18.1 Sequences of Transformations

Transcription

1 Name lass ate 1.1 Sequences of Transformations ssential Question: What happens when ou appl more than one transformation to a figure? plore ombining Rotations or Reflections transformation is a function that takes points on the plane and maps them to other points on the plane. Transformations can be applied one after the other in a sequence where ou use the image of the first transformation as the preimage for the net transformation. Resource ocker ind the image for each sequence of transformations. Using geometr software, draw a triangle and label the vertices,, and. Then draw a point outside the triangle and label it P. Rotate 3 around point P and label the image as. Then rotate 5 around point P and label the image as. Sketch our result. Make a conjecture regarding a single rotation that will map to. heck our conjecture, and describe what ou did. Houghton Mifflin Harcourt Publishing ompan Using geometr software, draw a triangle and label the vertices,, and. Then draw two intersecting lines and label them j and k. Reflect across line j and label the image as. Then reflect across line k and label the image as. Sketch our result. onsider the relationship between and. escribe the single t ransformation that maps to. How can ou check that ou are correct? Module 1 5 esson 1

2 Reflect 1. Repeat Step using other angle measures. Make a conjecture about what single transformation will describe a sequence of two rotations about the same center.. Make a conjecture about what single transformation will describe a sequence of three rotations about the same center. 3. iscussion Repeat Step, but make lines j and k parallel instead of intersecting. Make a conjecture about what single transformation will now map to. heck our conjecture and describe what ou did. plain 1 ombining Rigid Transformations In the plore, ou saw that sometimes ou can use a single transformation to describe the result of appling a sequence of two transformations. Now ou will appl sequences of rigid transformations that cannot be described b a single transformation. ample 1 raw the image of after the given combination of transformations. Reflection over line l then translation along v ν Step 1 raw the image of after a reflection across line l. abel the image. ν l Step Translate along v. abel this image. ν l l Houghton Mifflin Harcourt Publishing ompan Module 1 esson 1

3 1 rotation around point P, then translation along v, then reflection across line l ppl the rotation. abel the image. ppl the translation to. abel the image. P l ppl the reflection to. abel the image. ν Reflect. re the images ou drew for each eample the same size and shape as the given preimage? In what was do rigid transformations change the preimage? 5. oes the order in which ou appl the transformations make a difference? Test our conjecture b performing the transformations in Part in a different order.. or Part, describe a sequence of transformations that will take back to the preimage. Houghton Mifflin Harcourt Publishing ompan Your Turn raw the image of the triangle after the given combination of transformations. 7. Reflection across l then 9 rotation around. Translation along v then 1 rotation around point P point P then translation along u P l G P u v Module 1 7 esson 1

4 plain ombining Nonrigid Transformations ample raw the image of the figure in the plane after the given combination of transformations. (, ) ( 3_, 3_ ) (-, ) ( + 1, - ) 1. The first transformation is a dilation b a factor of 3. ppl the dilation. abel the image.. ppl the reflection of across the -ais. abel this image. 3. ppl the translation of. abel this image ' ' ' ' (, ) (3, ) ( 1_, - 1_ ) 1. The first transformation is a [horizontal/vertical] stretch b a factor of. ppl the stretch. abel the image.. The second transformation is a dilation b a factor of combined with a reflection. ppl the transformation to image.. abel the Reflect 9. If ou dilated a figure b a factor of, what transformation could ou use to return the figure back to its preimage? If ou dilated a figure b a factor of and then translated it right units, write a sequence of transformations to return the figure back to its preimage. 1. student is asked to reflect a figure across the -ais and then verticall stretch the figure b a factor of. escribe the effect on the coordinates. Then write one transformation using coordinate notation that combines these two transformations into one. - Houghton Mifflin Harcourt Publishing ompan Module 1 esson 1

