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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?
Verified Answer
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.
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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'.
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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?.
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