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# Is the area of the circle circumscribing triangle ABC less than 25π?

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Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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23 Feb 2018, 23:51
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Is the area of the circle circumscribing triangle ABC less than 25π?

1) Triangle ABC is a right angle triangle with sides length integers
2) Two of the Sides of Triangle ABC are 6 and 8

Source: www.GMATinsight.com

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Re: Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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24 Feb 2018, 09:40
GMATinsight wrote:
Is the area of the circle circumscribing triangle ABC less than 25π?

1) Triangle ABC is a right angle triangle with sides length integers
2) Two of the Sides of Triangle ABC are 6 and 8

Source: http://www.GMATinsight.com

circumscribing means containing a triangle that has its vertices on the circumference...

1) Triangle ABC is a right angle triangle with sides length integers
triangle with sides 3,4,5 .. ans YES here R<5
triangle with sides 12,16,20... NO here R>5
insuff

2) Two of the Sides of Triangle ABC are 6 and 8
if the angle between these TWO sides is nearly 180, that is almost a straight line, third side will be slightly < 8+6 so R>5
if the sides are 6,6,8, R<5...ans NO
insuff

combined
since sides are integers, 8 cannot be hypotenuse and sides will be 6,8,10
now if a side of 10 is a chord , R cannot be < 5 so area is NOT <25Pi
suff

C
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Re: Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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24 Feb 2018, 13:45
Statement 1: Not sufficient
Could be a triangle 3:4:5 and the circle area would be <25π
Could be a triangle 21:28:35 and the circle area would be >25π

Statement 2: Not sufficient
If you draw a 6:8:10 triangle you will see that the circle area will be = 25π
If the triangle is a 2√7:6:8 (8 being the hypothenuse) then the area of the circle will be < 25π

Both statements together: Sufficient
Because statement 1 says that the triangle has side lengths integers, the only possibility is a 6:8:10 triangle that the circle area = 25π. That's it, you have an answer
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Re: Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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25 Feb 2018, 02:27
Hi Chetan and danielarasan

When r is 5 area is 25π

Which is equal to 25π

Question is asking us to find less than 25π with that statement can't we say for sure that area will not be less than 25π

Will that not be sufficient?

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Re: Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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25 Feb 2018, 02:29
Sorry you are right it will be diameter

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Re: Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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27 Feb 2018, 12:45
GMATinsight wrote:
Is the area of the circle circumscribing triangle ABC less than 25π?

1) Triangle ABC is a right angle triangle with sides length integers
2) Two of the Sides of Triangle ABC are 6 and 8

Source: http://www.GMATinsight.com

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 3 variables for sides of the triangles and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
We can determine the circle from the triangle, which means if we have a specific triangle, we can determine the circumscribing circle.

Condition 1) & 2):
$$a^2 + b^2 = c^2$$ from the condition 1) where $$c$$ is a hypotenuse of the triangle.
$$a = 6$$ and $$b = 8$$ from the condition 2) and we have $$c = 10$$.

The diameter of the circumscribing circle is the length of the right triangle's hypotenuse.

Thus both conditions together are sufficient.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.
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Re: Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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28 Feb 2018, 23:46
GMATinsight wrote:
Is the area of the circle circumscribing triangle ABC less than 25π?

1) Triangle ABC is a right angle triangle with sides length integers
2) Two of the Sides of Triangle ABC are 6 and 8

Source: http://www.GMATinsight.com

IMO C
1st statement inform about that its a right triangle that means the diameter of the circle is the hypotenuse of the triangle but don't have any value so insufficient.
2nd statement we know about 2 side of the triangle but don't know anything about the relation between circle and triangle so insufficient
combining both we get diameter 10 and we can find the area of the triangle.
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Re: Is the area of the circle circumscribing triangle ABC less than 25π?  [#permalink]

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13 May 2018, 08:26
chetan2u wrote:
GMATinsight wrote:
Is the area of the circle circumscribing triangle ABC less than 25π?

1) Triangle ABC is a right angle triangle with sides length integers
2) Two of the Sides of Triangle ABC are 6 and 8

Source: http://www.GMATinsight.com

circumscribing means containing a triangle that has its vertices on the circumference...

1) Triangle ABC is a right angle triangle with sides length integers
triangle with sides 3,4,5 .. ans YES here R<5
triangle with sides 12,16,20... NO here R>5
insuff

2) Two of the Sides of Triangle ABC are 6 and 8
if the angle between these TWO sides is nearly 180, that is almost a straight line, third side will be slightly < 8+6 so R>5
if the sides are 6,6,8, R<5...ans NO
insuff

combined
since sides are integers, 8 cannot be hypotenuse and sides will be 6,8,10
now if a side of 10 is a chord , R cannot be < 5 so area is NOT <25Pi
suff

C

hi chetan2u

I didn't understand the logic you gave for Statement-2. As its not given that the triangle is a Right-angle so we can't apply D/2= Radius of the circle, concept here
Re: Is the area of the circle circumscribing triangle ABC less than 25π? &nbs [#permalink] 13 May 2018, 08:26
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