = 6. Since the diagonals of a rhombus bisect each other, the diagonals of the HOV lane diamond are 5 ft and 12 ft.

A = 12 d1d2 Use the formula for area of a rhombus.

= 12 (5)(12) Substitute 5 for d1 and 12 for d2.

= 30 Simplify.

The area of the HOV lane diamond is 30 ft2.

4. A rhombus has sides 10 cm long. If the longer diagonal is 16 cm, what is the area of the rhombus?

Got It?

Lesson CheckDo you know HOW?Find the area of each figure.

1. 2.

3. 4.

5. 6.

Do you UNDERSTAND? 7. Vocabulary Can a trapezoid and a parallelogram

with the same base and height have the same area? Explain.

8. Reasoning Do you need to know all the side lengths to find the area of a trapezoid?

9. Reasoning Can you find the area of a rhombus if you only know the lengths of its sides? Explain.

10. Reasoning Do you need to know the lengths of the sides to find the area of a kite? Explain.


4 m

6 m

10 m


15 in.

18 in.

27 in.


3 ft

5 ft


12 in.

12 in.


20 m10 m

10 m

10 m


3 cm

2 cm

1 cm

2 cm


h

b

h

b

How can you find the length of AB?AB is a leg of right △ABC. You can use the Pythagorean Theorem, a2 + b2 = c2, to find its length.


626 Chapter 10 Area

Practice and Problem-Solving Exercises

Find the area of each trapezoid

11. 12. 13.

14. Find the area of a trapezoid with bases 12 cm and 18 cm and height 10 cm.

15. Find the area of a trapezoid with bases 2 ft and 3 ft and height 13 ft.

16. Geography The border of Tennessee resembles a trapezoid with bases 340 mi and 440 mi and height 110 mi. Estimate the area of Tennessee by finding the area of the trapezoid.

Find the area of each trapezoid. If your answer is not an integer, leave it in simplest radical form.

17. 18. 19.

Find the area of each kite.

20. 21. 22.

Find the area of each rhombus.

23. 24. 25.

26. Think About a Plan A trapezoid has two right angles, 12-m and 18-m bases, and an 8-m height. Sketch the trapezoid and find its perimeter and area.

• Are the right angles consecutive or opposite angles? • How does knowing the height help you find the perimeter?

PracticeA See Problem 1.


21 in.

16 in.

38 in.


24.3 cm

8.5 cm

9.7 cm


6 ft18 ft

9 ft

See Problem 2.


6 ft

5 ft 3 ft


8 ft

60�

15 ft


6 m

10 m 8 m

See Problem 3.


2 in.

8 in.8 in.

8 in.


2 m3 m

3 m

4 m


4 ft 4 ft

6 ft

See Problem 4.


20 ft

30 ft


8 in.10 in.


6 m

5 m

ApplyB



27. Metallurgy The end of a gold bar has the shape of a trapezoid with the measurements shown. Find the area of the end.

28. Open-Ended Draw a kite. Measure the lengths of its diagonals. Find its area.

Find the area of each trapezoid to the nearest tenth.

29. 30. 31.

Coordinate Geometry Find the area of quadrilateral QRST.

32. 33. 34.

35. What is the area of the kite at the right?

90 m2 135 m2

108 m2 216 m2

36. a. Coordinate Geometry Graph the lines x = 0, x = 6, y = 0, and y = x + 4. b. What type of quadrilateral do the lines form? c. Find the area of the quadrilateral.

Find the area of each rhombus. Leave your answer in simplest radical form.

37. 38. 39.

40. Visualization The kite has diagonals d1 and d2 congruent to the sides of the rectangle. Explain why the area of the kite is 12 d1d2.

41. Draw a trapezoid. Label its bases b1 and b2 and its height h. Then draw a diagonal of the trapezoid.

a. Write equations for the area of each of the two triangles formed. b. Writing Explain how you can justify the trapezoid area formula using

the areas of the two triangles.

6.9 cm

9.2 cm

4.4 cm


4 cm

1 cm3 cm

3 cm


8 ft

9 ft30�


1.7 m

2.1 m

0.9 m

45�


y

xQ

T

S

2

�2

2 4

R


y

x

Q

S

R

T 2

2

4

4


x

Q

O

SR

T

y

2

�2 2

�2


9V2 m

45�

6 m


45�3 cm


60�

4 m


30�

8 in.


d1

d2

628 Chapter 10 Area

42. Algebra One base of a trapezoid is twice the other. The height is the average of the two bases. The area is 324 cm2. Find the height and the bases. (Hint: Let the smaller base be x.)

43. Sports Ty wants to paint one side of the skateboarding ramp he built. The ramp is 4 m wide. Its surface is modeled by the equation y = 0.25x2. Use the trapezoids and triangles shown to estimate the area to be painted.

44. In trapezoid ABCD at the right, AB } DC. Find the area of ABCD.

ChallengeC

x


O

y � 0.25x2

y

�2 �1 1 2

1


A B

CD

15 in.

20 in.

135� 30�

Mixed Review

48. Find the area of a right isosceles triangle that has one leg of length 12 cm.

49. A right isosceles triangle has area 112.5 ft2. Find the length of each leg.

50. Find the measure of an interior angle of a regular nonagon.

Get Ready! To prepare for Lesson 10-3, do Exercises 51–53.

Find the area of each regular polygon. Leave radicals in simplest form.

51. 52. 53.

See Lesson 10-1.

See Lesson 6-1.

See Lesson 8-2.


10 cm 10 ft


10 m


Standardized Test Prep

45. The area of a kite is 120 cm2. The length of one diagonal is 20 cm. What is the length of the other diagonal?

12 cm 20 cm 24 cm 48 cm

46. △ABC ∼ △XYZ . AB = 6, BC = 3, and CA = 7. Which of the following are NOT possible dimensions of △XYZ?

XY = 3, YZ = 1.5, ZX = 3.5 XY = 10, YZ = 7, ZX = 11

XY = 9, YZ = 4.5, ZX = 10.5 XY = 18, YZ = 9, ZX = 21

47. Draw an angle. Construct a congruent angle and its bisector.

SAT/ACT

ShortResponse

common core state standards 10-2 g-mg.a.1 mp 1, mp 3, mp 4...

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