# Pencil, highlighter, red pen, GP NB, packet, calculator Pencil, highlighter, red pen, GP NB, packet, calculator Given ABC ~ ABC, determine: A 11 B A) Ratio of similarity (small to large) 8 16 C A 12 C B) Length of BC +1 r= 8 =2 12 +1 3 B C) Ratio of the areas 2 Ar 3

4 Ar 9 2 +1 +1 +1 +1 2 = 16 3 B' C' 2(B' C' ) 48 +1 B' C' 24u S 65 Complete the number sequence patterns for the first ten terms. Look for a pattern that is not additive. A) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 B) 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 C) For any number, n, write a general term that describes each sequence. Also, write the name of each set of numbers. Part (A): n2 Part (B): n3 square numbers cubic numbers

This problem is an extension of S-56! Notice the entries in the first three columns in the first table are copies of the table for S-56. Each figure in the S66 is similar to others. If you choose any figure as your original figure, the others are either magnifications or reductions of the original (the new sides or edges are formed by multiplying the original length by a constant value). For example, if figure B is chosen as the original figure, then figure E is a 2.5 magnification and figure A is a 0.5 reduction. S 66 0.5 reduction 2.5 magnification Keep in mind magnification factor means constant multiplier. A) On the resource page, notice that the first 3 columns now refer to edges and faces. Complete the fourth column for the volumes of the cubes. B) Use the table in part (A) to complete the table of ratios on your resource page. Reduce the ratios to lowest terms. S 66 u u u u u u Ratio of Similarity u2 u2

u2 u2 u2 u u u u u3 u3 27u3 64u3 125u3 2 1 2 1 8 =2 4 1 4 1 8 1 4 1 5 1 3 2 4

1 5 1 3 2 16 =4 4 1 20 =5 4 1 12 =3 8 2 16 1 25 1 9 4 64 1 125 1 27 8 5 2 5 2 20 =5 8 2 25 4

125 8 S 67 Use your resource page to answer the following: A) Briefly state the relationship between the ratio of similarity (r) and the perimeter ratios. The perimeter ratios are the same as the ratio of similarity. B) Briefly state the relationship between the ratio of similarity (r) and the area ratios. Area ratios are the squares of ratio of similarity. C) Briefly state the relationship between the ratio of similarity (r) and the volume ratios. Volume ratios are the cube of the ratio of similarity. The r:r2 :r3 Theorem a If two figures are similar with a ratio of similarity b , then the following proportions for the small and large figures (which are enlargements or reductions of each other) are true: side area 2 Ratio of similarity r=a b a Sr r

b perimeter a Pr r b a A r r b 2 volume a Vr r b 3 3 For 2-D figures, the theorem refers to the ratios of sides, perimeters, and areas. For 3-D figures, the theorem refers to the ratios of edges or perimeters, areas of faces or total surface areas of solids, and ratios of volumes or weights. Given the following similar cubes, compute the following ratios from SMALL to LARGE: 5 2 ~ Ratio of similarity: r = 2

5 ratio of edges: S = 2 r (r:r :r Thm.) 5 2 2 ratio of surface areas: A = = 4 r 5 25 (r:r :r Thm.) 2 2 3 3 3 2 = 8 ratio of volumes: V = r 5 125 (r:r :r Thm.) 2 3 NOTE: You can also set the ratio LARGE to SMALL, but be consistent in writing large #s in the numerators and small #s in the denominators.

IN THIS CLASSyou will write the ratio SMALL to LARGE, unless you are told otherwise!!! S 68 ~ A) Suppose the ratio of the sides of the tetrahedra is 3:5. Compute the ratios of their surface areas. 2 9 r 3 5 Ar = 3 = 25 5 (r:r2:r3 Thm.) B) If the total surface area of the large tetrahedron is 240 square units, use ratios to solve for the total surface area of the small tetrahedron. A =9 240 25 25A 2160 A=86.4 u2 The TSA of the small tetrahedron is 86.4 u2. S 68 ~ r 3 5 C) If the volume of the small tetrahedron is 65 cubic units, compute the

volume of the large tetrahedron. Vr ( 3 )3 27 5 125 (r:r2:r3 Thm.) 65 = 27 V 125 27(V)=65(125) 27V 8125 V 300.93 u3 The volume of the large tetrahedron is about 300.93 u3. S 69 The 2 rectangular prisms are similar. The ratio of their vertical edges is 4:7. Use the r:r2:r3 Theorem to find the following without knowing the dimensions of the prisms. 4 r A) Find the ratio of their surface areas. 2 Ar = 4 = 16 49 7 7 B) Find the ratio of their volumes.

3 64 Vr = 4 = 343 7 (r:r2:r3 Thm.) (r:r2:r3 Thm.) C) Suppose the perimeter of the front face of the larger prism is 18 units. Find the perimeter of the front face of the small prism. 4 Pr 7 (r:r2:r3 Thm.) P =4 18 7 7P 72 P 10.29 u The perimeter of the front face of the small prism is about 10.29 u. S 69 4 r 7 A) Find the ratio of their surface areas. 2 Ar = 4 = 16 49 7 (r:r2:r3 Thm.)

D) Suppose the area of the front face of the larger prism is 15 square units. Find the area of the front face of the small prism. A = 16 15 49 49A 240 B) Find the ratio of their volumes. 3 64 Vr = 4 = 343 7 (r:r2:r3 Thm.) E) Suppose the volume of the small prism is 21 cubic units. Find the volume of the larger prism. 21= 64 V 343 64V 7203 A 4.90 u 2 The area of the front face of the small prism is about V 112.55 u3 The large prisms volume is about 112.55 u3.

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