Similarity (Copy)
1.
Triangle ABC is similar to triangle DEF.
AB = 6 cm, AC = 8 cm, BC = 10 cm.
DE = 9 cm.
(a) Find the scale factor of enlargement from ΔABC to ΔDEF.
(b) Find EF and DF.
2.
Two cylinders are similar. The height of the smaller cylinder is 12 cm and its radius is 4 cm. The radius of the larger cylinder is 10 cm.
(a) Find the scale factor of their lengths.
(b) Find the height of the larger cylinder.
(c) If the volume of the smaller cylinder is 603 cm³, find the volume of the larger cylinder.
3.
A cuboid has dimensions 2 cm × 3 cm × 4 cm. A similar cuboid has its shortest side measuring 5 cm.
(a) Find the scale factor.
(b) Find the dimensions of the larger cuboid.
(c) Find the volume of the larger cuboid.
4.
The areas of two similar triangles are in the ratio 9:25. The shorter side of the smaller triangle is 6 cm.
Find the corresponding side of the larger triangle.
5.
The lengths of two similar cones are in the ratio 3:5.
(a) Find the ratio of their surface areas.
(b) Find the ratio of their volumes.
6.
A model car is built to a scale of 1:20. The actual car is 4.6 m wide and 3.2 m high.
(a) Find the width of the model.
(b) Find the height of the model.
7.
Two spheres are similar. The radius of the smaller is 7 cm and the radius of the larger is 14 cm.
(a) Find the ratio of their surface areas.
(b) If the volume of the smaller is 1436 cm³, find the volume of the larger.
8.
Triangle XYZ is similar to triangle PQR.
XY = 5 cm, YZ = 8 cm, XZ = 10 cm.
PQ = 7.5 cm.
(a) Find PR and QR.
(b) Find the ratio of areas of the two triangles.
9.
The ratio of surface areas of two similar cones is 9:16. The slant height of the smaller cone is 12 cm. Find the slant height of the larger cone.
10.
A solid sphere of radius 9 cm is melted to form smaller spheres of radius 3 cm each.
(a) How many small spheres can be formed?
(b) If one sphere has a mass of 4 g per cm³, find the mass of one small sphere.
11.
A prism has volume 150 cm³. A similar prism has linear scale factor 2.
(a) Find the volume of the larger prism.
(b) Find the ratio of their surface areas.
12.
A right-angled triangle with sides 6 cm, 8 cm, 10 cm is similar to another triangle. The largest side of the other triangle is 25 cm.
(a) Find the scale factor.
(b) Find the lengths of the other two sides.
13.
Two rectangular water tanks are similar. The smaller tank holds 128 litres of water. The linear scale factor of enlargement is 3.
Find the capacity of the larger tank.
14.
Two triangles are similar. Their perimeters are 30 cm and 45 cm. The area of the smaller triangle is 72 cm².
Find the area of the larger triangle.
15.
A model of a building is made on a scale of 1:50. The real building is 25 m tall.
Find the height of the model in centimetres.
16.
Two cones are similar. The ratio of their volumes is 8:125.
(a) Find the ratio of their heights.
(b) If the smaller cone has height 6 cm, find the height of the larger cone.
17.
The areas of two similar kites are in the ratio 49:81. The shorter diagonal of the smaller kite is 14 cm.
Find the shorter diagonal of the larger kite.
18.
Two similar solids have lengths in the ratio 2:5.
(a) Find the ratio of their surface areas.
(b) Find the ratio of their volumes.
19.
A solid cone has height 24 cm and radius 7 cm. A similar cone has height 36 cm.
(a) Find the radius of the larger cone.
(b) Find the ratio of their volumes.
20.
A large sphere of volume 972 cm³ is melted down to make small spheres of radius 1 cm.
(a) Find the radius of the large sphere.
(b) Find the number of small spheres that can be made.
21.
The scale drawing of a garden is made where 1 cm represents 2 m. The actual garden is 30 m long and 20 m wide.
(a) Find the dimensions of the drawing.
(b) Find the ratio of the area of the drawing to the area of the garden.
22.
A cuboid has dimensions 3 cm × 4 cm × 5 cm. Another cuboid is similar and has volume 1080 cm³.
(a) Find the volume of the smaller cuboid.
(b) Find the linear scale factor.
(c) Find the dimensions of the larger cuboid.
23.
Two triangular prisms are similar. The smaller has surface area 45 cm², the larger has surface area 180 cm².
(a) Find the scale factor of their lengths.
(b) If the smaller has volume 135 cm³, find the volume of the larger.
24.
A model of a ship is made on a scale of 1:200. The real ship is 160 m long and 20 m wide.