5 Your Turn raw the image of the figure in the plane after the given combination of transformations. 11. (, ) ( - 1, - 1) (3, ) (-, -) 1. (, ) ( 3_, - ) ( - 5, + ) plain 3 Predicting the ffect of Transformations ample 3 Predict the result of appling the sequence of transformations to the given figure. MN is translated along the vector -, 3, reflected across the -ais, and then reflected across the -ais. Houghton Mifflin Harcourt Publishing ompan Predict the effect of the first transformation: translation along the vector -, 3 will move the figure left units and up 3 units. Since the given triangle is in Quadrant II, the translation will move it further from the - and -aes. It will remain in Quadrant II. - Predict the effect of the second transformation: Since the triangle is in Quadrant II, a reflection across the -ais will change the orientation and move the triangle into Quadrant I. - M - N Predict the effect of the third transformation: reflection across the -ais will again change the orientation and move the triangle into Quadrant IV. The two reflections are the equivalent of rotating the figure 1 about the origin. The final result will be a triangle the same shape and size as MN in Quadrant IV. It has been rotated 1 about the origin and is farther from the aes than the preimage. Module 1 9 esson 1

6 Square HIJK is rotated 9 clockwise about the origin and then dilated b a factor of, which maps (, ) (, ). Predict the effect of the first transformation: H I - - K - J - Predict the effect of the second transformation: - - The final result will be Your Turn Predict the result of appling the sequence of transformations to the given figure. 13. Rectangle GHJK is reflected across the -ais and translated along the vector 5,. K J TUV is horizontall stretched b a factor of 3, which maps (, ) ( 3, ), and then translated along the vector, U G H Houghton Mifflin Harcourt Publishing ompan T - V - Module 1 9 esson 1

7 laborate 15. iscussion How man different sequences of rigid transformations do ou think ou can find to take a preimage back onto itself? plain our reasoning. 1. Is there a sequence of a rotation and a dilation that will result in an image that is the same size and position as the preimage? plain our reasoning. 17. ssential Question heck-in In a sequence of transformations, the order of the transformations can affect the final image. escribe a sequence of transformations where the order does not matter. escribe a sequence of transformations where the order does matter. Houghton Mifflin Harcourt Publishing ompan Module 1 91 esson 1

8 valuate: Homework and Practice raw and label the final image of after the given sequence of transformations. 1. Reflect over the -ais and then translate b, -3. Online Homework Hints and Help tra Practice. Rotate 9 degrees clockwise about the origin and then reflect over the -ais Translate b,, rotate 9 degrees counterclockwise around, and reflect over the -ais.. Reflect over the -ais, translate b -3, -1, and rotate 1 degrees around the origin Houghton Mifflin Harcourt Publishing ompan - - Module 1 9 esson 1

9 raw and label the final image of after the given sequence of transformations. 5. (, ) (, 1_ 3 ) (-, -). (, ) ( - 3_, _ 3 ) ( +, - ) ( _ 3, - 3_ ) Predict the result of appling the sequence of transformations to the given figure. Houghton Mifflin Harcourt Publishing ompan 7. is translated along the vector -3, -1, reflected across the -ais, and then reflected across the -ais is translated along the vector -1, -3, rotated 1 about the origin, and then dilated b a factor of Module 1 93 esson 1

10 In ercises 9 1, use the diagram. ill in the blank with the letter of the correct image described. 9. is the result of the sequence: G reflected over a vertical line and then a horizontal line. G 1. is the result of the sequence: rotated 9 clockwise around one of its vertices and then reflected over a horizontal line. 11. is the result of the sequence: translated and then rotated 9 counterclockwise. 1. is the result of the sequence: rotated 9 counterclockwise and then translated. hoose the correct word to complete a true statement. 13. combination of two rigid transformations on a preimage will alwas/sometimes/never produce the same image when taken in a different order. 1. double rotation can alwas/sometimes/never be written as a single rotation. 15. sequence of a translation and a reflection alwas/sometimes/never has a point that does not change position. 17. sequence of rigid transformations will alwas/sometimes/never result in an image that is the same size and orientation as the preimage. 1. sequence of a reflection across the -ais and then a reflection across the -ais alwas/sometimes/never results in a 1 rotation of the preimage. 1. sequence of a rotation and a dilation will alwas/sometimes/never result in an image that is the same size and orientation as the preimage. 19. QRS is the image of MN under a sequence of transformations. an each of the following sequences be used to create the image, QRS, from the preimage, MN? Select es or no. a. Reflect across the -ais and then Yes No translate along the vector, -. b. Translate along the vector, - Yes No and then reflect across the -ais. c. Rotate 9 clockwise about the Yes No origin, reflect across the -ais, and then rotate 9 counterclockwise about the origin. d. Rotate 1 about the origin, reflect Yes No across the -ais, and then translate along the vector, -. S M R - - Q - N Houghton Mifflin Harcourt Publishing ompan Module 1 9 esson 1

