1-1. Nets and Drawings for Visualizing Geometry. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

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1 1-1 Nets and Drawings for Visualizing Geometry Vocabulary Review Identify each figure as two-dimensional or three-dimensional three-dimensional two-dimensional three-dimensional Vocabulary uilder polygon (noun) PHL ih gahn polygon Definition polygon is a two-dimensional figure with three or more sides, where each side meets exactly two other sides at their endpoints. Main Idea: polygon is a closed figure, so all sides meet. No sides cross each other. Examples: Triangles, rectangles, pentagons, hexagons, and octagons are polygons. Use Your Vocabulary Underline the correct word(s) to complete each sentence. 4. polygon is formed by two / three or more straight sides. 5. circle is / is not a polygon. 6. triangle / rectangle is a polygon with three sides. 7. The sides of a polygon are curved / straight. 8. Two / Three sides of polygon meet at the same point. ross out the figure(s) that are NOT polygons. 9. E D 10. M L Q N P 11. W R V X T S U hapter 1 2

2 Underline the correct word(s) to complete the sentence. 12. net is a two-dimensional / three-dimensional diagram that you can fold to form a two-dimensional / three-dimensional figure. 13. ircle the net that you can NOT fold into a cube. Use the net of a cube at the right for Exercises 14 and Suppose you fold the net into a cube. What number will be opposite each face? Suppose you fold the net into a cube. What number is missing from each view? Problem 1 Identifying a Solid rom a Net Got It? The net at the right folds into the cube shown. Which letters will be on the top and right side of the cube? 16. our of the five other letters will touch some side of ace when the net is folded into a cube. ross out the letter of the side that will NOT touch some side of ace. D E 17. Which side of the cube will that letter be on? ircle your answer. Top ottom Right Left ack 18. Use the net. Which face is to the right of ace? How do you know?. Explanations may vary. Sample: The left side of and the right side of are the same edge of the cube. 19. Use the net. Which face is on the top of the cube? How do you know? E. nswers may vary. Sample: E folds down to become the top of the cube. 3 Lesson 1-1

3 Problem 2 Drawing a Net rom a Solid Got It? What is a net for the figure at the right? Label the net with its dimensions. Write T for true or for false. T 20. Three of the faces are rectangles. 10 cm 10 cm 7 cm 4 cm 21. our of the faces are triangles. T 22. The figure has five faces in all. 23. Now write a description of the net. nswers may vary. Sample: The net has three rectangles and two triangles that fold to form the figure above. 24. ircle the net that represents the figure above. 10 cm 7 cm 10 cm 4 cm 10 cm 7 cm 10 cm 10 cm 7 cm 10 cm 4 cm 7 cm 7 cm 10 cm Problem 3 Isometric Drawing Got It? What is an isometric drawing of this cube structure? 25. The cube structure has edges that you can see and vertices that you can see. 26. The isometric dot paper shows 2 vertices and 1 edge of the cube structure. omplete the isometric drawing. hapter 1 4

4 Problem 4 Orthographic Drawing Got It? What is the orthographic drawing for this isometric drawing? 27. Underline the correct word to complete the sentence. If you built the figure out of cubes, you would use seven / eight cubes 28. ross out the drawing below that is NOT part of the orthographic drawing. Then label each remaining drawing. Write ront, Right, or Top. ront Right top right cross out front Lesson heck Do you UNDERSTND? Vocabulary Tell whether each drawing is isometric, orthographic, a net, or none. 29. Write dot paper, one view, three views or none. Then label each figure. Top one view Math Success heck off the vocabulary words that you understand. net isometric drawing orthographic drawing Rate how well you can use nets, isometric drawings, and orthographic drawings. Need to review ront three views Right ront dot paper Now I get it! Right none net orthographic isometric none 5 Lesson 1-1