Find the dimensions of the model in centimetres.
25.
Two pyramids are similar. The larger has volume 2500 cm³, the smaller has volume 320 cm³.
Find the linear scale factor of enlargement.
26.
Two similar cones have heights of 18 cm and 27 cm. The smaller has surface area 324 cm².
Find the surface area of the larger cone.
27.
The ratio of lengths of two similar prisms is 5:7. The smaller has volume 500 cm³.
Find the volume of the larger prism.
28.
A cylindrical candle has radius 3 cm and height 10 cm. A similar candle has height 15 cm.
Find the radius and volume of the larger candle.
29.
Two similar solids have volumes in the ratio 27:64. The surface area of the larger solid is 256 cm².
Find the surface area of the smaller solid.
30.
A cone has volume 200 cm³. A similar cone has linear scale factor 1.5.
Find the volume of the larger cone.
1.
Triangle ABC ~ Triangle DEF.
AB = 6 cm, AC = 8 cm, BC = 10 cm.
DE = 9 cm.
Solution:
- Scale factor = DE ÷ AB = 9 ÷ 6 = 1.5.
- EF corresponds to BC = 10 × 1.5 = 15 cm.
- DF corresponds to AC = 8 × 1.5 = 12 cm.
Answer: EF = 15 cm, DF = 12 cm.
2.
Two cylinders are similar.
Smaller: h = 12 cm, r = 4 cm.
Larger: r = 10 cm.
Solution:
(a) Scale factor = 10 ÷ 4 = 2.5.
(b) Height = 12 × 2.5 = 30 cm.
(c) Volume ratio = (2.5)³ = 15.625.
Volume = 603 × 15.625 = 9410 cm³ (approx).
Answer: Height = 30 cm, Volume ≈ 9410 cm³.
3.
Cuboid 2 × 3 × 4. Shortest side of similar cuboid = 5 cm.
Solution:
(a) Scale factor = 5 ÷ 2 = 2.5.
(b) New dimensions = 2.5 × (2, 3, 4) = 5, 7.5, 10.
(c) Volume = 5 × 7.5 × 10 = 375 cm³.
Answer: Dimensions 5 cm × 7.5 cm × 10 cm, Volume = 375 cm³.
4.
Areas ratio = 9:25, smaller side = 6 cm.
Solution:
- Side ratio = √(9:25) = 3:5.
- Larger side = 6 × (5/3) = 10 cm.
Answer: 10 cm.
5.
Length ratio = 3:5.
Solution:
(a) Surface area ratio = (3:5)² = 9:25.
(b) Volume ratio = (3:5)³ = 27:125.
Answer: (a) 9:25, (b) 27:125.
6.
Scale 1:20. Real: 4.6 m = 460 cm, 3.2 m = 320 cm.
Solution:
Width = 460 ÷ 20 = 23 cm.
Height = 320 ÷ 20 = 16 cm.
Answer: Width = 23 cm, Height = 16 cm.
7.
Spheres: radii 7 cm, 14 cm.
Solution:
(a) Surface area ratio = (7:14)² = 1:4.
(b) Volume ratio = (7:14)³ = 1:8.
Volume larger = 1436 × 8 = 11,488 cm³.
Answer: Ratio 1:4, Volume = 11,488 cm³.
8.
ΔXYZ ~ ΔPQR. XY = 5, PQ = 7.5.
Solution:
Scale factor = 7.5 ÷ 5 = 1.5.
PR = 10 × 1.5 = 15 cm.
QR = 8 × 1.5 = 12 cm.
Area ratio = (1.5)² = 2.25 = 9:4.
Answer: PR = 15 cm, QR = 12 cm, Ratio 9:4.
9.
Surface area ratio = 9:16, slant height = 12.
Solution:
Length ratio = √(9:16) = 3:4.
Larger slant height = 12 × (4/3) = 16 cm.
Answer: 16 cm.
10.
Sphere radius 9 → small spheres radius 3.
Solution:
Volume ratio = (9:3)³ = 27:1.
Number of small = 27.
Mass of one = Volume small × 4.
Small sphere volume = (4/3)π(3³) = 36π ≈ 113.1.
Mass ≈ 113.1 × 4 ≈ 452 g.
Answer: 27 spheres, 452 g each.
11.
Prism volumes.
Solution:
Volume ratio = 2³ = 8.
Larger volume = 150 × 8 = 1200.
Surface area ratio = 2² = 4.
Answer: Volume = 1200 cm³, Ratio 1:4.
12.
Right triangle sides 6–8–10, new hypotenuse = 25.
Solution:
Scale factor = 25 ÷ 10 = 2.5.