11 . teacher gave students this puzzle: I had a triangle with verte at (1, ) and verte at (3, ). fter two rigid transformations, I had the image shown. escribe and show a sequence of transformations that will give this image from the preimage. H.O.T. ocus on Higher Order Thinking 1. nalze Relationships What two transformations would ou appl to to get? How could ou epress these transformations with a single mapping rule in the form of (, ) (?,?)? Houghton Mifflin Harcourt Publishing ompan Image redits: JP/ Shutterstock. Multi-Step Muralists will often make a scale drawing of an art piece before creating the large finished version. muralist has sketched an art piece on a sheet of paper that is 3 feet b feet. a. If the final mural will be 39 feet b 5 feet, what is the scale factor for this dilation? b. The owner of the wall has decided to onl give permission to paint on the lower half of the wall. an the muralist simpl use the transformation (, ) 1 (, ) in addition to the scale factor to alter the sketch for use in the allowed space? plain. 3. ommunicate Mathematical Ideas s a graded class activit, our teacher asks our class to reflect a triangle across the -ais and then across the -ais. Your classmate gets upset because he reversed the order of these reflections and thinks he will have to start over. What can ou sa to our classmate to help him? Module 1 95 esson 1

12 esson Performance Task The photograph shows an actual snowflake. raw a detailed sketch of the arm of the snowflake located at the top left of the photo (1: on a clock face). escribe in as much detail as ou can an translations, reflections, or rotations that ou see. Then describe how the entire snowflake is constructed, based on what ou found in the design of one arm. Houghton Mifflin Harcourt Publishing ompan Image redits: Mark assino/superstock Module 1 9 esson 1

13 Name lass ate 1. Proving igures are ongruent Using Rigid Motions ssential Question: How can ou determine whether two figures are congruent? Resource ocker plore onfirming ongruence Two plane figures are congruent if and onl if one can be obtained from the other b a sequence of rigid motions (that is, b a sequence of reflections, translations, and/or rotations). landscape architect uses a grid to design the landscape around a mall. Use tracing paper to confirm that the landscape elements are congruent. Trace planter. escribe a transformation ou can use to move the tracing paper so that planter is mapped onto planter GH. What does this confirm about the planters? H G Trace pools JKM and NPQR. old the paper so that pool JKM is mapped onto pool NPQR. escribe the transformation. What does this confirm about the pools? M J K P Houghton Mifflin Harcourt Publishing ompan etermine whether the lawns are congruent. Is there a rigid transformation that maps MN to? What does this confirm about the lawns? Reflect M N Q R N 1. How do the sizes of the pairs of figures help determine if the are congruent? Module 1 97 esson

14 plain 1 etermining if igures are ongruent ample 1 Use the definition of congruence to decide whether the two figures are congruent. plain our answer. M J The two figures appear to be the same size and shape, so look for a rigid transformation that will map one to the other. - - K You can map onto JKM b reflecting over the -ais. This reflection is a rigid motion that maps to JKM, so the two figures are congruent. - The coordinate notation for the reflection is (, ) (-, ). - Y X The two figures appear to be the same/different Z You can map to XYZ b. This is/is not a rigid motion that maps to XYZ, so the two figures are/are not congruent. - The coordinate notation for the rotation is. Your Turn Use the definition of congruence to decide whether the two figures are congruent. plain our answer Y - Z - X W Z - - Y K J X - Houghton Mifflin Harcourt Publishing ompan Module 1 9 esson