5 1-2 Points, Lines, and Planes Vocabulary Review Draw a line from each net in olumn to the three-dimensional figure it represents in olumn. 1. olumn olumn Vocabulary uilder conjecture (noun, verb) kun JEK chur Main Idea: conjecture is a guess or a prediction. Definition: conjecture is a conclusion reached by using inductive reasoning. Use Your Vocabulary Write noun or verb to identify how the word conjecture is used in each sentence. 4. You make a conjecture that your volleyball team will win. noun 5. ssuming that your sister ate the last cookie is a conjecture. noun 6. You conjecture that your town will build a swimming pool. verb hapter 1 6

6 Key oncept Undefined and Defined Terms Write the correct word from the list on the right. Use each word only once Undefined or Defined Term Diagram Name point line line opposite rays plane point ray segment 9. plane P P 10. segment 11. ray 12. opposite rays, Draw a line from each item in olumn to its description in olumn. z olumn olumn 13. plane HGE intersection of and line z 14. plane EH 15. plane DE line through points and E 16. line y intersection of planes and G G 17. point plane containing points E,, and G Postulates 1 1, 1 2, 1 3, and omplete each postulate with line, plane, or point. Postulate 1-1 Through any two points there is exactly one 9. Postulate 1-2 If two distinct lines intersect, then they intersect in exactly one 9. Postulate 1-3 If two distinct planes intersect, then they intersect in exactly one 9. D H line point line E y x Postulate 1-4 Through any three noncollinear points there is exactly one 9. plane 7 Lesson 1-2

7 Write P if the statement describes a postulate or U if it describes an undefined term. U P U P P P 19. point indicates a location and has no size. 20. Through any two points there is exactly one line. 21. line is represented by a straight path that has no thickness and extends in two opposite directions without end. 22. If two distinct planes intersect, then they intersect in exactly one line. 23. If two distinct lines intersect, then they intersect in exactly one point. 24. Through any three nontcollinear points there is exactly one plane. Problem 2 Naming Segments and Rays Got It? Reasoning E ) and E ) form a line. re they opposite rays? Explain. or Exercises 25 29, use the line below. E 25. Draw and label points E and. Then draw E ) in one color and E ) in another color. 26. Do E ) and E ) share an endpoint? Yes / No 27. Do E ) and E ) form a line? Yes / No 28. re E ) and E ) opposite rays? Yes / No 29. Explain your answer to Exercise 28. nswers may vary. Sample: The rays point in opposite directions but they do not share an endpoint. Problem 3 inding the Intersection of Two Planes Got It? Each surface of the box at the right represents part of a plane. What are the names of two planes that intersect in * )? 30. ircle the points that are on * ) or in one of the two planes. D E G H 31. ircle another name for plane G. Underline another name for plane E. D G DH GH 32. Now name two planes that intersect in * ). nswers may vary. ccept any variation of E H D G E,, E, E,, G, G, and G. hapter 1 8

8 Problem 4 Using Postulate 1 4 Got It? What plane contains points L, M, and N? Shade the plane. 33. Use the figure below. Draw LM, LN, and MN as dashed segments. Then shade plane LMN. M L M J R N L K Q P J R K Q N P Underline the correct word to complete the sentence. 34. LM, LN, and MN form a triangle / rectangle. 35. Name the plane. nswers may vary. ccept any of LMN, MNL, NLM, MLN, LNM, NML Lesson heck Do you UNDERSTND? re ) and ) the same ray? Explain. Underline the correct symbol to complete each sentence. 36. The endpoint of ) is /. 37. The endpoint of ) is /. 38. Use the line. Draw and label points and. Then draw ) and ). Math Success Need to review 39. re ) and ) the same ray? Explain. No. They point in opposite directions and have different endpoints. heck off the vocabulary words that you understand. point line plane segment ray postulate axiom Rate how well you understand points, lines, and planes Explanations may vary. Sample: Now I get it! 9 Lesson 1-2

