11.2 Areas of Trapezoids and Kites

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Investigating g Geometry ACTIVITY Use before Lesson 11.2 11.

A trapezoid or a kite can be cut out and rearranged to form a parallelogram.

Cut out two congruent trapezoids. Label as sown. Create a parallelogram Arrange te two trapezoids from Step 1 to form a parallelogram. Ten tape tem togeter.

Label te oter diagonal. Cut triangles Cut out te kite. Cut along to form two congruent triangles. Ten cut one triangle along part of to form two rigt triangles.

D R A W C O N C L U S I O N S Use your observations to complete tese eercises 1.

1 Investigating g Geometry ACTIVITY Use before Lesson Areas of Trapezoids and Kites MATERIALS grap paper straigtedge scissors tape Q U E S T I O N How can you use a parallelogram to find oter areas? A trapezoid or a kite can be cut out and rearranged to form a parallelogram. E X L O R E 1 Use two congruent trapezoids to form a parallelogram STE 1 STE 2 b 1 b 1 b2 b1 Draw a trapezoid Fold grap paper in alf and draw a trapezoid. Cut out two congruent trapezoids. Label as sown. Create a parallelogram Arrange te two trapezoids from Step 1 to form a parallelogram. Ten tape tem togeter. E X L O R E 2 Use one kite to form a rectangle STE 1 STE 2 STE 3 Draw a kite Draw a kite and its perpendicular diagonals. Label te diagonal tat is a line of symmetry. Label te oter diagonal. Cut triangles Cut out te kite. Cut along to form two congruent triangles. Ten cut one triangle along part of to form two rigt triangles. Create a rectangle Turn over te rigt triangles. lace eac wit its ypotenuse along a side of te larger triangle to form a rectangle. Ten tape te pieces togeter. D R A W C O N C L U S I O N S Use your observations to complete tese eercises 1. In Eplore 1, ow does te area of one trapezoid compare to te area of te parallelogram formed from two trapezoids? Write epressions in terms of b 1,, and for te base, eigt, and area of te parallelogram. Ten write a formula for te area of a trapezoid. 2. In Eplore 2, ow do te base and eigt of te rectangle compare to and? Write an epression for te area of te rectangle in terms of and. Ten use tat epression to write a formula for te area of a kite Areas of Trapezoids, Rombuses, and Kites 729

11.2 Areas of Trapezoids, Rombuses, and Kites Before You found areas of triangles and parallelograms.

So you can solve find te a problem area of a in free-trow sports, as lane, in Eample as in Eample 1. 1. Key Vocabulary eigt of a trapezoid diagonal, p.

542 As you saw in te Activity on page 729, you can use te area formula for a parallelogram to develop area formulas for oter special quadrilaterals.

Trapezoid Kite Rombus base diagonals eigt base diagonals THEOREM For Your Notebook THEOREM 11.

736 (b 1 1 ) E X A M L E 1 Find te area of a trapezoid BASKETBALL Te free-trow lane on an international basketball court is saped like a trapezoid.

2 11.2 Areas of Trapezoids, Rombuses, and Kites Before You found areas of triangles and parallelograms. Now You will find areas of oter types of quadrilaterals. Wy? So you can solve find te a problem area of a in free-trow sports, as lane, in Eample as in Eample Key Vocabulary eigt of a trapezoid diagonal, p. 507 bases of a trapezoid, p. 542 As you saw in te Activity on page 729, you can use te area formula for a parallelogram to develop area formulas for oter special quadrilaterals. Te areas of te figures below are related to te lengts of te marked segments. Te eigt of a trapezoid is te perpendicular distance between its bases. Trapezoid Kite Rombus base diagonals eigt base diagonals THEOREM For Your Notebook THEOREM 11.4 Area of a Trapezoid Te area of a trapezoid is one alf te product of te eigt and te sum of te lengts of te bases. b 1 roof: E. 40, p. 736 (b 1 1 ) E X A M L E 1 Find te area of a trapezoid BASKETBALL Te free-trow lane on an international basketball court is saped like a trapezoid. Find te area of te free-trow lane. 3.6 m ANOTHER WAY In a trapezoid, te average of te lengts of te bases is also te lengt of te midsegment. So, you can also find te area by multiplying te midsegment by te eigt. Solution Te eigt of te trapezoid is 5.8 meters. Te lengts of te bases are 3.6 meters and 6 meters. (b 1 1 ) Formula for area of a trapezoid 5 1 } 2 (5.8)( ) Substitute 5.8 for, 3.6 for b 1, and 6 for Simplify. c Te area of te free-trow lane is about 27.8 square meters. 5.8 m 6 m 730 Capter 11 Measuring Lengt and Area