15 plain inding a Sequence of Rigid Motions The definition of congruence tells ou that when two figures are known to be congruent, there must be some sequence of rigid motions that maps one to the other. ample The figures shown are congruent. ind a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations ou use. PQR JKM Q R P - M K J - Map to PQR with a rotation of 1 around the origin, followed b a horizontal translation. Rotation: (, ) (-, -) Translation: (, ) ( + 1, ) Map to JKM with a followed b a. : (, ) : (, ), Reflect. How is the orientation of the figure affected b a sequence of transformations? Houghton Mifflin Harcourt Publishing ompan Your Turn The figures shown are congruent. ind a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations ou use. 5. JKM WXYZ. PQRST M K J S W - X T - R P Z Y Q Module 1 99 esson

16 plain 3 Investigating ongruent Segments and ngles ongruence can refer to parts of figures as well as whole figures. Two angles are congruent if and onl if one can be obtained from the other b rigid motions (that is, b a sequence of reflections, translations, and/or rotations.) The same conditions are required for two segments to be congruent to each other. ample 3 etermine which angles or segments are congruent. escribe transformations that can be used to verif congruence. and are congruent. The transformation is a translation. There is no transformation that maps to either of the other angles. _ and are congruent. sequence of transformations is a and a translation. There is no transformation that maps to either of the other segments. Your Turn 7. etermine which segments and which angles are congruent. escribe transformations that can be used to show the congruence. G H laborate. an ou sa two angles are congruent if the have the same measure but the segments that identif the ras that form the angle are different lengths? 9. iscussion an figures have congruent angles but not be congruent figures? Houghton Mifflin Harcourt Publishing ompan 1. ssential Question heck-in an ou use transformations to prove that two figures are not congruent? Module 1 9 esson

17 valuate: Homework and Practice Use the definition of congruence to decide whether the two figures are congruent. plain our answer. Give coordinate notation for the transformations ou use. Online Homework Hints and Help tra Practice K J W Z - - X Y - G 3.. P T -Q - R - S - X Z - - Y - N - M Houghton Mifflin Harcourt Publishing ompan 5. P Q S - R - Module 1 91 esson

18 The figures shown are congruent. ind a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations ou use.. RSTU WXYZ 7. T S - R U - - Z W Y X GH PQRST 9. WXY G H W - - S - - T X - R Y P - Q etermine which of the angles are congruent. Which transformations can be used to verif the congruence? Houghton Mifflin Harcourt Publishing ompan Module 1 9 esson

19 etermine which of the segments are congruent. Which transformations can be used to verif the congruence? Use the definition of congruence to decide whether the two figures are congruent. plain our answer. Give coordinate notation for the transformations ou use. 1. W 15. X Y J K K J N M Houghton Mifflin Harcourt Publishing ompan 1. R U 17. S T - G H - K X Y W M - Module 1 93 esson

20 The figures shown are congruent. ind a sequence of transformations for the indicated mapping. Give coordinate notation for the transformations ou use. 1. Map PQRST to GH. 19. Map WXYZ to JKM. T S P M - - W Q - G H Z R - X Y J K. Map PQRSTU to. 1. Map to KM. - R Q S P T U M - K -. etermine whether each pair of angles is congruent or not congruent. Select the correct answer for each lettered part. a. and ongruent Not congruent b. and ongruent Not congruent c. and ongruent Not congruent d. and ongruent Not congruent e. and ongruent Not congruent Houghton Mifflin Harcourt Publishing ompan Module 1 9 esson