9 1-3 Measuring Segments Vocabulary Review Draw an example of each. nswers may vary. Samples are shown. 1. point 2. * ) 3. D ) D Vocabulary uilder segment (noun) SEG munt Definition: segment is part of a line that consists of two endpoints and all points between them. segment HJ H J Main Idea: You name a segment by its endpoints. Use Your Vocabulary omplete each sentence with endpoint, endpoints, line, or points. 4. ray has one line contains infinitely many segment has two segment is part of a 9. endpoint points endpoints line Place a check if the phrase describes a segment. Place an if it does not. 8. Earth s equator 9. the right edge of a book s cover 10. one side of a triangle Postulate 1 5 Ruler Postulate Every point on a line can be paired with a real number, called the coordinate of the point. hapter 1 10

10 Problem 1 Measuring Segment Lengths Got It? What are UV and SV on the number line? 11. Label each point on the number line with its coordinate. S U V P P P P P P P P 12. ind UV and SV. Write a justification for each statement. UV Definition of distance SV UV 5 24 Subtract. SV UV 5 4 ind the absolute value. SV 5 18 Postulate 1 6 Segment ddition Postulate If three points,, and are collinear and is between and, then 1 5. Given points,, and are collinear and is between and, complete each equation and 5 4, so and and 5 7, so and 5 5. Problem 2 Using the Segment ddition Postulate Got It? In the diagram, JL What are JK and KL? 15. Write a justification for each statement. JK 1 KL 5 JL (4x 1 6) 1 (7x 1 15) x x 5 99 x 5 9 Segment ddition Postulate Substitute. Simplify. Subtract 21 from each side. Divide each side by 11. J 4x 6 7x You know that JK 5 4x 1 6 and KL 5 7x Use the value of x from Exercise 15 to to find JK and KL. ind JK and KL. K L 4(9) and 7(9) JK 5 42 and KL Lesson 1-3

11 Problem 3 omparing Segment Lengths Got It? Use the diagram below. Is congruent to DE? D E In Exercises 18 and 19, circle the expression that completes the equation j u u u u u u 19. DE 5 j u u u u 20. fter simplifying, 5 5 and DE Is congruent to DE? Explain. Explanations may vary. Sample: No. Segments with different lengths are not congruent. The midpoint of a segment is the point that divides the segment into two congruent segments. Use the number line below for Exercises D E G H I J K Point is halfway between points and J. 23. The midpoint of E is point. 24. Point H divides EK into two congruent segments. 25. ind the midpoint of each segment. Then write the coordinate of the midpoint. Midpoint oordinate G DH K 26. ind the coordinate of the midpoint of each segment. oordinate of midpoint segment with segment with endpoints at 24 and 2 endpoints at 22 and ircle the expression that relates the coordinate of the midpoint to the coordinates of the endpoints. x 1 1 x 2 D (x 1 1 x 2 ) 2 (x 1 2 x 2 ) 2 hapter 1 12

12 Problem 4 Using the Midpoint Got It? U is the midpoint of TV. What are TU, UV, and TV? 28. Use the justifications at the right to complete the steps below. 8x 11 12x 1 T U V Step 1 ind x. TU 5 UV Definition of midpoint 8x x 2 1 Substitute. 8x x dd 1 to each side x Subtract 8x from each side. 3 5 x Divide each side by 4. Step 2 ind TU and UV. TU 5 8? Substitute 3 for x. UV 5 12? Substitute. Step 3 ind TV. TV 5 TU 1 UV Definition of midpoint Substitute Simplify. Lesson heck Do you UNDERSTND? Vocabulary Name two segment bisectors of PR. Underline the correct word or symbol to complete each sentence. P Q R S T 29. bisector / midpoint may be a point, line, ray, or segment The midpoint of PR is point P / Q / R. 31. Line l passes through point P / Q / R. 32. Two bisectors of PR are Q and <. Math Success heck off the vocabulary words that you understand. congruent segments coordinate midpoint segment bisector Rate how well you can find lengths of segments. Need to review Now I get it! 13 Lesson 1-3