THEOREMS For Your Notebook ANOTHER WAY THEOREM 11.5 Area of a Rombus Remember tat a rombus is also a parallelogram, so you can also use te formula A 5 b.

roof: E. 41, p. 736 E X A M L E 2 Find te area of a rombus READ DIAGRAMS MUSIC Rombus QRS represents one of te inlays on te guitar in te poto. Find te area of te inlay.

3 THEOREMS For Your Notebook ANOTHER WAY THEOREM 11.5 Area of a Rombus Remember tat a rombus is also a parallelogram, so you can also use te formula A 5 b. Te area of a rombus is one alf te product of te lengts of its diagonals. Justification: E. 39, p. 735 THEOREM 11.6 Area of a Kite Te area of a kite is one alf te product of te lengts of its diagonals. roof: E. 41, p. 736 E X A M L E 2 Find te area of a rombus READ DIAGRAMS MUSIC Rombus QRS represents one of te inlays on te guitar in te poto. Find te area of te inlay. Solution 9 mm Wen you read a diagram, look for information you need to find. Te diagram gives te lengts of } QN and }N, but not te lengts of } QS and } R. STE 1 Find te lengt of eac diagonal. Te diagonals of a rombus bisect eac oter, so QN 5 NS and N 5 NR. QS 5 QN 1 NS mm R 5 N 1 NR mm 12 mm S N R STE 2 Find te area of te rombus. Let represent QS and represent R. Formula for area of a rombus 5 1 } 2 (18)(24) Substitute Simplify. c Te area of te inlay is 216 square millimeters. GUIDED RACTICE for Eamples 1 an Find te area of te figure ft 2. 4 ft 6 in m 8 ft 14 in. 40 m 11.2 Areas of Trapezoids, Rombuses, and Kites 731

E X A M L E 3 Standardized Test ractice ELIMINATE CHOICES In Eample 3, you can eliminate coices A and B because in eac case, one diagonal is not twice as long as te oter diagonal.

, 12 in. Solution Draw and label a diagram. Let be te lengt of one diagonal. Te oter diagonal is twice as long, so label it 2. Use te formula for te area of a kite to find te value of.

4 E X A M L E 3 Standardized Test ractice ELIMINATE CHOICES In Eample 3, you can eliminate coices A and B because in eac case, one diagonal is not twice as long as te oter diagonal. One diagonal of a kite is twice as long as te oter diagonal. Te area of te kite is square inces. Wat are te lengts of te diagonals? A 6 in., 6 in. B 8.5 in., 8.5 in.c 8.5 in., 17 in.d6 in., 12 in. Solution Draw and label a diagram. Let be te lengt of one diagonal. Te oter diagonal is twice as long, so label it 2. Use te formula for te area of a kite to find te value of. 2 Formula for area of a kite } 2 ()(2) Substitute for A, for, an for Simplify Find te positive square root of eac side. Te lengts of te diagonals are 8.5 inces an(8.5) 5 17 inces. c Te correct answer is C.ABCD E X A M L E 4 Find an area in te coordinate plane CITY LANNING You ave a map of a city park. Eac grid square represents a meter by meter square. Find te area of te park. y B C Solution STE 1 Find te lengts of te bases and te eigt of trapezoid ABCD. b 1 5 BC m O A E D 5 AD m 5 BE m STE 2 Find te area of ABCD. (b 1 1 ) 5 1 } 2 (50)( ) c Te area of te park is 2750 square meters. GUIDED RACTICE for Eamples 3 and 4 4. Te area of a kite is 80 square feet. One diagonal is 4 times as long as te oter. Find te diagonal lengts. 5. Find te area of a rombus wit vertices M(1, 3), N(5, 5), (9, 3), and Q(5, 1). 732 Capter 11 Measuring Lengt and Area

11.2 EXERCISES SKILL RACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 9, 17, and 35 5 STANDARDIZED TEST RACTICE Es. 2, 15, 30, 39, and 42 1.

