Exploring Some Geometric Themes Class 8 Extra Questions Maths Part 2 Chapter 4

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Class 8 Exploring Some Geometric Themes Extra Questions

Class 8 Maths Chapter 4 Exploring Some Geometric Themes Extra Questions

Exploring Some Geometric Themes Extra Questions Class 8

Question 1.
In the Sierpinski Triangle, find the ratio of the area removed in the nth step to the area remaining after the (n – 1)^th step. Is this ratio the same for all steps?
Solution:
From each triangle, \(\frac {1}{4}\) of its area is removed at each step.
Therefore, the fraction removed from the whole figure is always \(\frac {1}{4}\) of the area present in the previous step.
Hence, the ratio of the area removed in the nth step to the area remaining after the (n – 1)^th step is 1 : 4.
This ratio does not depend on n, so it is constant for all steps.

Question 2.
An equilateral triangle of side length 1 unit is used to form a Koch snowflake. By what factor does the perimeter increase from one step to the next? Using this factor, find the perimeter at step 3.
Solution:
From the construction of the Koch snowflake, the perimeter is multiplied by \(\frac {4}{3}\) at each step.
Since the perimeter at step 0 is 3 units, the perimeter at step 3 = \(3 \times \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3}=\frac{64}{9}\) units.

Question 3.
In a Koch snowflake construction, the length of each small side at step 2 is \(\frac {1}{9}\) units.
(a) What was the length of each side at step 1?
(b) What will be the length of each side at step 4?
Solution:
(a) Since, at each step, the sidelength becomes one-third of the previous step.
It is given that the length of each small side at step 2 = \(\frac {1}{9}\) units
So, length of each small side at step 1 = \(\frac {1}{9}\) × 3 = \(\frac {1}{3}\) units

(b) From step 2 to step 4 is two steps, so we need to divide the sidelength of step 2 by 3 twice.
\(\frac{1}{9} \div 3 \div 3=\frac{1}{9} \times \frac{1}{3} \times \frac{1}{3}=\frac{1}{81}\) units

Question 4.
Draw the top, front, and side views of the following solid.

Solution:
The following are the top, front, and side views of the given solid.

Question 5.
In the given figure, p is the projection of l. If EB = 20 cm and the length of l is 101 cm, then find the length of its projection p.

Solution:
In the right-angled triangle AEB, using the Baudhayana Theorem, we get
l² = p² + EB²
⇒ p² = l² – EB²
⇒ p² = 101² – 20²
⇒ p² = 10201 – 400
⇒ p² = 9801
⇒ p = √9801 = 99
Hence, the length of the projection is 99 cm.

Question 6.
Find the number of remaining squares at Step 3 in Sierpinski carpet.
Solution:
The total number of remaining squares at step n in Sierpinski carpet = 8″.
Then, the total number of remaining squares in Step 3
= 8³ = 8 × 8 × 8
= 512

Question 7.
How many solid triangles are present at Step 2?
Solution:
We know that the total number of remaining triangles at step n in Sierp inski gasket = 3ⁿ.
Then, the total number of remaining triangles at Step 2 = 3² = 9

Question 8.
A Koch Snowflake begins with an equilateral triangle, where each side is 12 cm long. Calculate the total perimeter of the shape after Step 2 of the construction.
Solution:
The initial shape is an equilateral triangle having three sides, each of length 12 cm.
P₀ = 3 × 12 = 36 cm
According to the theory, the perimeter at step n is (\(\frac{4}{3}\))ⁿ times the original perimeter.

The formula is P_n = P₀ × \(\left(\frac{4}{3}\right)^n\)
∴ P₂ = 36 × (\(\frac{16}{9}\))
= 36 × \(\frac{16}{9}\)
⇒ P₂ = 4 × 16
= 64 cm
After Step 2, the new perimeter of the shape is 64 cm.

Question 9.
Write the number of faces, edges and vertices of each shape in the table.
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Solution:
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Question 10.
A spider is at one corner of a rectangular box of dimensions 10 cm x 6 cm x 4 cm. It wants to reach the diagonally opposite corner on the outer surface. Find the shortest path.
im-3
Solution:
According to question, the position of spider and final position by A and B in the given figure.
In ∆ACB, by Baudhayana-Pythagoras theorem,
AB² = AC² + BC² =(10 + 6)² + 4²
= 162 + 16
= 256 + 16
⇒ AB = \(\sqrt{272}\)
Hence, thee distance between spider and her final position is \(\sqrt{272}\) cm.

Question 11.
Identify the correct nets from the following which is valid to make cylinder as.
im-4
Solution:
A cylinder net must have one rectangle which forms the curved surface two circles which form the top and bottom
So, in the given figures, nets (ii) and (iii) are two valid nets to make cylinders.