13 1-4 Measuring ngles Vocabulary Review Write T for true or for false. 1. ) names a ray with endpoints and. T 2. You name a ray by its endpoint and another point on the ray. Vocabulary uilder angle (noun, verb) NG gul Other Word orms: angular (adjective), angle (verb), angled (adjective) Definition: n angle is formed by two rays with the same endpoint. Use Your Vocabulary Name the rays that form each angle. 3. Definition n angle is formed by two rays with the same endpoint. The rays are the sides of the angle. The endpoint is the vertex of the angle. 4. ) ) ) ) and and Key oncept ngle How to Name It You can name an angle by its vertex a point on each ray and the vertex a number Diagram 1 hapter 1 14

14 or Exercises 5 8, use the diagram in the Take Note on page 14. Name each part of the angle. 5. the vertex 6. two points that are NOT the vertex 7. the sides and and 8. Name the angle three ways. by its vertex by a point on each side and the vertex by a number ) ) l l l1 Problem 1 Naming ngles Got It? What are two other names for lkml? 9. ross out the ray that is NOT a ray of /KML. MK ) MJ ) ML ) 10. ircle all the possible names of /KML. J K 1 2 L M /1 /2 /JKL /JMK /JML /KMJ /LMK Key oncept Types of ngles 11. Draw your own example of each type of angle. ngles may vary. Samples are given. acute right obtuse straight 0, x, 90 x , x, 180 x In the diagram, ml 5 70 and mle Describe each angle as acute, right, obtuse or straight. Give an angle measure to support your description. 12. / 13. /D 14. /G 15. /H acute; 708 obtuse; 1108 right; 908 straight; 1808 E H D G 15 Lesson 1-4

15 Problem 2 Measuring and lassifying ngles Got It? What are the measures of /LKH, /HKN, and /MKH in the art below? lassify each angle as acute, right, obtuse, or straight. J L M H inches Write the measure of each angle. Then classify each angle. K /LKH /HKN /MKH N acute straight obtuse Problem 3 Using ongruent ngles Got It? Use the photo at the right. If m/ 5 49, what is m/de? 17. / has 1 angle mark(s). 18. The other angle with the same number of marks is / DE. 19. Underline the correct word to complete the sentence. The measure of / and the measure of the angle in Exercise 18 are equal / unequal. 20. m/de 5 49 Postulate 1 8 ngle ddition Postulate If point is in the interior of /O, then m/o 1 m/o 5 m/o. 21. Draw /T with point L in the interior and /L and /LT. ngles may vary. Sample: 22. omplete: m/l 1 m/ LT 5 m/ T D E J G L T H K L M hapter 1 16

16 Problem 4 Using the ngle ddition Postulate Got It? /DE is a straight angle. What are m/de and m/e? (11x 12) (2x 10) 23. Write a justification for each statement. D E m/de straight angle measures m/de 1 m/e (11x 2 12) 1 (2x 1 10) x x x 5 14 ngle ddition Postulate Substitute. Simplify. dd 2 to each side. Divide each side by Use the value of x to find m/de and m/e. m/de 5 11x ( 14 ) m/e 5 2x (14) Lesson heck Do you know How? lgebra If mld 5 85, what is an expression to represent ml? 25. Use the justifications at the right to complete the statements below. m/ 1 m/d 5 m/d m/ 1 x 5 85 Substitute. Math Success ngle ddition Postulate m/ 1 x 2 x x Subtract x from each side. Need to review m/ 5 852x heck off the vocabulary words that you understand Simplify. Now I get it! 1 x acute angle obtuse angle right angle straight angle Rate how well you can classify angles. D 17 Lesson 1-4

