Name : Math 5CST - Review of Transformations, Equivalence & Similarity The cylinders below are similar solids. V? V 7 cm A b 6 cm A b 44 cm The area of the base of the smaller cylinder is 6 cm, and its volume is 7 cm. The area of the base of the larger cylinder is 44 cm. What is the volume of the larger cylinder to the nearest cubic centimetre? The Chunky Meal in a Can soup is available in two sizes: the single serving size and the family size. The cans are similar solids. Single Family The area of the base of the smaller can is 6 cm and its volume is 40 cm. The area of the base of the larger can is 5 cm. What is the volume of the larger can to the nearest cubic centimetre? What are the dimensions of a cube which is equivalent to the larger can?
Two similar rectangular prisms are shown below. cm 8 cm Which of the following is TRUE? A) The ratio of the total areas is :. B) The ratio of the volumes is 4:9. C) The ratio of the widths is :. D) The ratio of the perimeters of the tops is 4:9. 4 A layer cake is made up of three similar cylinders. The total height of the cake is cm. The height of the top layer is 4 cm. Its radius is 5 cm. The radius of the bottom layer is 5 cm.? 5 cm 5 cm 4 cm cm What is the radius of the cake s middle layer?
5 The ratio of the lateral areas of two similar cones is What is the ratio of the volumes of these cones? 5. 4 A) 5 C) 5 8 B) 5 4 D) 65 6 6 In the figure below, rhombuses ABCD and EFGH are congruent and intersect at points P and Q. Rhombus EPCQ is similar to rhombuses ABCD and EFGH. B P F A E C G D Q H In addition: m AB 9 cm m QH cm The area of rhombus EPCQ is 480 cm. What is the area of the shaded region of this figure? 7 The square-based right prisms given below are similar. The volume of the bigger prism is 080 cm. The edges of its base each measure 6 cm. The height of the smaller prism is cm. cm What is the volume of the smaller prism? 6 cm Volume: 080 cm Volume:? A) 7.648 cm C) 7.8 cm B) 69. cm D) 4 cm
8 The two right circular cones below are similar. The height of the smaller cone is 5 cm and its volume is 45 cm. The height of the larger cone is 0 cm and its volume is 60 cm. h s 5 cm h l 0 cm V s 45 cm V l 60 cm What is the ratio of the areas of the two cones? 9 In the Cartesian plane below, triangle B is the image of triangle A. y Which composite of transformations maps triangle A onto triangle B? B 0 A x A) h (O, ) r (O, 80 ) C) t (0, 4) s y h (O, ) B) t (0, 4) h (O, -) D) h (O, ) t (-4, 0) s y
0 Similar triangles ABC and PQR are shown below. Triangle PQR is the image of triangle ABC. A P C 4 units B R Q 6 units Which of the following compositions was applied to make triangle PQR the image of triangle ABC? A) B) C) A reflection with respect to side AC followed by a dilatation with dilatation point C and scale factor A rotation of 80 about centre C followed by a dilation with dilatation point C and scale factor A translation that makes Q the image of B followed by a dilatation with dilatation point Q and scale factor - - D) A dilatation with dilatation point A and scale factor followed by a translation that makes R the image of B In the Cartesian plane below, triangle B is the image of triangle A. y A 0 B x Which isometry maps triangle A onto triangle B? A) t (, 8) r (O, 90 ) C) t (-, -8) r (O, -90 ) B) t (, 8) r (O, -90 ) D) t (-, -8) r (O, 90 )
Two prisms are similar. The volume of prism A is 7 times greater than the volume of prism B. Given the following statements:. The height of prism A is times the height of prism B.. The perimeter of the base of prism A is 6 times the perimeter of the base of prism B.. The lateral area of prism A is 9 times the lateral area of prism B. Which of the statements are TRUE? A) and only C) and only B) and only D), and Two prisms with square bases are similar. One prism is 5 cm high and has a volume of 960 cm. The side of the base of the other prism measures 6 cm. How many of the small prisms will fit inside the large prism? 4 The two cylinders on the right are similar. The ratio of their areas is. 5 r 6 cm r? The radius of the small cylinder is 6 cm. What is the radius of the large cylinder? A) 00 cm C) 60 cm B) 50 cm D) 0 cm
5 The area of triangle ABC is 96 cm. A' A similarity transformation applied to triangle ABC, produces triangle A'B'C', shown on the right. What is the perimeter of triangle ABC? cm 5 cm C' 4 cm B' 6 The dimensions of two similar prisms are in the ratio of. The volume of the smaller prism is 50 cm. What is the volume of the larger prism? 7 Two right cylinders are similar. The area of the base of the small cylinder is 9π cm. The volume of the big cylinder is 50π cm and its height is 0 cm. What is the height of the small cylinder? 8 Two cylindrical water glasses are of different sizes. One glass, whose radius is cm, has a maximum capacity of 5π cm. The other glass, cm high, has a maximum capacity of 47π cm. Are these glasses similar solids? 9 A right triangle has a height of 5 cm and a base of 0 cm. A second triangle, similar to the first, has an area of 600 cm. What are the measures of base b and height h of the second triangle? What are the dimensions of a square equivalent to the second triangle? 0 A factory makes a certain washing machine model that has a cylindrical tub with a radius of cm and a height of 60 cm, as shown in the adjacent diagram. There is a demand for a similar smaller model. 60 cm cm What must the dimensions of the tub of the smaller model be if its volume is to be 4 60.09 cm?