Describe wat you know about te segments and angles formed by te intersecting diagonals. EXAMLE 1 on p. 730 for Es. 3 6 FINDING AREA Find te area of te trapezoid. 3. 8 4. 5. 7.6 6 5 11 6 4.8 6.

5 11.2 EXERCISES SKILL RACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Es. 9, 17, and 35 5 STANDARDIZED TEST RACTICE Es. 2, 15, 30, 39, and VOCABULARY Copy and complete: Te perpendicular distance between te bases of a trapezoid is called te? of te trapezoid. 2. WRITING Sketc a kite and its diagonals. Describe wat you know about te segments and angles formed by te intersecting diagonals. EXAMLE 1 on p. 730 for Es. 3 6 FINDING AREA Find te area of te trapezoid DRAWING DIAGRAMS Te lengts of te bases of a trapezoid are 5.4 centimeters an0.2 centimeters. Te eigt is 8 centimeters. Draw and label a trapezoid tat matces tis description. Ten find its area. EXAMLE 2 on p. 731 for Es FINDING AREA Find te area of te rombus or kite ERROR ANALYSIS Describe and correct te error in finding te area cm 13 cm 12 cm 19 cm A 5 } 1 (13)( ) cm 2 12 cm (12)(21) cm 2 5 cm 16 cm EXAMLE 3 on p. 732 for Es MULTILE CHOICE One diagonal of a rombus is tree times as long as te oter diagonal. Te area of te rombus is 24 square feet. Wat are te lengts of te diagonals? A 8 ft, 11 ft B 4 ft, 12 ft C 2 ft, 6 ft D 6 ft, 24 ft 11.2 Areas of Trapezoids, Rombuses, and Kites 733

ALGEBRA Use te given information to find te value of. 16. Area 5 8 ft 2 17. Area 5 300 m 2 18. Area 5 0 y 22 ft 20 m yd m 14 ft EXAMLE 4 on p. 732 for Es.

6 ALGEBRA Use te given information to find te value of. 16. Area 5 8 ft Area m Area 5 0 y 22 ft 20 m yd m 14 ft EXAMLE 4 on p. 732 for Es COORDINATE GEOMETRY Find te area of te figure. 19. y 20. y y ALGEBRA Find te lengts of te bases of te trapezoid described. 22. Te eigt is 3 feet. One base is twice as long as te oter base. Te area is 13.5 square feet. 23. One base is 8 centimeters longer tan te oter base. Te eigt is 6 centimeters and te area is 54 square centimeters. FINDING AREA Find te area of te saded region OEN-ENDED MATH Draw tree eamples of trapezoids tat matc tis description: Te eigt of te trapezoid is 3 units and its area is te same as te area of a parallelogram wit eigt 3 units and base 8 units. VISUALIZING Sketc te figure. Ten determine its perimeter and area. 31. Te figure is a trapezoid. It as two rigt angles. Te lengts of its bases are 7 an5. Its eigt is Te figure is a rombus. Its side lengt is 13. Te lengt of one of its diagonals is 24. B 33. CHALLENGE In te diagram sown at te rigt, ABCD is a parallelogram and BF Find te area of ~ABCD. Eplain your reasoning. (Hint: Draw auiliary lines troug point A and troug point D tat are parallel to } EH.) A C D 9 E 8 F 3 G H WORKED-OUT SOLUTIONS on p. WS1 5 STANDARDIZED TEST RACTICE

ROBLEM SOLVING EXAMLE 1 on p. 730 for E. 34 34. TRUCKS Te windsield in a truck is in te sape of a trapezoid. Te lengts of te bases of te trapezoid are 70 inces and 79 inces. Te eigt is 35 inces.

MULTI-STE ROBLEM As sown, a baseball stadium s playing field is saped like a pentagon. To find te area of te playing field sown at te rigt, you can divide te field into two smaller polygons. a. Classify te two polygons.

VISUAL REASONING Follow te steps in parts (a) (c). a. Analyze Copy te table and etend it to include a column for n 5 5. Complete te table for n 5 4 and n 5 5. Rombus number, n 1 2 3 4 Diagram?