17 1-5 Exploring ngle Pairs Vocabulary Review Use a word from the list below to complete each sentence. Use each word just once. interior rays vertex 1. The 9 of an angle is the region containing all of the points between the two sides of the angle. 2. When you use three points to name an angle, the 9 must go in the middle. 3. The sides of /QRS are 9 RS and RQ. interior vertex rays Use the figure below for Exercises 4 7. Identify each angle as acute, right, obtuse, or straight. 4. /SRV 5. /TRS obtuse 6. /TRQ 7. /VRQ right Vocabulary uilder conclusion (noun) kun KLOO zhun Other Word orms: conclude (verb) Definition: conclusion is the end of an event or the last step in a reasoning process. Use Your Vocabulary acute straight omplete each sentence with conclude or conclusion. 8. If it rains, you can 9 that soccer practice will be canceled. 9. The last step of the proof is the 9. S T U Q R V conclude conclusion hapter 1 18

18 Key oncept Types of ngle Pairs ngle Pair djacent angles Vertical angles omplementary angles Supplementary angles Definition Two coplanar angles with a common side, a common vertex, and no common interior points Two angles whose sides are opposite rays Two angles whose measures have a sum of 90 Two angles whose measures have a sum of 180 Draw a line from each word in olumn to the angles it describes in olumn. olumn olumn 10. supplementary /1 and /2 11. adjacent /2 and /3 12. vertical /2 and /5 13. complementary /3 and / Problem 1 Identifying ngle Pairs Got It? Use the diagram at the right. re le and ld vertical angles? Explain. 14. The rays of /E are E ) and ) / ). 15. The rays of /D are ) and D ) / ). omplete ) each ) statement. 16. E ) and ) are opposite rays. 17. and D are opposite rays. 18. re /E and /D vertical angles? Yes / No Problem 2 Making onclusions rom a Diagram Got It? an you conclude that TW O WV from the diagram? Explain. 19. ircle the items marked as congruent in the diagram. PW and WQ /TWQ and /PWT TW and WV /TWQ and /VWQ P E 20. an you conclude that TW > WV? Why or why not? Explanations may vary. Sample: T W V 28 Q D ongruence marks are on TW and WV. 19 Lesson 1-5

19 Postulate 1 9 Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 21. If / and / form a linear pair, then m/ 1 m/ Problem 3 inding Missing ngle Measures Got It? Reasoning lkpl and ljpl are a linear pair, mlkpl 5 2x 1 24, and mljpl 5 4x How can you check that mlkpl 5 64 and mljpl 5 116? 22. What is one way to check solutions? Place a in the box if the response is correct. Place an in the box if it is incorrect. L (2x 24) (4x 36) K P J Draw a diagram. If it looks good, the solutions are correct. Substitute the solutions in the original problem statement. 23. Use your answer(s) to Exercise 22 to check the solutions. mlkpl x x 5 40 x 5 20 ml JPL x x 5 80 x How does your check show that you found the correct angle measurements? nswers may vary. Sample: You solved correctly because you found the same solution to both equations. Problem 4 Using an ngle isector to ind ngle Measures ) Got It? KM bisects ljkl. If mljkl 5 72, what is mljkm? 25. Write a justification for each step. m/jkm 5 m/mkl Definition of angle bisector m/jkm 1 m/mkl 5 m/jkl ngle ddition Postulate 2m/JKM 5 m/jkl Substitute. Overmatter m/jkm m/jkl Divide each side by 2. hapter 1 20

20 26. omplete. m/jkl 5 72, so m/jkm Now complete the diagram below. J 36 M 36 K L Lesson heck Do you UNDERSTND? Error nalysis Your friend calculated the value of x below. What is her error? 4x + 2x = 180 6x = 180 x = 30 4x 2x 28. ircle the best description of the largest angle in the figure. acute obtuse right straight 29. omplete: 4x 1 2x What is your friend s error? Explain. nswers may vary. Sample: She thought a right angle measures 1808, so she set the sum of the angle measures equal to 180. Math Success heck off the vocabulary words that you understand. angle complementary supplementary angle bisector vertical Rate how well you can find missing angle measures. Need to review Now I get it! 21 Lesson 1-5

