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Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral is
- a)12
- b)24
- c)36
- d)48
Correct answer is option 'B'. Can you explain this answer?
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Four angles of a quadrilateral are in G.P. Whose common ratio is an in...
Let the angles be a, ar, ar 2, ar 3.
Sum of the angles = a ( r 4- 1 ) /r -1 = a ( r 2 + 1 ) ( r + 1 ) = 360
a< 90 , and ar< 90, Therefore, a ( 1 + r ) < 180, or ( r 2 + 1 ) > 2
Therefore, r is not equal to 1. Trying for r = 2 we get a = 24 Therefore, The angles are 24, 48, 96 and 192.
Sum of the angles = a ( r 4- 1 ) /r -1 = a ( r 2 + 1 ) ( r + 1 ) = 360
a< 90 , and ar< 90, Therefore, a ( 1 + r ) < 180, or ( r 2 + 1 ) > 2
Therefore, r is not equal to 1. Trying for r = 2 we get a = 24 Therefore, The angles are 24, 48, 96 and 192.
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Four angles of a quadrilateral are in G.P. Whose common ratio is an in...
Given:
- The four angles of a quadrilateral are in geometric progression (G.P).
- The common ratio of the G.P is an integer.
- Two of the angles are acute, while the other two are obtuse.
To find:
The measure of the smallest angle of the quadrilateral.
Approach:
Let the four angles of the quadrilateral be a, ar, ar^2, and ar^3, where a is the smallest angle and r is the common ratio of the G.P.
We know that the sum of the angles of any quadrilateral is 360 degrees.
Therefore, the sum of the four angles of the given quadrilateral is:
a + ar + ar^2 + ar^3 = 360
Simplifying this equation, we get:
a(1 + r + r^2 + r^3) = 360
Since a, r, and (1 + r + r^2 + r^3) are all integers, the value of a must be a factor of 360.
To find the smallest possible value of a, we need to find the smallest factor of 360 that satisfies the given conditions.
Calculation:
The prime factorization of 360 is:
360 = 2^3 * 3^2 * 5
To find the smallest factor of 360, we need to consider the prime factors in ascending order.
- For a to be the smallest possible angle, it should be a factor of 360.
- Since a is an acute angle, it must be less than 90 degrees.
Considering the prime factors in ascending order, we have:
- If a = 2, the remaining factors of 360 are 2^2 * 3^2 * 5 = 180, which is not a factor of 360.
- If a = 3, the remaining factors of 360 are 2^3 * 3 * 5 = 120, which is not a factor of 360.
- If a = 5, the remaining factors of 360 are 2^3 * 3^2 = 72, which is not a factor of 360.
- If a = 2 * 3 = 6, the remaining factor of 360 is 2^2 * 5 = 20, which is not a factor of 360.
- If a = 2 * 5 = 10, the remaining factor of 360 is 2^3 * 3^2 = 72, which is not a factor of 360.
- If a = 3 * 5 = 15, the remaining factor of 360 is 2^3 * 3 = 24, which is not a factor of 360.
- If a = 2 * 3 * 5 = 30, the remaining factor of 360 is 2^3 = 8, which is a factor of 360.
Therefore, the smallest possible value for a is 30.
Conclusion:
The measure of the smallest angle of the quadrilateral is 30 degrees, which corresponds to option 'B'.
- The four angles of a quadrilateral are in geometric progression (G.P).
- The common ratio of the G.P is an integer.
- Two of the angles are acute, while the other two are obtuse.
To find:
The measure of the smallest angle of the quadrilateral.
Approach:
Let the four angles of the quadrilateral be a, ar, ar^2, and ar^3, where a is the smallest angle and r is the common ratio of the G.P.
We know that the sum of the angles of any quadrilateral is 360 degrees.
Therefore, the sum of the four angles of the given quadrilateral is:
a + ar + ar^2 + ar^3 = 360
Simplifying this equation, we get:
a(1 + r + r^2 + r^3) = 360
Since a, r, and (1 + r + r^2 + r^3) are all integers, the value of a must be a factor of 360.
To find the smallest possible value of a, we need to find the smallest factor of 360 that satisfies the given conditions.
Calculation:
The prime factorization of 360 is:
360 = 2^3 * 3^2 * 5
To find the smallest factor of 360, we need to consider the prime factors in ascending order.
- For a to be the smallest possible angle, it should be a factor of 360.
- Since a is an acute angle, it must be less than 90 degrees.
Considering the prime factors in ascending order, we have:
- If a = 2, the remaining factors of 360 are 2^2 * 3^2 * 5 = 180, which is not a factor of 360.
- If a = 3, the remaining factors of 360 are 2^3 * 3 * 5 = 120, which is not a factor of 360.
- If a = 5, the remaining factors of 360 are 2^3 * 3^2 = 72, which is not a factor of 360.
- If a = 2 * 3 = 6, the remaining factor of 360 is 2^2 * 5 = 20, which is not a factor of 360.
- If a = 2 * 5 = 10, the remaining factor of 360 is 2^3 * 3^2 = 72, which is not a factor of 360.
- If a = 3 * 5 = 15, the remaining factor of 360 is 2^3 * 3 = 24, which is not a factor of 360.
- If a = 2 * 3 * 5 = 30, the remaining factor of 360 is 2^3 = 8, which is a factor of 360.
Therefore, the smallest possible value for a is 30.
Conclusion:
The measure of the smallest angle of the quadrilateral is 30 degrees, which corresponds to option 'B'.
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Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral isa)12b)24c)36d)48Correct answer is option 'B'. Can you explain this answer?
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Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral isa)12b)24c)36d)48Correct answer is option 'B'. 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 Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral isa)12b)24c)36d)48Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral isa)12b)24c)36d)48Correct answer is option 'B'. Can you explain this answer?.
Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral isa)12b)24c)36d)48Correct answer is option 'B'. 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 Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral isa)12b)24c)36d)48Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Quant 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral isa)12b)24c)36d)48Correct answer is option 'B'. Can you explain this answer?.
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