Chapter 9 Extra Practice Answer Key
Get Ready
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1. a) congruent b) similar c), d) neither
2. b), c), d); each pair of lengths are equivalent to the ratio 1:3
3. No. Proportional lengths must be multiples of each other.
4. Answers will vary. a) 25 cm, 2.5 cm b) 4 km, 5 km c) 50 mm, 25 mm
d) 6 m, 9 m
5. a) 40 b) 4 c) 1 d) 0
6. a) 4 b) 16 c) 5 d) 10
7. a) pentagonal prism b) hexagonal pyramid c) trapezoidal prism
d) triangular prism
8. a) 7 vertices, 12 edges, 7 faces
b) 24 vertices, 36 edges, 14 faces
c) 5 vertices, 8 edges, 5 faces
d) 10 vertices, 15 edges, 7 faces
9. a) quadragonal pyramid b) It clearly describes the shape of the base (square) not just the number of sides.
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9.1 Dilatations
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1. Corresponding angles are congruent. Corresponding sides are parallel and proportional. Perimeters are proportional. Area of the image equals the area of the pre-image multiplied by the square of the scale factor.
2. a) to d) The diagonal in the images and pre-image is proportional.
3. a),b)
c)A = A', B = B', C = C'. A'B' = 2AB, B'C' = 2BC, C'A' = 2CA,
AB || A'B', AC || A'C', BC || B'C'
d) yes e) 1:4 f) 1:2; each side in the image is twice as long as the corresponding side in the pre-image
g) centre P with scale factor 0.5
4. a)
b) Same as corresponding angles in the original. c) Twice as long.
d) Four times as great.
5. a),b)See P'Q'R'S' in part d).
c)P = P', Q = Q', R = R'. S = S'. P'Q' = 2PQ, Q'R' = 2QR, R'S' = 2RS, S'T' = 2ST, PQ || P'Q', QR || Q'R', RS || R'S', ST || S'T'
d)
e) The quadrilaterals fit neatly inside one another.
6. a),b) Diagrams may vary.
c) Yes, they are parallel. The ratios of the sides are the same as the scale factors.
d) The area of each image is two scale factors times the area of the original. For example, for the scale factor 3, the area of the image is 9 times the area of the original.
e) Diagrams may vary.The lines go through the corresponding vertex of each image. If I drew other images with the same centre, the diagonals would also pass through each vertex of those images.
7. a),b)
c)A = A', B = B', C = C'; A'B' = 2.5AB, B'C' = 2.5BC, C'A' = 2.5CA
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9.2 Properties of Similar Figures
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1. a) For quadrilaterals, you need to determine that all corresponding angles in both quadrilaterals are equal.
b) For triangles, you need to determine that all corresponding angles in both triangles are equal or that all corresponding sides in both triangles are in the same ratio.
2. Yes. P is common to both triangles and PST = 85 = PRQ. So, two pairs of corresponding angles are congruent, therefore PST PRQ.
3. Since the sum of interior angles of a pentagon is 540°, we can calculate the unknown angle measure in each pentagon. We see that all corresponding angles in the pentagons are equal. The pentagons are similar.
4. ABCD and MNOP are similar. Corresponding sides are in proportion. Since both are rectangles, corresponding angles are equal.
5. approximately 53.3 m
6. Answers may vary.
7. a) KM = 1.8, ST = 5.25, RT = 6.3
b) HI = 10, IJ = 7.5, JK = 12.5, KL = 15, FL = 22.5
8. 6.2 cm
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9.3 Cones and Cylinders
1. A is not a cone. It does not have an apex at which all line segments from the base meet. B, C, and D are cones; B is also a pyramid because all its faces are polygons.
2. B is not a cylinder. Corresponding points on the bases are not connected by line segments on the surface. A, C, and D are cylinders; A and D are also prismsbecause all their faces are polygons.
3. A is a cone. It has a base and an apex at which all line segments from the base meet. B is not a cone. Some line segments connecting points on the base to the apex are not on the surface.
4. a) hexagonal prism b) square pyramid
5. The polygon in the name refers to the shape of the base. You can use the shape of the base to visualize or sketch the prism, and count vertices, edges, and faces. You can also count planes of symmetry and find order of rotational symmetry.
6. a) 7 vertices, 12 edges, 7 faces.
b) Five faces will be triangles and the base is a hexagon.
c) six planes of symmetry: one through each pair of opposite vertices and edges on the base; order of rotational symmetry 6: one for each plane of symmetry through the base
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9.4 Draw Polyhedra
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1. Advantages: shows three dimensions, shows lengths of component shapes/ Disadvantages: does not show depth, requires dot paper, may require two or more views to show entire object.
2. Advantages: clearly shows composition of object.
Disadvantages: requires several views to show entire object, does not show three dimensions, requires dot paper.
3. Advantages: shows three dimensions and depth, does not require special paper.
Disadvantages: may not show entire object clearly, does not show accurate lengths of sides.
4. A rectangular prism with a small triangular prism on top of it is beside a large triangular prism.
5.
6. a) Drawings may vary.
b) For each isometric drawing you could measure accurately: length, width, height; for each perspective drawing no measurements could be made accurately.
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9 Review
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1. a) yes; it is similar b) all angles are the same in the image and the original, and all sides are in the ratio 2:1
c) A'B'C'D' d) Draw a line through each pair of corresponding vertices (e.g., A and A'). The point where the lines intersect is the centre of dilatation.
e) 1:2; the sides are in the ratio 1:2
2. a) It must have one polygonal base and an apex that is connected to the points on the edge of that base by line segments on the outside of the object.
b) It must have two parallel polygonal bases, and corresponding points on the two bases must be connected by line segments on the outside of the object.
c) A right prism has bases that meet the sides at 90° angles. The bases of an oblique prism do not meet the sides at 90° angles.
3. a) yes; it has two parallel congruent bases b) yes, it all its faces are polygons c) no; it has more than five faces
d) right; the bases meet the sides at 90° angles e) Diagrams may vary.
f) The perspective drawing shows realistic depth, but does not accurately show the lengths of the sides.
4. a) AC = 1.87, EF = 20, DF = 23.38
b) VW = 21 cm, WX = 10.5, XY = 30, UV = 33
5. a) congruent; all angles are equal and side AC is common b)EHG is congruent to FGH. Two pairs of sides are equal and the angles contained by those sides are equal.
6. a) cylinders; they have parallel circular bases and lines joining the edges of those bases are on the outside of the boxes b) similar; they are all circles, but have different diameters
7.
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9 Practice Test
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1. D 2. A 3. B 4. B 5. C
6. a) The gardens that measure 1.5 m 6 m and 2 m 8 m are similar. The sides are in the same ratio (1:4). b) Answers may vary, 4 m 12 m and 5 m 15 m
7. a) no, corresponding sides are not all proportional b), d) Diagrams may vary. c) Answers may vary.
e) trapezoidal prism
8. a)
b) 18.9 m c) 6 m
9. a) hexagonal prism; it has two parallel hexagonal bases and lines connecting corresponding points on the edges of those bases along the outside of its sides
b)
c) The isometric drawing accurately portrays the side lengths of the prism.
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