21 3. If and WXYZ are congruent, then can be mapped to WXYZ using a rotation and a translation. etermine whether the statement is true or false. Then eplain our reasoning. Y X W Z. Which segments are congruent? Which are not congruent? plain. 5. Which angles are congruent? Which are not congruent? plain. G H. The figures shown are congruent. ind a sequence of transformations that will map G to QRSTU. Give coordinate notation for the transformations ou use. 7. The figures shown are congruent. ind a sequence of transformations that will map MN to XYZ. Give coordinate notation for the transformations ou use. Houghton Mifflin Harcourt Publishing ompan G - -U T S Q R Y. Which sequence of transformations does not map a figure onto a congruent figure? plain.. Rotation of 1 about the origin, reflection across the -ais, horizontal translation (, ) ( +, ). Reflection across the -ais, combined translation (, ) ( -5, + ). Rotation of 1 about the origin, reflection across the -ais, dilation (, ) (, ). ounterclockwise rotation of 9 about the origin, reflection across the -ais, combined translation (, ) ( -11, - 1) Z - N M X Module 1 95 esson

22 9. The figures shown are congruent. ind a sequence of transformations that will map GH to VWXYZ. Give coordinate notation for the transformations ou use. H G Z - - W Y V X 3. How can ou prove that two arrows in the reccling smbol are congruent to each other? 31. The cit of St. ouis was settled b the rench in the mid 17s and joined the United States in 13 as part of the ouisiana Purchase. The cit flag reflects its rench histor b featuring the fleur-de-lis. How can ou prove that the left and right petals are congruent to each other? 3. raw onclusions Two students are tring to show that the two figures are congruent. The first student decides to map G to PQRST using a rotation of 1 around the origin, followed b the translation (, ) (, + ). The second student believes the correct transformations are a reflection across the -ais, followed b the vertical translation (, ) (, - ). re both students correct, is onl one student correct, or is neither student correct? T S P Q - - R G - Houghton Mifflin Harcourt Publishing ompan Image redits: Imagelub/Gett Images; tlaspi/shutterstock - Module 1 9 esson

23 33. Justif Reasoning Two students are tring to show that the two figures are congruent. The first student decides to map G to RSTU using a rotation of 1 about the origin, followed b the vertical translation (, ) (, + ). The second student uses a reflection across the -ais, followed b the vertical translation (, ) (, + ), followed b a reflection across the -ais. re both students correct, is onl one student correct, or is neither student correct? R S T G U H.O.T. ocus on Higher Order Thinking 3. ook for a Pattern ssume the pattern of congruent squares shown in the figure continues forever. Write rules for rigid motions that map square onto square 1, square onto square, and square onto square Write a rule for a rigid motion that maps square - onto square n. Houghton Mifflin Harcourt Publishing ompan 35. nalze Relationships Suppose ou know that is congruent to and that is congruent to GHJ. an ou conclude that is congruent to GHJ? plain. 3. ommunicate Mathematical Ideas lla plotted the points (, ), (, ), and (, ). Then she drew _ and _. Give two different arguments to eplain wh the segments are congruent. Module 1 97 esson

24 esson Performance Task The illustration shows how nine congruent shapes can be fitted together to form a larger shape. ach of the shapes can be formed from Shape #1 through a combination of translations, reflections, and/or rotations escribe how each of Shapes 9 can be formed from Shape #1 through a combination of translations, reflections, and/or rotations. Then design a figure like this one, using at least eight congruent shapes. Number the shapes. Then describe how each of them can be formed from Shape #1 through a combination of translations, reflections, and/or rotations. Houghton Mifflin Harcourt Publishing ompan Module 1 9 esson

25 Name lass ate 1.3 orresponding Parts of ongruent igures re ongruent ssential Question: What can ou conclude about two figures that are congruent? Resource ocker plore ploring ongruence of Parts of Transformed igures You will investigate some conclusions ou can make when ou know that two figures are congruent. old a sheet of paper in half. Use a straightedge to draw a triangle on the folded sheet. Then cut out the triangle, cutting through both laers of paper to produce two congruent triangles. abel them and, as shown. Houghton Mifflin Harcourt Publishing ompan Place the triangles net to each other on a desktop. Since the triangles are congruent, there must be a sequence of rigid motions that maps to. escribe the sequence of rigid motions. The same sequence of rigid motions that maps to maps parts of to parts of. omplete the following. _ What does Step tell ou about the corresponding parts of the two triangles? Wh? Module 1 99 esson 3

