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

Question

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

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chetan2u · Answer

circumscribing means containing a triangle that has its vertices on the circumference... Area5 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

danielarasan · Answer

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

jrk23 · Answer

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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jrk23 · Answer

Sorry you are right it will be diameter

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MathRevolution · Answer

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.

Therefore, the answer is C.

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.

rishabhmishra · Answer

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.
so c is the answer

cruiseav · Answer

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
Thanks in advance..

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

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GMAT Data Sufficiency (DS) Questions

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

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