Other sides = 6 × 2.5 = 15, 8 × 2.5 = 20.
Answer: 15 cm, 20 cm.
13.
Tanks ratio linear = 3.
Solution:
Volume ratio = 3³ = 27.
Larger capacity = 128 × 27 = 3456 litres.
Answer: 3456 L.
14.
Perimeter ratio = 30:45 = 2:3.
Solution:
Area ratio = (2:3)² = 4:9.
Area = 72 × (9/4) = 162 cm².
Answer: 162 cm².
15.
Scale 1:50. Height 25 m = 2500 cm.
Solution:
Model = 2500 ÷ 50 = 50 cm.
Answer: 50 cm.
16.
Volume ratio 8:125.
Solution:
Length ratio = ∛(8:125) = 2:5.
If small height = 6 → large = 6 × (5/2) = 15 cm.
Answer: 15 cm.
17.
Area ratio = 49:81.
Solution:
Length ratio = √(49:81) = 7:9.
Large diagonal = 14 × (9/7) = 18 cm.
Answer: 18 cm.
18.
Lengths 2:5.
Solution:
Surface area ratio = 4:25.
Volume ratio = 8:125.
Answer: 4:25, 8:125.
19.
Cones 24,7 and 36,h.
Solution:
Scale factor = 36 ÷ 24 = 1.5.
Radius = 7 × 1.5 = 10.5 cm.
Volume ratio = (1.5)³ = 3.375.
Answer: Radius = 10.5 cm, Volume ratio = 27:8.
20.
Volume = 972.
Solution:
V = (4/3)πr³ = 972.
r³ = (972 × 3)/(4π) ≈ 232.
r ≈ 6.2 cm.
Small volume = (4/3)π(1³) ≈ 4.19.
Number = 972 ÷ 4.19 ≈ 232.
Answer: r ≈ 6.2 cm, 232 spheres.
21.
Scale: 1 cm = 2 m.
Solution:
Drawing: 30 ÷ 2 = 15 cm, 20 ÷ 2 = 10 cm.
Drawing area = 15 × 10 = 150 cm².
Real area = 30 × 20 = 600 m² = 60000 dm² = 6,000,000 cm².
Ratio = 150:6,000,000 = 1:40,000.
Answer: 15 cm × 10 cm, ratio 1:40,000.
22.
Cuboid 3 × 4 × 5.
Solution:
Volume small = 60 cm³.
Scale factor = ∛(1080 ÷ 60) = ∛18 ≈ 2.62.
Dimensions ≈ 7.86, 10.48, 13.1 cm.
Answer: ≈ 7.9 × 10.5 × 13.1 cm.
23.
Surface areas 45,180.
Solution:
Area ratio = 1:4.
Length ratio = √(1:4) = 1:2.
Volume ratio = 1:8.
Volume = 135 × 8 = 1080.
Answer: Scale factor 2, Volume = 1080 cm³.
24.
Scale 1:200. Real: 160 m, 20 m = 16,000 cm, 2000 cm.
Solution:
Model = 16,000 ÷ 200 = 80 cm, 2000 ÷ 200 = 10 cm.
Answer: 80 cm × 10 cm.
25.
Volumes 2500:320.
Solution:
Ratio = 2500 ÷ 320 = 7.8125 ≈ 195:25.
Length ratio = ∛(2500/320) = ∛7.8125 ≈ 1.99 ≈ 2.
Answer: Scale factor ≈ 2.
26.
Heights 18:27 = 2:3.
Solution:
Area ratio = (2:3)² = 4:9.
Large area = 324 × (9/4) = 729.
Answer: 729 cm².
27.
Volumes.
Solution:
Length ratio = 5:7.
Volume ratio = (5:7)³ = 125:343.
Large volume = 500 × (343/125) = 1372.
Answer: 1372 cm³.
28.
Candle radius 3, h = 10 → h = 15.
Solution:
Scale factor = 15 ÷ 10 = 1.5.
Radius = 3 × 1.5 = 4.5 cm.
Volume ratio = (1.5)³ = 3.375.
Small volume = π × 3² × 10 = 90π ≈ 283 cm³.
Large = 283 × 3.375 ≈ 955 cm³.
Answer: r = 4.5 cm, V ≈ 955 cm³.
29.
Volume ratio 27:64.
Solution:
Length ratio = ∛(27:64) = 3:4.
Area ratio = (3:4)² = 9:16.
If large area = 256, small = 256 × (9/16) = 144.
Answer: 144 cm².
30.
Volume 200, scale 1.5.
Solution:
Volume ratio = (1.5)³ = 3.375.
Large volume = 200 × 3.375 = 675.
Answer: 675 cm³.