Answer Key
Chapter 4.5 Dilations
4.5 Dilations Exercise 1 ππ β² ( ππππ , ππππ ) Exercise 2 60% because this scale factor is less than 1. Exercise 3 Because πΆπΆππ β² πΆπΆππ = 6 14 , the scale factor is ππ = 37 . So the dilation is a reduction. Exercise 4 Because πΆπΆππ β² πΆπΆππ = 24 9 , the scale factor is ππ = 83 . So the dilation is an enlargement. Exercise 5 Because πΆπΆππ β² πΆπΆππ = 9 15 ,the scale factor is ππ = 35 . So the dilation is a reduction. Exercise 6 Because πΆπΆππ β² πΆπΆππ = 28 8 , the scale factor is ππ = 72 . So the dilation is an enlargement. Exercise 7 Exercise 8
Exercise 9 Exercise 10
Exercise 11 Exercise 12 Exercise 13
Exercise 14 Exercise 15 ( ππ , ππ ) β (3 ππ , 3 ππ ) ππ (6, β 1) β ππ β² (18, β 3) ππ ( β 2, β 4) β ππ β² ( β 6, β 12) ππ (1,2) β ππ β² (3,6)
Exercise 16 ( ππ , ππ ) β οΏ½ 65 ππ , 65 πποΏ½ π΄π΄ (0,5) β π΄π΄ β² (0,6) π΅π΅ ( β 10, β 5) β π΅π΅ β² ( β 12, β 6) πΆπΆ (5, β 5) β πΆπΆ β² (6, β 6) Exercise 17 ( ππ , ππ ) β οΏ½ 23 ππ , 23 πποΏ½ ππ (9, β 3) β ππ β² (6, β 2) ππ (6,0) β ππ β² (4,0) ππ (3,9) β ππ β² (2,6) ππ (0,0) β ππ β² (0,0)
Exercise 18 ( ππ , ππ ) β (0.25 ππ , 0.25 ππ ) π½π½ (4,0) β π½π½ β² (1,0) πΎπΎ ( β 8,4) β πΎπΎ β² ( β 2,1) πΏπΏ (0, β 4) β πΏπΏ β² (0, β 1) ππ (12, β 8) β ππ β² (3, β 2) Exercise 19 ( ππ , ππ ) β οΏ½β 15 ππ , β 15 πποΏ½ π΅π΅ ( β 5, β 10) β π΅π΅ β² (1,2) πΆπΆ ( β 10,15) β πΆπΆ β² (2, β 3) π·π· (0,5) β π·π· β² (0, β 1)
Exercise 20 ( ππ , ππ ) β ( β 3 ππ , β 3 ππ ) πΏπΏ (0,0) β πΏπΏ β² (0,0) ππ ( β 4,1) β ππ β² (12, β 3) ππ ( β 3, β 6) β ππ β² (9,18)
Exercise 21 ( ππ , ππ ) β ( β 4 ππ , β 4 ππ ) π π ( β 7, β 1) β π π β² (28,4) ππ (2,5) β ππ β² ( β 8, β 20) ππ ( β 2, β 3) β ππ β² (8,12) ππ ( β 3, β 3) β ππ β² (12,12) Exercise 22 ( ππ , ππ ) β ( β 0.5 ππ , β 0.5 ππ ) ππ (8, β 2) β ππ β² ( β 4,1) ππ (6,0) β ππ β² ( β 3,0) ππ ( β 6,4) β ππ β² (3, β 2) ππ ( β 2,2) β ππ β² (1, β 1)
Exercise 23 The scale factor is the ratio of the length of the corresponding sides of the image and preimage. ππ = πΆπΆππ β² πΆπΆππ = 3 12 = 14 Exercise 24 If P(x,y) is the preimage of a point, then its image after dilation at the origin (0,0) with scale factor k is the point ππ β² ( ππππ , ππππ ). Therefore k=2. Exercise 25 ππ = 15 9 = 53 ππ = 35 ππ 53 = 35 ππ 5 ππ = 105 ππ = 21 Exercise 26 ππ = 2814 = 2 2 = 12 ππ 2 ππ = 12 ππ = 6 Exercise 27 ππ = 32 ππ = ππ 2 32 = ππ 2 3 = ππ Exercise 28
ππ = 7 28 = 14 ππ = 4 ππ 14 = 4 ππ ππ = 16 Exercise 29 ππ = 5 2.5 = 2 ππππ ππ = 7 3.5 = 2 Exercise 30 ππ = 10 8.5 = 100 85 = 2013 Exercise 31 image length actual length = ππ image length 60 = 5 πΌπΌπππΌπΌπΌπΌπΌπΌ πππΌπΌπππΌπΌππβ = 300 ππππ Exercise 32 image length actual length = ππ image length 4.5 = 10 πΌπΌπππΌπΌπΌπΌπΌπΌ πππΌπΌπππΌπΌππβ = 45 ππππ Exercise 33 image length actual length = ππ image length 47 = 20 πΌπΌπππΌπΌπΌπΌπΌπΌ πππΌπΌπππΌπΌππβ = 980 ππππ Exercise 34 image length actual length = ππ image length 12 = 15 πΌπΌπππΌπΌπΌπΌπΌπΌ πππΌπΌπππΌπΌππβ = 180 ππππ Exercise 35 Grasshopper: image length actual length = 15 2 = 7.5 Black beetle: image length actual length = 4.20.6 = 7
