Geometry AP Book 8, Part 2: Unit 7

Transcription

1 Geometry P ook 8, Part 2: Unit 7 P ook G8-7 page base # s V F C n n-agon n 2n n n ; 5; 8; 5; No. a) = 24 8 e) ii) top, and faces iii) bottom, and faces f) ; 8 = 24 g) times; From f), we see that each vertex is at the intersection of three faces. This means that each vertex has been triple counted. INVSTIGTION. 4; 6 4 = 24.,, top bottom, faces at each edge = 2 C. 12; 12 2 = 24. Yes 4. a),, C,, F (only doesn t), F, F ONUS a) Cube triangular pyramid with faces that are all equilateral triangles. In other pyramids, all faces but the base are triangles (e.g. a square pyramid has a square base). no other pyramid has all congruent sides, so they can t be Platonic solids. No, it isn t; lthough they are all congruent triangles, of the faces meet at the top and bottom vertices, while 4 come together at the other three vertices. 5. a) triangular faces: tetrahedron, octahedron, icosahedron square faces: cube pentagonal faces: dodecahedron faces: tetrahedron, cube, dodecahedron 4 faces: octahedron 5 faces: icosahedron 6. F Vf Fv equation V T 4 4 = V 4 C = V 8 O = 4 V = V 20 I = 5 V 12 F f Fe equation T = 2 6 C = 2 12 O = = 2 0 I = a) T C O I V F V + F th row = rd row + 2 or V + F = + 2 S F V Hold? n n + 2 n 2n 9. a) = 2 Yes: = Yes: = Yes: = Yes: 2n + n + 2 = n + 2 (12 5) + (20 6) = 180 edges No, it s not; ach of the ball s edges is shared by 2 faces, so the actual number of edges will be less. 2; 2; = = V = e) V = 60 f) No, its faces aren t congruent: they re a mix of pentagons and hexagons. P ook G8-8 page a) Teacher to check. Teacher to check. 2. nswers will vary Sample description: Translate the shape 2 units or repeatedly. Translate row 1 unit up or down repeatedly. Sample description: Translate the shape 1 unit up or down repeatedly. Rotate each shape in the column 180 around its top vertex. Translate both columns together 4 units or repeatedly.. a) i) Reflection ii) Translation iii) Rotation Yes; 180 rotation around point Q, translate 2 units up 180 rotation around point R Yes, it tessellates. escriptions will vary teacher to check. Sample description: 180 rotation around point P (onto shape 4). Together, 1 and 4 create a rectangle, which tessellates using translations. COPYRIGHT 2011 JUMP MTH: NOT TO COPI V-40 nswer Keys for P ook 8.2

2 Geometry P ook 8, Part 2: Unit 7 (continue 4. xplanations may vary Sample strategy explanation: nswers may vary Sample answers: a) C O M 1 (i) 120 CW to, and (ii) 120 CCW to F. Together, these shapes create a hexagon. Reflect this hexagon in line, and continue reflecting in the vertical sides of the hexagon to create a row of hexagons. 2. = 10 = 19 C = 24 = 46 The fifth angle equals 540 minus the sum of the other four angles, so: = 540 ( ) = = 228 Start with shape 1: rotate shape 180 around point, then translate it 2 units. Rotate two shapes together 90 CCW around point. Follow this same process with shape 2, translating 4 units down. Follow this same process with shape, translating 8 units down. Continue with ever larger Ls to tessellate further. 5. a) nswers may vary Sample answers: i) G H ii) C iii) iv) i) Reflection in mirror line M 1 ii) Reflection in mirror line M 2 The two reflections above (in order) will take shape to shape C. single rotation of 120 CCW about centre O will have the same result. 7. nswers may vary Sample answers: F M 2 C Translate the whole row using arrow. Continue translations repeatedly. P ook G8-9 page a) = (r + s + t) + (u + v + w) + (x + y + z) = 180 = 540 Measured angles: = 54 = 142 C = 64 = 150 = 10 Sum = 540 nswers will vary based on labelling. r t u x. 60 ; xplanations will vary Sample explanation: Four angles fit around a point, and 4 90 = Sidra s total is 60 greater than the sum of the pentagon s interior angles alone. Rather than triangles, she divided the pentagon into 5 triangles. Correct total: 180 = 540. Sidra s total: = 900. The 900 comes from the sum of the interior angles plus the extra 60 in the five angles around the point at the centre of the shape. COPYRIGHT 2011 JUMP MTH: NOT TO COPI v) F vi) G vii) F viii) nswers for and will vary but must be based on shape pairs given in a). Teacher to check. Sample answers based on vii) and viii) above: F: Reflected in vertical line through (, 0) and then translated 6 units down. : Rotated 180 around point (0, 4) and then translated 4 units up. a) C or C e) (reflect to C, then translate C to ) f) (reflect to, then rotate to ) g) C (translate C to, then rotate to ) ONUS nswers may vary Sample answer: Rotate shape around centre twice: s v w + + C + + = r + (t + u + x) + z + (w + y) + (s + v) = (r + s + t) + (u + v + w) + (x + y + z) = 180 = 540 Measured angles: = 10 = 19 C = 77 = 121 = 100 Sum = 540 y z 5. a) 60 We know from Question that the sum of angles around a point or vertex is 60, and 6 60 = 60. Four squares will fit: 4 90 = 60. e) 6. Teacher to check. NOT: ivisions can originate at any vertex. 4 triangles 5 triangles 6 triangles nswer Keys for P ook 8.2 V-41