21 1-6 asic onstructions Vocabulary Review Draw a line from each word in olumn to its symbol or picture in olumn. olumn 1. congruent 2. point olumn W S 3. ray G 4. vertex 5. intersection of segments O W P Vocabulary uilder perpendicular (adjective) pur pun DIK yoo lur Definition: Perpendicular means at right angles to a given line or plane. Example: Each corner of this paper is formed by perpendicular edges of the page. Non-Examples: cute, obtuse, and straight angles do not have perpendicular rays. Use Your Vocabulary 6. ircle the figure that shows perpendicular segments. s s hapter 1 22

22 Problem 1 onstructing ongruent Segments Got It? Use a straightedge to draw XY. Then construct RS so that RS 5 2XY. 7. student did the construction at the right. Describe each step of the construction. Step 1 Use a straightedge to draw XY. X Y Step 2 Draw a ray with endpoint R. Step 3 Draw an arc with compass point at R and opening XY. R S Step 4 Step 5 Draw another arc with the compass point at the intersection. Label the point of intersection S. Problem 2 onstructing ongruent ngles Got It? onstruct l so that ml 5 2ml at the right. 8. Use arc or compass to complete the sentence(s) in each step. In the large box, construct /. Step 1 Use a straightedge to construct a ray with endpoint. Step 6 Draw R. R Step 5 Use the same compass setting. Put the? point on point T. Draw an? and label its intersection with the first? as point R. compass / arc / arc T Step 2 With your? point on vertex, draw a(n)? that intersects both sides of. Label the points of intersection and. compass / arc S Step 3 Use the same compass setting. Put the? point on point. Draw a long? and label its intersection with the ray as S. compass / arc Step 4 Open the? to the length of. With the compass point on point S, draw an?. Label where this arc intersects the other arc as point T. compass / arc 23 Lesson 1-6

23 perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint. 9. ircle the drawing that shows the perpendicular bisector of a segment. E E E Problem 3 onstructing the Perpendicular isector Got It? Draw ST. onstruct its perpendicular bisector. 10. Error nalysis student s construction of the perpendicular bisector of ST is shown below. Describe the student s error. S T nswers may vary. Sample: The student did not make the opening of the arc drawn from points S and T greater than 1 2 ST. 11. Do the construction correctly in the box below. X S M T Y hapter 1 24

24 Problem 4 onstructing the ngle isector Got It? Draw obtuse lxyz. Then construct its bisector YP ). 12. Obtuse /XYZ is drawn in the box at the right. omplete the flowchart and do each step of the construction. X P Step 1 Put the compass point on vertex. Draw an arc that intersects the sides of Y. Label the points of Y intersection and. Y Z Step 2 Put the compass point on point and draw an arc. With the same / a different compass setting, draw an arc using point. e sure the arcs intersect. Label the point where the two arcs intersect P. Step 3 Draw YP. Lesson heck Do you UNDERSTND? Vocabulary What two tools do you use to make constructions? Draw a line from each task in olumn to the tool used in olumn. olumn olumn 13. measure lines compass 14. measure angles protractor 15. construct arcs ruler 16. construct lines straightedge Math Success heck off the vocabulary words that you understand. straightedge compass construction perpendicular bisector Rate how well you can construct angles and bisectors. Need to review Now I get it! 25 Lesson 1-6

25 1-7 Midpoint and Distance in the oordinate Plane Vocabulary Review Use the figure at the right for Exercises 1 6. Write T for true or for false. 1. Points and are both at the origin. y T 2. If 5, then is the midpoint of The midpoint of E is. D 4. The Pythagorean Theorem can be used for any triangle. 5. Point is at (6, 0). 10 E 5 x 6. Point E has a y-coordinate of 28. Vocabulary uilder midpoint (noun) MID poynt Definition: midpoint of a segment is a point that divides the segment into two congruent segments. Use Your Vocabulary Use the figure at the right for Exercises The midpoint of E is G( 0, 2.5 ). 8. The midpoint of is ( 0, 0 ), or the origin. 9. The midpoint of D is ( 0.5, 0 ). 4 4 E 2 4 y G D O 4 x hapter 1 26