Question 12.
Which among the following nets can be used to form cuboids?
im-5
Solution:
A valid cuboid net must have 6 faces arranged so that opposite faces can close without overlape.
(i) Valid (faces fold properly to form a cuboid)
(ii) Valid (standard net pattern)
(iii) Not valid (two long strips cannot fold to close the top and bottom faces)
(iv) Valid (all faces meet correctly)
So, (i), (ii) and (iv) can be used to make cuboids.

Question 13.
Dice are cubes with dots on each face. Opposite faces of a dice always have a total of seven dots on them. Here are two nets to make dice (cubes), the numbers inserted in each square indicate the number of dots in the box. Insert suitable numbers in the blanks, remembring that the sum of number on the opposite faces of the dice should be 7.
im-6
Solution:
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Question 14.
Can this be a net for a die? Explain your answer.
im-8
Solution:
Yes, this figure is a net of a die because it has six squares arranged, so they can fold to form a cube without overlapping. Also, the numbers are placed correctly as on a standard die, where opposite faces add upto 7. Hence, it forms a proper die.

Question 15.
Match the nets with appropriate solids.
im-9
Solution:
(i) → (d), (ii) → (a), (iii) → (c), (iv) → (b)

Question 16.
For each solid given below, the three views (a), (b) and (c) are given. Identify for each solid the corresponding top, front and side views.
im-10
Solution:
(i) (a) Top
(b) Side or front
(c) Side or front

(ii) (a) Side
(b) Front
(c) Top

(iii) (a) Top
(b) Side
(c) Front

Question 17.
Draw a view of each solid as seen from the direction indicated.
im-11
Solution:
The view of each solid as seen from the direction indicated by the arrow are given below.

Question 18.
A bulb is kept burning just right above the following solids. Name the shape of the shadows obtained in each case. Attempt to give a rough sketch of the shadow.
im-12
Solution:
When a bulb is kept burning just right above the given solid then
(i) the shape of the shadow of a pipe looks like a circle.
(ii) the shape of the shadow of a cone looks like a triangle.

Question 19.
From the following shadows of some 3D-objects. When seen under the lamp, identify the solid(s) that match each shadow.
im-13
Solution:
The given shadows can be obtained in the case of the following objects.
(i) A notebook or a cuboid shape.
(ii) A cube shaped object such as dice.

Question 20.
State whether the following statements are True or False.
(i) The cone can cast a shadow in the shape of a triangle.
(ii) The cone can cast a shadow in the shape of a rectangle.
Solution:
(i) True, the cone cast a shadow in the shape of a triangle.
(ii) False, the cone cannot cast a shadow in the shape of a rectangle.

Question 21.
The dimensions of a cuboid are 6 cm, 3 cm and 2 cm. Draw different isometric sketches of this cuboid.
Solution:
Given, dimensions of a cuboid are 6 cm, 3 cm and 2 cm. There different isometric sketches of cuboid are as follows,
(i) When length and height of front face are 6 cm and 3 cm.
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(ii) When length and height of front face are 2 cm and 6 cm.
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(iii) When length and height of front face are 3 cm and 6 cm.
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Question 22.
Three cubes each with 3 cm edge are placed side by side to form a cuboid. Sketch an isometric sketch of this cuboid.
Solution:
An isometric sketch of a cuboid formed by placing three cubes with 3 cm edge side by side is given below.
im-17
Then, length of cuboid = 3 + 3 + 3 = 9 cm,
breadth of cuboid = 3 cm
and height of cuboid = 3 cm

When some combined shapes are observed then some of the parts remain hidden from the viewer. Therefore, visualising solid shapes is a very useful skill through, which we can see the hidden parts of a solid shape.

Question 23.
Count the number of cubes in following arrangements.
im-18
Solution:
Number cubes in the given arrangements are as follows,
(i) 36
(ii) 20
(iii) 46

Question 24.
Two cubes each with 3 cm edge are placed one above the other. What would the dimensions of the resulting cuboid be?
im-19
Solution:
Two cubes each with 3 cm edge are placed one above the other then
length of the resulting cuboid = 3 cm,
breadth of resulting cuboid = 3 cm
and height of the resulting cuboid = 3 + 3 = 6 cm

Question 25.
Two dice are placed side by side as shown.
im-20
Can you find the total numbers on the faces opposite to the shown faces?
(i) 6 + 2
(ii) 3 + 4
Solution:
We know that opposite faces of a die always have a total of seven dots on them.
(i) Here, the face opposite to the face having 6 dots is the face having 1 dot and the face opposite to the face having 2 dots is the face having 5 dot.
So, the total number on the face opposite to 6 + 2 is 1 + 5 i.e. 6.

(ii) The face opposite to the face having 3 dots is the face having 4 dots and face opposite to the face having
4 dots is the face having 3 dots.
So, the total number on the face opposite to 3 + 4 = 4 + 3 i.e. 7.

Question 26.
Two cubes with 3 cm edge are placed side by side to form a cuboid. Try to make a oblique sketch and say what could be its length, breadth and height?
Solution:
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From the oblique sketch, it is clear that length = 6 cm, breadth = 3 cm and height = 3 cm.

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