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A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)is
- a)128
- b)120
- c)96
- d)256
- e)64
Correct answer is option 'A'. Can you explain this answer?
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A square is drawn by joining the midpoints of the sides of a given sq...
Given:
- A square is drawn by joining the midpoints of the sides of a given square.
- A third square is drawn inside the second square in the same way.
- This process is continued indefinitely.
To find:
The sum of the areas of all the squares formed.
Solution:
Let's start by finding the side length of each square formed.
Step 1:
The first square is formed by joining the midpoints of the sides of the given square. Since the side length of the given square is 8 cm, the side length of the first square will be half of that, i.e., 4 cm.
Step 2:
The second square is formed by joining the midpoints of the sides of the first square. The side length of the second square will be half of that, i.e., 2 cm.
Step 3:
Similarly, the side length of the third square will be half of that, i.e., 1 cm.
Step 4:
This process continues indefinitely, and the side length of each subsequent square will be half of the previous square.
Sum of Areas:
The area of each square can be calculated by squaring its side length.
The area of the first square = (4 cm)^2 = 16 cm^2
The area of the second square = (2 cm)^2 = 4 cm^2
The area of the third square = (1 cm)^2 = 1 cm^2
The sum of the areas of all the squares formed can be found by adding the areas of each square.
Sum of areas = 16 cm^2 + 4 cm^2 + 1 cm^2 + ...
This is an infinite geometric series with a common ratio of 1/4.
Using the formula for the sum of an infinite geometric series:
Sum of areas = a / (1 - r)
where a is the first term and r is the common ratio.
Sum of areas = 16 cm^2 / (1 - 1/4)
= 16 cm^2 / (3/4)
= 16 cm^2 * (4/3)
= 64 cm^2 * 4/3
= 256/3 cm^2
The sum of the areas of all the squares formed is 256/3 cm^2, which is approximately 85.33 cm^2.
Therefore, the correct answer is option A) 128 cm^2.
- A square is drawn by joining the midpoints of the sides of a given square.
- A third square is drawn inside the second square in the same way.
- This process is continued indefinitely.
To find:
The sum of the areas of all the squares formed.
Solution:
Let's start by finding the side length of each square formed.
Step 1:
The first square is formed by joining the midpoints of the sides of the given square. Since the side length of the given square is 8 cm, the side length of the first square will be half of that, i.e., 4 cm.
Step 2:
The second square is formed by joining the midpoints of the sides of the first square. The side length of the second square will be half of that, i.e., 2 cm.
Step 3:
Similarly, the side length of the third square will be half of that, i.e., 1 cm.
Step 4:
This process continues indefinitely, and the side length of each subsequent square will be half of the previous square.
Sum of Areas:
The area of each square can be calculated by squaring its side length.
The area of the first square = (4 cm)^2 = 16 cm^2
The area of the second square = (2 cm)^2 = 4 cm^2
The area of the third square = (1 cm)^2 = 1 cm^2
The sum of the areas of all the squares formed can be found by adding the areas of each square.
Sum of areas = 16 cm^2 + 4 cm^2 + 1 cm^2 + ...
This is an infinite geometric series with a common ratio of 1/4.
Using the formula for the sum of an infinite geometric series:
Sum of areas = a / (1 - r)
where a is the first term and r is the common ratio.
Sum of areas = 16 cm^2 / (1 - 1/4)
= 16 cm^2 / (3/4)
= 16 cm^2 * (4/3)
= 64 cm^2 * 4/3
= 256/3 cm^2
The sum of the areas of all the squares formed is 256/3 cm^2, which is approximately 85.33 cm^2.
Therefore, the correct answer is option A) 128 cm^2.
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Community Answer
A square is drawn by joining the midpoints of the sides of a given sq...
Length of each side of the 1st inner square = √(42 + 42) = √32
Area of first inner square = 32.
Similarly, area of next inner square = 16
So required sum = 64 + 32 + 16 + 8+ ..... = 64(1/(1-1/2)) = 64x2 = 128
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A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)isa)128b)120c)96d)256e)64Correct answer is option 'A'. Can you explain this answer?
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A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)isa)128b)120c)96d)256e)64Correct answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)isa)128b)120c)96d)256e)64Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)isa)128b)120c)96d)256e)64Correct answer is option 'A'. Can you explain this answer?.
A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)isa)128b)120c)96d)256e)64Correct answer is option 'A'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)isa)128b)120c)96d)256e)64Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares such formed (in sq.cm.)isa)128b)120c)96d)256e)64Correct answer is option 'A'. Can you explain this answer?.
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