26 Reflect 1. If ou know that, what si congruence statements about segments and angles can ou write? Wh?. o our findings in this plore appl to figures other than triangles? or instance, if ou know that quadrilaterals JKM and PQRS are congruent, can ou make an conclusions about corresponding parts? Wh or wh not? M J K S P Q R plain 1 orresponding Parts of ongruent igures re ongruent The following true statement summarizes what ou discovered in the plore. orresponding Parts of ongruent igures re ongruent If two figures are congruent, then corresponding sides are congruent and corresponding angles are congruent. ample 1. ind the given side length or angle measure. m Step 1 ind the side that corresponds to _. Since,. Step ind the unknown length. =, and =. cm, so =. cm. Step 1 ind the angle that corresponds to. Since,.. cm 3.7 cm 3.5 cm 73 5 Houghton Mifflin Harcourt Publishing ompan Step ind the unknown angle measure. m = m, and m =, so m =. Module 1 91 esson 3

27 Reflect 3. iscussion The triangles shown in the figure are congruent. an ou conclude that JK QR? plain. K Q R J P Your Turn STU VWX. ind the given side length or angle measure. 1 S 1 ft T W 3 3 ft 3 ft 1 X. SU 5. m S U V plain ppling the Properties of ongruence Rigid motions preserve length and angle measure. This means that congruent segments have the same length, so UV XY implies UV = XY and vice versa. In the same wa, congruent angles have the same measure, so J K implies m J = m K and vice versa. Houghton Mifflin Harcourt Publishing ompan Properties of ongruence Refleive Propert of ongruence Smmetric Propert of ongruence Transitive Propert of ongruence ample If, then. If and, then.. ind the given side length or angle measure. Since,. Therefore, =. Write an equation. Subtract 3 from each side. 3 + = 5 = (3 + ) in. (5 + 11) 5 in. ( + ) (5) in. 3 ivide each side b. = So, = 3 + = 3 () + = 1 + = in. Module esson 3

28 m Since,. Therefore, m = m. Write an equation. 5 + = + Subtract 5 from each side. 11 = + Subtract from each side. = So, m = ( + ) = ( + ) =. Your Turn Quadrilateral GHJK quadrilateral MNP. ind the given side length or angle measure. G ( + 3) cm (9 + 17) H P 1 cm (1) K ( 13) cm J N (11-1) M. M 7. m H plain 3 ample 3 Using ongruent orresponding Parts in a Proof Write each proof. Given: Prove: is the midpoint of _. Statements Given. Reasons. orresponding parts of congruent figures are congruent. Houghton Mifflin Harcourt Publishing ompan 3. is the midpoint of _. 3. efinition of midpoint. Module 1 91 esson 3

29 Given: Quadrilateral JKM quadrilateral NPQR; J K J K R Q Prove: J P M N P Statements 1. Quadrilateral JKM quadrilateral NPQR 1. Reasons. J K. 3. K P 3.. J P. Your Turn Write each proof. V. Given: SVT SWT _ Prove: ST bisects VSW. S T W Statements Reasons Houghton Mifflin Harcourt Publishing ompan 9. Given: Quadrilateral quadrilateral GH; _ Prove: GH _ Statements H Reasons G Module esson 3

30 laborate 1. student claims that an two congruent triangles must have the same perimeter. o ou agree? plain. 11. If PQR is a right triangle and PQR XYZ, does XYZ have to be a right triangle? Wh or wh not? 1. ssential Question heck-in Suppose ou know that pentagon is congruent to pentagon GHJK. How man additional congruence statements can ou write using corresponding parts of the pentagons? plain. valuate: Homework and Practice 1. anielle finds that she can use a translation and a reflection to make quadrilateral fit perfectl on top of quadrilateral WXYZ. What congruence statements can anielle write using the sides and angles of the quadrilaterals? Wh? Online Homework Hints and Help tra Practice Y Z X W GHJ. ind the given side length or angle measure. ft 3 19 ft 11 5 J H 31 ft G Houghton Mifflin Harcourt Publishing ompan. JH 3. m Module 1 91 esson 3

