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The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.
- a)35 cm2
- b)44 cm2
- c)72 cm2
- d)58 cm2
Correct answer is option 'C'. Can you explain this answer?
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The mid-points of the adjacent sides of a square are joined. Again the...
The side of the outermost square is 6 cm. Area of the square one in G.P with first term 36 and common ratio = 1/2
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The mid-points of the adjacent sides of a square are joined. Again the...
Let the side of the outermost square be $s$. Then its diagonal is $s\sqrt{2}=6$, so $s=3\sqrt{2}$.
The first figure formed by joining the midpoints of adjacent sides is a square whose side length is $\frac{s}{\sqrt{2}}=\frac{3\sqrt{2}}{2}$. Its area is $\left(\frac{3\sqrt{2}}{2}\right)^2=\frac{27}{4}$.
The second figure formed by joining the midpoints of adjacent sides of the first figure is a square whose side length is $\frac{s}{2}$. Its area is $\left(\frac{s}{2}\right)^2=\frac{18}{4}=4.5$.
The third figure formed by joining the midpoints of adjacent sides of the second figure is a square whose side length is $\frac{s}{2\sqrt{2}}=\frac{3}{2}$. Its area is $\left(\frac{3}{2}\right)^2=\frac{9}{4}$.
This process continues indefinitely, with each successive figure being a square whose side length is half the previous square's side length. The sum of the areas of all these squares is therefore:
$$\frac{27}{4}+4.5+\frac{9}{4}+\frac{9}{16}+\frac{1}{4}+\frac{1}{64}+\cdots$$
This is an infinite geometric series with first term $\frac{27}{4}$ and common ratio $\frac{1}{8}$. Therefore, the sum is:
$$\frac{\frac{27}{4}}{1-\frac{1}{8}}=\frac{\frac{27}{4}}{\frac{7}{8}}=\frac{27}{4}\cdot\frac{8}{7}=\boxed{12}$$
The first figure formed by joining the midpoints of adjacent sides is a square whose side length is $\frac{s}{\sqrt{2}}=\frac{3\sqrt{2}}{2}$. Its area is $\left(\frac{3\sqrt{2}}{2}\right)^2=\frac{27}{4}$.
The second figure formed by joining the midpoints of adjacent sides of the first figure is a square whose side length is $\frac{s}{2}$. Its area is $\left(\frac{s}{2}\right)^2=\frac{18}{4}=4.5$.
The third figure formed by joining the midpoints of adjacent sides of the second figure is a square whose side length is $\frac{s}{2\sqrt{2}}=\frac{3}{2}$. Its area is $\left(\frac{3}{2}\right)^2=\frac{9}{4}$.
This process continues indefinitely, with each successive figure being a square whose side length is half the previous square's side length. The sum of the areas of all these squares is therefore:
$$\frac{27}{4}+4.5+\frac{9}{4}+\frac{9}{16}+\frac{1}{4}+\frac{1}{64}+\cdots$$
This is an infinite geometric series with first term $\frac{27}{4}$ and common ratio $\frac{1}{8}$. Therefore, the sum is:
$$\frac{\frac{27}{4}}{1-\frac{1}{8}}=\frac{\frac{27}{4}}{\frac{7}{8}}=\frac{27}{4}\cdot\frac{8}{7}=\boxed{12}$$
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The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.a)35 cm2b)44 cm2c)72 cm2d)58 cm2Correct answer is option 'C'. Can you explain this answer?
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The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.a)35 cm2b)44 cm2c)72 cm2d)58 cm2Correct answer is option 'C'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.a)35 cm2b)44 cm2c)72 cm2d)58 cm2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.a)35 cm2b)44 cm2c)72 cm2d)58 cm2Correct answer is option 'C'. Can you explain this answer?.
The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.a)35 cm2b)44 cm2c)72 cm2d)58 cm2Correct answer is option 'C'. Can you explain this answer? for Quant 2024 is part of Quant preparation. The Question and answers have been prepared according to the Quant exam syllabus. Information about The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.a)35 cm2b)44 cm2c)72 cm2d)58 cm2Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The mid-points of the adjacent sides of a square are joined. Again the mid-points of the adjacent sides of the newly formed figure are connected and this process is repeated again and again. Calculate the sum of the areas of all such figures given that the diagonal of outermost square is 6√2cm.a)35 cm2b)44 cm2c)72 cm2d)58 cm2Correct answer is option 'C'. Can you explain this answer?.
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