Honeybee: image length actual length = 75 16 5 8 = 7.5 Monarch butterfly: image length actual length = 29.25 3.9 = 7.5 Therefore grasshoppers, honeybees and monarch butterοΏ½lies were looked at using the same magnifyi Exercise 36 Building β³ ABC and β³ ABC dilatation β³ A β² B β² C β² . We connect each point to its corresponding image in order to construct the dilation's center. The center of dilation is where the three lines converge. Exercise 37 Small sides: 84 = ππ 1 Large sides: 10 6 = ππ 2 Because the scale factor is not equal, the photo can't be dilated to fit the frame. Exercise 38 image length actual length = ππ = 13 Hence, the large figure is the actual figure and the other one is the dilated figure. Exercise 39 Since the larger triangle is the dilation of the smaller triangle, 62 = 2 ππ + 8 ππ + 1 3 = 2 ππ + 8 ππ + 1 3 ππ + 3 = 2 ππ + 8 ππ = 6
Since angle measures are preserved, 3y β 34 = y + 16 3y β y = 16 + 34 2y = 50 y = 25 Exercise 40 To understand why a scale factor of 2 is equivalent to 200%, we need to first understand what scale factor means in mathematics. A scale factor is a number that scales, or multiplies, a quantity or measurement by a certain amount. For example, if we have a line segment that is 5 units long and we want to scale it by a factor of 2, we would multiply 5 by 2 to get a new length of 10 units. Now, let's consider the percentage scale. Percentages are a way of expressing a number as a fraction of 100. For example, 50% is the same as 50/100 or 0.5. To convert a scale factor to a percentage, we can simply multiply it by 100. For example, if our scale factor is 2, we can multiply it by 100 to get 200%, which is the same as saying that we are scaling our quantity by a factor of 2. In mathematical notation, we can express this as: scale factor of 2 = 2 200%=2Γ100 So, a scale factor of 2 is the same as 200% because they both represent multiplying a quantity or measurement by a factor of 2. Exercise 41 The original figure is closer to the centre of dilation. Exercise 42 The dilated figure is closer to the centre of dilation. Exercise 43 The original figure is closer to the centre of dilation. Exercise 44 The dilated figure is closer to the centre of dilation. Exercise 45 a. Length ππ β² π΄π΄ β² is 2 times longer than OA. b. OA and ππ β² π΄π΄ β² are parallel. Exercise 46 a. Length π΄π΄ β² π΅π΅ β² is equal to half of the length of AB. b. AB and π΄π΄ β² π΅π΅ β² are parallel. Exercise 47 image length actual length = ππ 9 12 β 12 = 9 144 = 1 16 Hence the scale factor is 1 16 . Exercise 48
Yes! When the scale factor k=-1, the figure rotates 180 0 but the size and shape don't change. Exercise 49 a. Length of the rectangle = 5-(-3)=8 and width of the rectangle =3-(-1)=4 Perimeter = 2*8+2*4=24 units Area = 8*4=32 square units b. Length of the dilated rectangle = 3*8=24 and width of the dilated rectangle =3*4=12 Perimeter = 2*24+2*12=72 units Area = 24*12 =288 square units The perimeter of the dilated rectangle =3 (perimeter of the initial rectangle ) Area of the dilated rectangle = 9 (area of the initial rectangle) c. Length of the dilated rectangle = 84 = 2 and width of the dilated rectangle = 44 = 1 Perimeter = 2*2+2*1=6 units Area = 2*1 =2 square units The perimeter of the dilated rectangle = (perimeter of the initial rectangle )/4 Area of the dilated rectangle = (area of the initial rectangle)/16 