A similarity transformation with a ratio of - is applied to a certain solid. How many exact copies of the initial solid will the new solid hold? The two right prisms with square bases represented below are similar. A L 0 cm h cm c 5 cm The lateral area of the smaller prism is 0 cm and one side of its base measures 5 cm. If the height of the larger prism is cm, what is its volume? What is the radius of a sphere equivalent to the larger prism? Quadrilateral A'B'C'D' is the result of a similarity transformation applied to quadrilateral ABCD. B cm A A' 6 cm cm B' C 4 cm 4 cm D C' 8 cm Identify and describe the composition of transformations applied to quadrilateral ABCD. D'
Math 54 - Review of Transformations, Equivalence & Similarity Answers 44 cm Ratio of areas of bases 6 cm Volume larger 7 cm 7 9 9 Scale factor Ratio of volumes 7 Volume larger 7 7 cm 944 cm Volume area of base height A A 6 5 Therefore, h h 6 5 4 5 and v v 4 5 Answer The volume of the larger can is 469 cm. An equivalent cube would have dimensions 7.77 cm x 7.77 cm x 7.77 cm C 64 5 64 40 x 468.75 5 x 4 The lengths of corresponding segments of similar figures are proportional. Height of the cake s bottom layer Height of the bottom layer Height of the top layer Height of the bottom layer 4 cm Height of the cake s bottom layer is cm Height of the cake s middle layer Radius of the bottom layer Radius of the top layer 5 cm 5 cm Height of cake Height of bottom layer Height of top layer cm cm 4 cm 6 cm Radius of the cake s middle layer Radius of the middle layer Height of the middle layer Radius of the top layer Height of the top layer Radius of the middle layer 6 cm 5 cm 4 cm Radius of the cake s middle layer 7.5 cm Answer: The radius of the cake s middle layer is 7.5 cm.
5 6 7 C The sides of a rhombus are congruent. Since rhombuses ABCD and EFGH are congruent, their corresponding segments are congruent. m EH m AB 9 cm Similarity ratio for rhombuses EFGH and EPCQ m EH k m EQ 9 cm m EH m QH 9 cm 9 cm cm 9 cm 6 cm Area of rhombus EFGH Area of rhombus EFGH k Area of rhombus EPCQ Area of rhombus EFGH 480 cm Area of rhombus EFGH 080 cm Area of the shaded region Area of rhombus EFGH Area of rhombus EPCQ 080 cm 600 cm Answer: The area of the shaded region of this figure is 600 cm. B 480 cm 8 The ratio of the areas is /4 or 4. 9) C 0) C ) C ) B
Measure of one side of the st prism s h V 960 s 5 s 64 Ratio of similarity: st prism to nd prism Ratio of similarity: st prism to nd prism s s 6 8 Ratio of volumes of the prisms 8 V 960 s h 5 V V 8 s 6 or K 8 Answer: 8 of the small prisms would fit in the large prism. 4 D 5 Area of triangle A'B'C' b h A 4 A 6 cm Ratio of similarity (k) Area of ABC Area of A'B'C' 96 6 6 Since the ratio between the areas is equal to the square of the ratio of similarity, k 6 4 Perimeter of triangle ABC Perimeter of ABC Perimeter of A'B'C' Perimeter of ABC 4 Perimeter of ABC 4 48 Answer: The perimeter of triangle ABC is 48 cm. 4
6 7 The volume of the larger prism is 84.75 cm. Radius of small cylinder A b πr 9π πr r Radius of big cylinder V πr h 50π π r 0 r 5 Height of small cylinder Radius of small cylinder Radius of big cylinder height of small cylinder height of big cylinder h 5 0 h 6 Answer: The height of the small cylinder is 6 cm. 8 Height of the first glass V πr h 5π π h h 5 cm Ratio of similarity If V V h h, then the two glasses are similar. Since 5π 47π 5, the two glasses are not similar. Answer: They are not similar because the ratio of their volumes is not equal to the cube of the ratio of their heights.
9 Area of first triangle 5 0 Area 50 Ratio of areas k 600 50 Ratio of similarity k 4 4 Dimensions of second triangle h 5 0 b 0 40 Result The measure of height h is 0 cm. The measure of base b is 40 cm. Dimensions of a square which is equivalent: 4.49 cm x 4.49 cm 0 Volume of the tub of the large model V l πr l h l π 60 V l 8 6,54 Similarity ratio V V s l 4 60.09 8 6.54 0.667 Radius of tub of smaller model r s 0.667 r l r s 0.667 r s 4 Height of the tub of the smaller model h s 0.667 h l h s 0.667 60 h s 40.0 Answer The radius of the tub measures 4 cm and its height is 40 cm. Accept radius in interval [4, 4.007] and height in interval [40, 40.0].
The new solid will hold 7 copies of the old solid. Measure of the height of small prism A L 4c h 0 4 5 h h 6 Volume of small prism (V s ) V s A b h V s 5 5 6 V s 50 Volume of large prism (V l ) 6 50 V 50 V V l 00 Answer The volume of the larger prism is 00 cm. The radius of a sphere equivalent to this is 6.59 cm. Identification of transformation h s: or s h: Description of transformation Reflection in segment AD Similarity transformation with centre A, k or Similarity transformation with centre A, k Reflection in segment AD