3 Geometry P ook 8, Part 2: Unit 7 (continue e) 7 triangles f) 8 triangles S T xp I Sum I n n (n 2) 180 (n 2) INVSTIGTION. V Sum I ach I = = = = = 15. Increase; Yes. s the chart above shows, the interior angles increase as the number of sides/vertices in the regular polygon increases. C. a) It gives the number of copies of the regular polygon that fit around a common vertex. Recall that, to tessellate, a polygon must fit around a common vertex with no gaps or overlaps. For this to happen, its interior angle x must divide evenly into 60.. x 60 x Only equilateral triangles, squares and regular hexagons will tessellate (since 60 x is a whole number for 60, 90 and 120 respectively). F. From the table in, we see that the measures of the interior angles form an increasing sequence. Therefore we know that the measure of an interior angle of a regular polygon with more than 8 sides is more than 15, but less than 180. This means that placing two copies of the polygon around a vertex will not fill 60, and the remaining gap will be less than 90. This is smaller than the angle of a third copy of the polygon, so placing a third copy will create an overlap. s such, we can t produce a tessellation. 7. a) 18 ach polygon consists of seven hexagons, so it is actually this original hexagon that is tessellating: P ook G8-40 page Like a square, a rectangle has four 90 angles. Since four rectangles fit evenly around a common point, it tessellates: = 2. a) The adjacent angles in any parallelogram add to 180, so it can tessellate using only translations. Sample answer: First, translate the parallelogram below a units to the. Then translate this whole row down b units in the direction of the slanted side. a b Rotation; Specifically, a 180 rotation around the midpoint of one of its sides: Yes; Once it is rotated to form a parallelogram as in, the triangle will tessellate as described in a).. a) In i) and ii) below, students can use a few transformation combinations that are correct, such as: a translation then a reflection, or a 180 rotation followed by a translation, etc. Teacher to check. i) ii) Like in a), the transformations used may vary teacher to check. Teacher to check. 4. a) b d c a 2 1 a c d b b x c 4 a d Since the angles around a vertex add to 60, we know: x = 60 c d b ut a, b, c and d are the interior angles of the quadrilateral, so add to 60. x = a qual sides are marked in a) above. The fourth copy is shown in a) above. To 1: translation up and To 2: 180 rotation around the midpoint of their common side To : 180 rotation around the midpoint of its side e) In a quadrilateral, the interior angles a, b, c and d always add to 60. s such, they will also fit evenly around a common point. 5. a) No; ngles in a regular octagon equal 15, and 60 (2 15 ) = 90. This is not enough space to accommodate a third octagon. COPYRIGHT 2011 JUMP MTH: NOT TO COPI V-42 nswer Keys for P ook 8.2

4 Geometry P ook 8, Part 2: Unit 7 (continue COPYRIGHT 2011 JUMP MTH: NOT TO COPI = 90 (see above for explanation) Square 6. a) Square; 1 cm Kong also needs to use a square, but his will have sides that are 2 cm long. 7. a) Since the sum of interior angles in a pentagon is 540, we know that: + = = 10 ut they are equal, so = = 155. and round and : 8. a) and don t divide into 60 but: + + C = = 60 e) Prediction: Yes 540 2(120 ) a = = 00 = 100 x = 60 2a = 60 2(100 ) = 160 ut x = y = z, x = 160 y = 160 z = 160 No, she can t. The angles in the pentagons are 100 and 120, which can t be combined to add +to 160 : = = = a) = 540 ( ) = 170 This pentagon will tessellate since = 60. = = 540 2(120 ) 70 2 = 20 2 = 115 This pentagon won t tessellate: its angles can t be combined in any way to add up to 60. C = 540 4(120 ) = 60 This pentagon will tessellate since = 60. Tessellations may vary teacher to check. Sample tessellations: 10. a) i) Unknown angles 720 2(60 ) = 4 = 150 ii) Unknown angles 720 (108 ) = = 12 i) Correct prediction: It tessellates since = 60. ii) Correct prediction: It won t tessellate since no combination of 108 and 12 adds up to 60. Teacher to check. escriptions may vary teacher to check. Sample answer: For i), translate the hexagon up or down 1 unit repeatedly. Then translate this column of hexagons or, while shifting it down half a unit so it fits into the triangular hole that s been created. 11. a) and : and C: C and F: nswers may vary Sample tessellations: and : is an octagon with all equal angles a = 15 is a isosceles triangle its two other angles = 45 Since = 180, four s and one form a rectangle, which will tessellate. and C: is a scalene triangle with one 0 angle its rd angle = 60 C is an octagon with four 120 angles b = 4 = 150 Since = 180 and = 180, four s and one C form a rectangle, which will tessellate. and F: is an equilateral triangle all three angles = 60 F is a regular hexagon c = 120 Since = 180, and F will form a variety of shapes that will tessellate.,, and F 12. a) Yes F F nswer Keys for P ook 8.2 V-4