26 Key oncept Midpoint ormulas On a Number Line The coordinate of the midpoint M of a b with endpoints at a and b is. 2 In the oordinate Plane Given (x 1, y 1 ) and (x 2, y 2 ), the coordinates of the x ( 1 x 2 y 1 y 2 midpoint of are M, 2 2 ) ind the coordinate of the midpoint M of each segment with the given endpoints on a number line. 10. endpoints 5 and endpoints 23 and endpoints 210 and endpoints 28 and omplete the diagram below O y ( 4) 2 2 (1 ) 17 2 (17, 4) (, ) 9 3 (1, 2) x Problem 2 Endpoint oordinates ( 3, 5 ) x x inding an Endpoint Got It? The midpoint of has coordinates (4, 29). Endpoint has coordinates (23, 25). What are the coordinates of? 15. omplete the equations below. x Midpoint ormula x y (, 2 2 ) Solve two equations. Midpoint oordinates y ( 4, 9 ) 5 y y The coordinates of endpoint are ( 11, 13 ). 27 Lesson 1-7

27 ormula The Distance ormula The distance between two points (x 1, y 1 ) and (x 2, y 2 ) is d 5 "(x 2 2 x 1 ) 2 1 (y 2 2 y 1 ) 2. The Distance ormula is based on the Pythagorean Theorem. y y 2 d y 2 y 1 y 1 x 2 x 1 x O x 1 x 2 c a a 2 b 2 c 2 b Use the diagrams above. Draw a line from each triangle side in olumn to the corresponding triangle side in olumn. olumn olumn 17. y 2 2 y 1 a 18. x 2 2 x 1 b 19. distance, d c Problem 3 inding Distance S( 2, 14) y Got It? SR has endpoints S(22, 14) and R(3, 21). What is SR to the nearest tenth? 20. omplete the diagram at the right. 21. Let S(22, 14) be (x 1, y 1 ) and let R(3, 21) be (x 2, y 2 ). Use the justifications and complete the steps below to find SR (22) 5 d 5 Q x 2 x Ä 1 R 2 1 Q 2 y 1 R 2 2 y 2 Use the Distance ormula. SR 5 Q 3 2 (22)R 2 1 Q R 2 Substitute. Ä 5 Q 5 R 2 1 Q 15 R 2 Subtract. Ä Simplify powers. Ä dd. Ä R(3, 1) x < 15.8 Use a calculator. hapter 1 28

28 Problem 4 inding Distance Got It? On a zip-line course, you are harnessed to a cable that travels through the treetops. You start at Platform and zip to each of the other platforms. How far do you travel from Platform D to Platform E? Each grid unit represents 5 m. y O D x E 22. The equation is solved below. Write a justification for each step. d 5 "(x 2 2 x 1 ) 2 1 (y 2 2 y 1 ) 2 DE 5 "( ) 2 1 ( ) 2 5 " (235) 2 5 " "1325 Use the Distance ormula. Substitute. Simplify. 23. To the nearest tenth, you travel about 36.4 m. Lesson heck Do you UNDERSTND? Reasoning How does the Distance ormula ensure that the distance between two different points is positive? 24. radical symbol with no sign in front of it indicates a positive / negative square root. 25. Now answer the question. nswers may vary. Sample: The radical in the Distance ormula represents a positive square root. Math Success heck off the vocabulary words that you understand. midpoint distance coordinate plane Rate how well you can use the Midpoint and Distance ormulas. Need to review Now I get it! 29 Lesson 1-7

29 1-8 Perimeter, ircumference, and rea Vocabulary Review 1. ross out the shapes that are NOT polygons. 2. Write the name of each figure. Use each word once. triangle square rectangle circle circle rectangle square triangle Vocabulary uilder consecutive (adjective) kun SEK yoo tiv Definition: onsecutive means following in order without interruption. Related Word: sequence Example: The numbers 2, 4, 6, 8,... are consecutive even numbers. Non-Example: The numbers 1, 3, 2, 5, 4,... are NOT consecutive numbers. Use Your Vocabulary Draw a line from each sequence of letters in olumn to the next consecutive letter in olumn. olumn 3. L, M, N, O,... R 4. V, U, T, S,... I 5.,, E, G. P olumn hapter 1 30