7 ROBLEM SOLVING EXAMLE 1 on p. 730 for E TRUCKS Te windsield in a truck is in te sape of a trapezoid. Te lengts of te bases of te trapezoid are 70 inces and 79 inces. Te eigt is 35 inces. Find te area of te glass in te windsield. EXAMLE 2 on p. 731 for E INTERNET You are creating a kite-saped logo for your scool s website. Te diagonals of te logo are 8 millimeters and 5 millimeters long. Find te area of te logo. Draw two different possible sapes for te logo. 36. DESIGN You are designing a wall anging tat is in te sape of a rombus. Te area of te wall anging is 432 square inces and te lengt of one diagonal is 36 inces. Find te lengt of te oter diagonal. 37. MULTI-STE ROBLEM As sown, a baseball stadium s playing field is saped like a pentagon. To find te area of te playing field sown at te rigt, you can divide te field into two smaller polygons. a. Classify te two polygons. b. Find te area of te playing field in square feet. Ten epress your answer in square yards. Round to te nearest square foot. 145 ft 179 ft 450 ft 315 ft 322 ft 38. VISUAL REASONING Follow te steps in parts (a) (c). a. Analyze Copy te table and etend it to include a column for n 5 5. Complete te table for n 5 4 and n 5 5. Rombus number, n Diagram? Area, A 2 4 6? b. Use Algebra Describe te relationsip between te rombus number n and te area of te rombus. Ten write an algebraic rule for finding te area of te nt rombus. c. Compare In eac rombus, te lengt of one diagonal ( ) is 2. Wat is te lengt of te oter diagonal ( ) for te nt rombus? Use te formula for te area of a rombus to write a rule for finding te area of te nt rombus. Compare tis rule wit te one you wrote in part (b). 39. SHORT RESONSE Look back at te Activity on page 729. Eplain ow te results for kites in Eplore 2 can be used to justify Teorem 11.5, te formula for te area of a rombus Areas of Trapezoids, Rombuses, and Kites 735

ROVING THEOREMS 11.4 AND 11.6 Use te triangle area formula and te triangles in te diagram to write a plan for te proof. 40. Sow tat te area A of te trapezoid 41.

8 ROVING THEOREMS 11.4 AND 11.6 Use te triangle area formula and te triangles in te diagram to write a plan for te proof. 40. Sow tat te area A of te trapezoid 41. Sow tat te area A of te kite sown is 1 } 2 (b 1 1 ). sown is 1 } 2. R S b 1 R S 42. EXTENDED RESONSE You will eplore te effect of moving a diagonal. A B C A B C Moved } BD closer to C. Didn t move it up or down. D original kite D still a kite a. Investigate Draw a kite in wic te longer diagonal is orizontal. Suppose tis diagonal is fied and you can slide te vertical diagonal left or rigt and up or down. You can keep sliding as long as te diagonals continue to intersect. Draw and identify eac type of figure you can form. b. Justify Is it possible to form any sapes tat are not quadrilaterals? Eplain. c. Compare Compare te areas of te different sapes you found in part (b). Wat do you notice about te areas? Eplain. 43. CHALLENGE James A. Garfield, te twentiet president of te United States, discovered a proof of te ytagorean Teorem in His proof involved te fact tat a trapezoid can be formed from two congruent rigt triangles and an isosceles rigt triangle. Use te diagram to sow tat a c 2. b a c c b a MIXED REVIEW Solve for te indicated variable. Write a reason for eac step. (p. 5) 44. d 5 rt; solve for t 45. b; solve for l 1 2w; solve for w REVIEW repare for Lesson 11.3 in E Find te angle measures of an isosceles triangle if te measure of a base angle is 4 times te measure of te verte angle. (p. 264) 48. In te diagram at te rigt, n QR, n STU. Te perimeter of n STU is 81 inces. Find te eigt and te perimeter of n QR. (p. 372) 20 R S T U 736 EXTRA RACTICE for Lesson 11.2, p. 916 ONLINE QUIZ at classzone.com

8.5 Area of Parallelograms

8.5 Area of Parallelograms 8. Area of Parallelograms Goal Find te area of parallelograms. Key Words ase of a parallelogram eigt of a parallelogram parallelogram p. 0 romus p. Geo-Activity Use a straigtedge to draw a line troug one