31 KMN PQRS. ind the given side length or angle measure. S K.1 cm N 75.9 cm 79 M R P Q. m R 5. PS TUV. ind the given side length or angle measure. (5 + 7) cm V U ( - 1) ( - 1) cm (3 + ) () ( + ) cm T. 7. m U G KMN. ind the given side length or angle measure. G N ( - 9) in. ( + 1) (5 - ) ( + 3)in. K ( + 9) in. M. G 9. m Houghton Mifflin Harcourt Publishing ompan GHJ PQR and PQR STU. omplete the following using a side or angle of STU. Justif our answers. _ 1. GH 11. J 1. GJ = 13. m G = Module esson 3

32 Write each proof. 1. Given: Quadrilateral PQTU quadrilateral QRST _ Prove: QT bisects PR. _ P Q R U T S Statements Reasons 15. Given: _ Prove: bisects and bisects _. Statements Reasons 1. Given: Pentagon pentagon GHJK; Prove: K G H J K Statements Reasons Houghton Mifflin Harcourt Publishing ompan Module 1 91 esson 3

33 . ind the given side length or angle measure. J M m 1. m 19. The figure _ shows _ the dimensions of two cit parks, where RST XYZ and YX YZ. cit emploee wants to order new fences to surround both parks. What is the total length of the fences required to surround the parks? R X S 3 ft T 1 ft Z Y Houghton Mifflin Harcourt Publishing ompan Image redits: Ken rown/+/gett Images. tower crane is used to lift steel, concrete, and building materials at construction sites. The figure shows part of the horizontal beam of a tower crane, in which G H HG G 59 H 7 a. Is it possible to determine m GH? If so, how? If not, wh not? b. member of the construction crew claims that _ is twice as long as _. o ou agree? plain. Module esson 3

34 1. Multi-Step compan installs triangular pools at hotels. ll of the pools are congruent and JK MNP in the figure. What is the perimeter of each pool? J 1 ft M ( - ) ft P K (5 + 1) ft N ( - 15) ( ). Kendall and va la out the course shown below for their radio-controlled trucks. In the figure,. The trucks travel at a constant speed of 15 feet per second. How long does it take a truck to travel on the course from to to to? Round to the nearest tenth of a second. 3 ft 5 ft ft 3. MNP QRS. etermine whether each statement about the triangles is true or false. Select the correct answer for each lettered part. N 5 mm P M ( - 33) Q 5 mm a. QRS is isosceles. True alse _ b. MP is longer than MN. True alse c. m P = 5 True alse d. The perimeter of QRS is 1 mm. True alse e. M Q True alse R S ( + 19) 3 mm Houghton Mifflin Harcourt Publishing ompan Image redits: (T)Image ollective/lam, () Oleksi Maksmenko Photograph/lam Module 1 91 esson 3

35 H.O.T. ocus on Higher Order Thinking. Justif Reasoning Given that, =.7 ft, and = 3. ft, is it possible to determine the length of _? If so, find the length and justif our steps. If not, eplain wh not. 5. plain the rror student was told that GHJ RST and was asked to find GH. The student s work is shown below. plain the error and find the correct answer. Student's Work 5 - = - 5 G (5 - ) m J H S T ( - 5) m ( + 3) m R - = = GH = 5 - = 5 (3) - = 13 m. ritical Thinking In, m = 55, m = 5, and m = 75. In, m = 5, and m = 5. Is it possible for the triangles to be congruent? plain. Houghton Mifflin Harcourt Publishing ompan 7. nalze Relationships PQR SQR and RS RT. student said that point R appears to be the midpoint of PT. _ Is it possible to prove this? If so, write the proof. If not, eplain wh not. Statements Reasons Q P S R T Module esson 3

36 esson Performance Task The illustration shows a Yankee Puzzle quilt. a. Use the idea of congruent shapes to describe the design of the quilt. Houghton Mifflin Harcourt Publishing ompan Image redits: Ken rown/+/gett Images _ b. plain how the triangle with base _ can be transformed to the position of the triangle with base. c. plain how ou know that =. Module 1 9 esson 3