Review Exercise 9

Review Exercise 9 Solutions

Comprehensive syllabus checks, MCQs, diagonals count, jugs/bottles capacity scaling, and compound tessellation areas.

Question 1 MCQ Quiz
Choose the correct option for each question. Perform the interactive self-grading quiz below to master similar figures concepts.
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Question 2 Interior sum
If the sum of the interior angles of a polygon is 1080°, how many sides does the polygon have?
Solution Step-by-Step
The sum of the interior angles formula is:
Sum = (n - 2) × 180°
Substitute Sum = 1080°:
(n - 2) × 180° = 1080°
n - 2 = 1080°180° = 6
n = 6 + 2 = 8
Thus, the polygon has 8 sides (regular octagon).
Question 3 Similar Bottles
Two similar bottles are such that one is twice as high as the other. What is the ratio of their surface areas and their capacities?
Solution Step-by-Step
Since the bottles are similar, let their height ratio be h1 : h2 = 2 : 1.
(i) Ratio of surface areas:
The ratio of surface areas corresponds to the square of the height ratio:
A1A2 = (h1h2)2 = (21)2 = 4:1
(ii) Ratio of capacities (volumes):
The ratio of capacities corresponds to the cube of the height ratio:
C1C2 = (h1h2)3 = (21)3 = 8:1
Thus, the surface area ratio is 4:1 and the capacity ratio is 8:1.
Question 4 Model Car Ratios
Each dimension of a model car is 110 of the corresponding car dimension. Find the ratio of:
(a) the areas of their windscreens
(b) the capacities of their boots
(c) the widths of the cars
(d) the number of wheels they have
Solution Step-by-Step
The linear scale factor from model to actual car is k = 1/10.
(a) Ratio of the areas of their windscreens:
Area Ratio = k2 = (110)2 = 1:100
(b) Ratio of the capacities of their boots:
Capacity Ratio = k3 = (110)3 = 1:1000
(c) Ratio of the widths of the cars:
Since width is a linear dimension, the ratio equals the linear scale factor:
Width Ratio = k = 1:10
(d) Ratio of the number of wheels:
Both cars have exactly 4 wheels, as wheels count is not scaled geometrically:
Wheels Ratio = 4 : 4 = 1:1
Question 5 Three Jugs
Three similar jugs have heights 8 cm, 12 cm, and 16 cm. If the smallest jug holds 0.5 liters, find the capacities of the other two.
Solution Step-by-Step
Let the heights of the three jugs be h1 = 8 cm, h2 = 12 cm, and h3 = 16 cm. The capacity of the smallest jug is V1 = 0.5 liters.
(a) Capacity of the second jug (V2):
V2V1 = (h2h1)3
V20.5 = (128)3 = (1.5)3 = 3.375
V2 = 0.5 × 3.375 = 1.6875 liters ≈ 1.69 liters
(b) Capacity of the third jug (V3):
V3V1 = (h3h1)3
V30.5 = (168)3 = 23 = 8
V3 = 0.5 × 8 = 4 liters
Question 6 Three Glasses
Three similar drinking glasses have heights 7.5 cm, 9 cm and 10.5 cm. If the tallest glass holds 343 millilitres, find the capacities of the other two.
Solution Step-by-Step
Let the heights be h1 = 7.5 cm, h2 = 9 cm, and h3 = 10.5 cm. The capacity of the tallest glass is V3 = 343 ml.
(a) Capacity of the first glass (V1):
V1V3 = (h1h3)3
V1343 = (7.510.5)3 = (57)3 = 125343
V1 = 343 × 125343 = 125 ml
(b) Capacity of the second glass (V2):
V2V3 = (h2h3)3
V2343 = (910.5)3 = (67)3 = 216343
V2 = 343 × 216343 = 216 ml
Question 7 Model Car Dimensions
A toy manufacturer produces model cars which are similar in every way to the actual cars. If the ratio of the door area of the model to the door area of the car is 1 cm² to 2500 cm², find:
(a) the ratio of their lengths
(b) the ratio of the capacities of their petrol tanks
(c) the width of the model, if the actual car is 150 cm wide
(d) the area of the rear window of the actual car if the area of the rear window of the model is 3 cm².
Solution Step-by-Step
The ratio of door areas is A1 : A2 = 1 : 2500.
(a) Ratio of their lengths:
The linear ratio is the square root of the area ratio:
Linear Ratio = √(12500) = 1:50
(b) Ratio of capacities of petrol tanks:
The volume ratio is the cube of the linear ratio:
Volume Ratio = (1/50)3 = 1:125,000
(c) Width of the model car:
Since width is a linear dimension, it scales by the linear ratio 1:50:
Widthmodel = Widthactual / 50 = 150 / 50 = 3 cm
(d) Area of the rear window of the actual car:
Areas scale by the area ratio 2500:
Areaactual = 2500 × Areamodel = 2500 × 3 = 7,500 cm²
Question 8 Coffee Jars
The ratio of the areas of two similar labels on two similar jars of coffee is 144 : 169. Find the ratio of:
(a) the heights of the two jars
(b) their capacities
Solution Step-by-Step
Given the area ratio of the labels is A1 : A2 = 144 : 169.
(a) Ratio of the heights of the two jars:
The height (linear) ratio is the square root of the area ratio:
Height Ratio = √(144169) = 12:13
(b) Ratio of their capacities:
The volume ratio is the cube of the linear ratio:
Capacity Ratio = (12/13)3 = 1728 : 2197
Question 9 Floor Tessellation
A tessellation of tiles on a floor has been made using a repeating pattern of a regular hexagon, six squares, and six equilateral triangles. Find the total area of a single pattern with side length 0.5 metre of each polygon.
Solution Step-by-Step
The repeating pattern contains: 1 regular hexagon, 6 squares, and 6 equilateral triangles. The side length is s = 0.5 m.
(i) Area of 6 squares:
Area of 1 square = s2 = 0.52 = 0.25 m²
Area of 6 squares = 6 × 0.25 = 1.5 m²
(ii) Area of 6 equilateral triangles:
Area of 1 triangle = √34 × s2 = √34 × 0.25 ≈ 0.10825 m²
Area of 6 triangles = 6 × 0.10825 ≈ 0.65 m²
(iii) Area of 1 regular hexagon:
Area of hexagon = 6 × Area of 1 triangle = 6 × 0.10825 ≈ 0.65 m²
(iv) Total area of the single repeating pattern:
Total Area = Areasquares + Areatriangles + Areahexagon
= 1.5 + 0.65 + 0.65 = 2.8 m²
Thus, the total area of the pattern is 2.8 m².
Interactive Sandbox Universal Similarity Calculator
Scaling Relationships:
Linear ratio (k) = -
Area ratio (k²) = -
Volume ratio (k³) = -
Type any ratio in the boxes. The calculator will automatically deduce the remaining ratios using the geometric scaling laws:
• Area scales as k²
• Volume/Capacity scales as k³