d. The perimeter of the dilated rectangle =k (perimeter of the initial rectangle ) Area of the dilated rectangle = ππ 2 (area of the initial rectangle) where k is the scale factor. Exercise 50 When you reduce a page and place it on the original page, there is always a point that is in the same place on both pages. This is because when you reduce the page, all of the points on the original page are scaled down by the same factor. Therefore, the relative distances between the points on the original page are preserved in the reduced page. To understand this concept mathematically, let's consider a point P on the original page with coordinates (x,y), and let's assume that the page is reduced by a factor of r. This means that the coordinates of P on the reduced page will be (rx,ry). We can see that the relative distances between the points on the original page are preserved in the reduced page by comparing the distance between two points on the original page with the distance between the corresponding points on the reduced page. Let's consider two points P and Q on the original page with coordinates ( ππ 1 , ππ 1 ) and ( ππ 2 , ππ 2 ) respectively. The distance between these two points can be calculated using the distance formula: οΏ½ ( ππ 2 β ππ 1 ) 2 + ( ππ 2 β ππ 1 ) 2 Now, let's consider the corresponding points on the reduced page with coordinates οΏ½ππ π₯π₯ 1 , ππ π¦π¦ 1 οΏ½ and οΏ½ππ π₯π₯ 2 , ππ π¦π¦ 2 οΏ½ . The distance between these two points can also be calculated using the distance formula: οΏ½οΏ½ππ π₯π₯ 2 β ππ π₯π₯ 1 οΏ½ 2 + οΏ½ππ π¦π¦ 2 β ππ π¦π¦ 1 οΏ½ 2
We can simplify this expression using the fact that ππ π₯π₯ 1 = ππ π₯π₯ ππ and ππ π¦π¦ 1 = ππ π¦π¦ ππ , and similarly for the other point: οΏ½οΏ½ π₯π₯ 2 βπ₯π₯ 1 ππ οΏ½ 2 + οΏ½ π¦π¦ 2 βπ¦π¦ 1 ππ οΏ½ 2 We can see that the expression on the right-hand side is simply the original distance formula divided by r. This means that the distance between the two points on the reduced page is scaled down by the same factor as the page itself, so the relative distances between points are preserved. Therefore, we can conclude that there is always a point that is in the same place on both the original and reduced pages, as long as the reduction is done by the same factor in both dimensions. Exercise 51 π΄π΄ β² (4,4), π΅π΅ β² (4,12) πΌπΌππππ πΆπΆ β² (10,4) Exercise 52 ( ππ , ππ ) β ( ππ , ππ β 4) π΄π΄ (2, β 1) β π΄π΄ β² (2, β 5) π΅π΅ (0,4) β π΅π΅ β² (0,0) πΆπΆ ( β 3,5) β πΆπΆ β² ( β 3,1) Exercise 53 ( ππ , ππ ) β ( ππ β 1, ππ + 3) π΄π΄ (2, β 1) β π΄π΄ β² (1,2) π΅π΅ (0,4) β π΅π΅ β² ( β 1,7) πΆπΆ ( β 3,5) β πΆπΆ β² ( β 4,8) Exercise 54 ( ππ , ππ ) β ( ππ + 3, ππ β 1) π΄π΄ (2, β 1) β π΄π΄ β² (5, β 2) π΅π΅ (0,4) β π΅π΅ β² (3,3) πΆπΆ ( β 3,5) β πΆπΆ β² (0,4) Exercise 55
( ππ , ππ ) β ( ππ β 2, ππ ) π΄π΄ (2, β 1) β π΄π΄ β² (0, β 1) π΅π΅ (0,4) β π΅π΅ β² ( β 2,4) πΆπΆ ( β 3,5) β πΆπΆ β² ( β 4,5) Exercise 56 ( ππ , ππ ) β ( ππ + 1, ππ β 2) π΄π΄ (2, β 1) β π΄π΄ β² (3, β 3) π΅π΅ (0,4) β π΅π΅ β² (1,2) πΆπΆ ( β 3,5) β πΆπΆ β² ( β 2,3) Exercise 57 ( ππ , ππ ) β ( ππ β 3, ππ + 1) π΄π΄ (2, β 1) β π΄π΄ β² ( β 1,0) π΅π΅ (0,4) β π΅π΅ β² ( β 3,5) πΆπΆ ( β 3,5) β πΆπΆ β² ( β 6,6)
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