30 Key oncept Perimeter, ircumference, and rea 6. Label the parts of each of the figures below. Square Triangle Rectangle ircle s s a b c h h b d r P 5 4s P 5 a 1 b 1 c P 5 2b 1 2h 5pd or 5 2pr 5 s bh 5 bh 5pr2 Problem 1 inding the Perimeter of a Rectangle Got It? You want to frame a picture that is 5 in. by 7 in. with a 1-in.-wide frame. What is the perimeter of the picture? 7. The picture is 5 in. by 7 in. 8. ircle the formula that gives the perimeter of the picture. P 5 4s P 5 2b 1 2h P 5 a 1 b 1 c 5pd 9. Solve using substitution. P 5 2b 1 2h 5 2(5) 1 2(7) The perimeter of the picture is 24 in. Problem 2 inding ircumference Got It? What is the circumference of a circle with radius 24 m in terms of π? 11. Error nalysis t the right is one student s solution. What error did the student make? nswers may vary. Sample: The student used a diameter of 24 m instead of a radius of 24 m. 12. ind the correct circumference. 5 2pr 5 2p(24) 5 48p 24 m c = πd c = π(24) c = 24π 31 Lesson 1-8

31 Problem 3 inding Perimeter in the oordinate Plane Got It? Graph quadrilateral JKLM with vertices J(23, 23), K(1, 23), L(1, 4), and M(23, 1). What is the perimeter of JLKM? 13. Graph the quadrilateral on the coordinate plane at the right. 14. Use the justifications at the right to find the length of each side. JK 5 P P 5 4 Simplify. Use the Ruler Postulate. KL 5 P P Use the Ruler Postulate. 5 7 Simplify. 5 M 4 3 J 2 y 5 4 L O K 4 5 x JM 5 P P Use the Ruler Postulate. 5 4 Simplify. ML 5 (1 2 (23)) 2 1 (4 2 1 ) 2 Use the Distance ormula. Ä 5 ( 4 ) Simplify within parentheses. Ä 5 ( 16 ) 1 ( 9 ) Simplify powers. Ä 5 ( 25 ) dd. Ä 5 5 Take the square root. 15. dd the side lengths to find the perimeter. JK 1 KL 1 JM 1 ML The perimeter of JKLM is 20 units. Problem 5 inding rea of a ircle Got It? The diameter of a circle is 14 ft. What is its area in terms of p? 17. Label the diameter and radius of the circle at the right. 18. Use the formula 5pr 2 to find the area of the circle in terms of p. 5pr 2 5p(7) p 19. The area of the circle is 49 p ft 2. Key oncept Postulate 1 10 rea ddition Postulate ft 7 ft 20. The area of a region is the sum / difference of the areas of its nonoverlapping parts. hapter 1 32

32 2 3 Problem 6 inding rea of an Irregular Shape Got It? Reasoning The figure below shows one way to separate the figure at the left. What is another way to separate the figure? 3 cm 9 cm 1 3 cm 3 cm 2 6 cm 3 9 cm 3 cm 3 cm 21. Draw segments to show two different ways to separate the figure. Separate the left-hand figure into three squares. Drawings will vary. Samples are given. 3 cm 3 cm 1 9 cm 9 cm Lesson heck Do you UNDERSTND? ompare and ontrast Your friend can t remember whether 2pr computes the circumference or the area of a circle. How would you help your friend? Explain. 22. Underline the correct word(s) to complete each sentence. rea involves units / square units. ircumference involves units / square units. The formula 2pr relates to area / circumference because it involves units / square units. Math Success heck off the vocabulary words that you understand. perimeter Need to review area Rate how well you can find the area of irregular shapes Now I get it! 33 Lesson 1-8