5 Geometry P ook 8, Part 2: Unit 7 (continue Yes No, her shape won t tessellate; In order to tessellate (that is, to fit the gap with angle, Nellie must place the two cut out shapes so they will end up side by side when tessellated (a + b =. This is impossible here; the rectangles will overlap. P ook G8-41 page Teacher to check. 2. a) Teacher to check. First, students must rotate their shape180 around the midpoint from a). This will result in a parallellogram. fter creating this initial parallelogram, students may choose to use a variety of transformations to form the tessellation. Teacher to check. Sample tessellation: To form a parallelogram, I rotated my shape 180 around the point marked. From there, I used translations to create my full tessellation. 4. a) Teacher to check. To eliminate the curved sides, you have to place the shapes using a 120 rotation. However, when you place six shapes (eliminating all the curved sides), they make a loop with a hexagonal hole in the middle: The hole cannot be filled with this shape, so the shape does not tessellate. 5. will tessellate: is congruent to (just rotated CCW slightly) so, yes, it will tessellate. C will tessellate: P ook G8-42 page Teacher to check. 2. Teacher to check. 4. Teacher to check f) g) h) 5. a) 6. Teacher to check 7. Teacher to check C,,, 8. P ook G8-4 page Teacher to check 2. a) Circle: the last (4 th ) view The side is shaded here: From this sketch, we can see that the layer heights, from to back, are 2, 1,. We can also see that only the two bottom corners have multiple layers.. Teacher to check 4. Teacher to check v. a) Teacher to check.. 9. nswers will vary 5., top, COPYRIGHT 2011 JUMP MTH: NOT TO COPI V-44 nswer Keys for P ook 8.2

6 Geometry P ook 8, Part 2: Unit 7 (continue 6. P ook G8-44 page m 8 m f) 25 m 25 m side view 10 m side view 8 m g) 4. h) side view 2. width height : 1 cm 1 cm : 1 cm 2 cm : 1 cm 2 cm width height : 2 m 1 m : 2 m 2 m ONUS e) : 1 m 2 m width height : 2 cm 25 mm : 2 cm 7 mm : 25 mm 7 mm. ONUS bottom view COPYRIGHT 2011 JUMP MTH: NOT TO COPI 7. nswers will vary 5 cm 5 cm 25 mm cm cm 25 mm e) 5. a) nswer Keys for P ook 8.2 V-45

7 Geometry P ook 8, Part 2: Unit 7 (continue Circle: e) T L m R F F k k R T L m. a) (the structure in the top, corner) f) INVSTIGTION 1. Teacher to check built structure. xplanation and most helpful view will vary teacher to check. i) 5. Turn the shape vertically 90 CCW (so that the face becomes the bottom face).. before 90 CCW 180 CCW C. It rotated 90 counterclockwise each time. 270 CCW ii) i) 6. Teacher to check drawings. Without the thick lines, the and side views are reflections of one another in a vertical line. INVSTIGTION 2. Teacher to check built structure: ii) For example: a). and C. L F R k before P ook G8-45 page a) iii) 7. nswers will vary P ook G8-46 page CW 180 CW 270 CW 1. Teacher to check built structure. back back top bottom side view 2. a) top bottom In the questions below, the following face codes are used: T = top m = bottom F = k = back R = L = 1. a) T R m L F F k k R m L T T T m m F L k R R F L k 2. a) T T m m F R k L R k L F T T m m F k k F R L L R T m m T F F k k R L L R 90 CCW. In each row, the and side views (without the thick lines) are reflections of each other in a vertical line. This is also true of the back and s.. When you move down from one row to the next (not including the last row), the views all shift 1 cell to the. This makes sense since the structure is rotating 90 each time. COPYRIGHT 2011 JUMP MTH: NOT TO COPI V-46 nswer Keys for P ook 8.2

8 Geometry P ook 8, Part 2: Unit 7 (continue F. 180 CCW rotation has the same effect as a 180 CW rotation (since = 60 ), so its views will match the rd row. 270 CCW rotation has the same effect as a 90 CW rotation (since = 60 ), so its views will match the 2 nd row. COPYRIGHT 2011 JUMP MTH: NOT TO COPI nswer Keys for P ook 8.2 V-47