Is the area of the triangle ABC more than 12 sq. units?

Question

(1) The GCD of the sides AB and AC is 2.
(2) The LCM of the sides AB and AC is 12.

Question made by me.

bharatdasaka · Answer

A is not sufficient as we will have a Number of AB, AC lengths that can have a GCD of 2. For Eg: 4,6 8,10. For Different AB AC values, the range of the third side will vary.

B is also not sufficient as we will have a number of AB AC lengths that can have LCM 12. For Eg: 4,3 6,12 1,12

If we take A and B together we will have : AB * AC=24. Now if AB and AC are Base and height of the triangle, then we will have the Area as 12. which is not more than 12(as per question)

if we do not have AB and AC Base and height then, AB and AC should be 4 and 6 respectively. That means AC should lie between 2 and 10. If we take AC=8, then using sqrt   formula we will have area less than 12. If we have AC =3, then we will have the area nearly 5. IE, always, the area will be less than 12.

There fore the answer should be C. Let me know if that the correct answer

aceacharya · Answer

The answer should be E
As both the statements speak of only two sides of the triangle. But the area of any triangle with only 2 fixed sides can theoritically have a range between 0  to infinity  to 1/2 A x B. Therefore both options should not be sufficient to answer the question.

Zarrolou · Answer

Hi aceacharya, this is not quite true.

If you have 2 sides of a triangle (say 6 and 9) the gratest area you can obtain is the case in which those sides form a 90° angle.
So in this case the greatest area is 6*9/2=27 (not infinite)

And also the area cannot be 0, otherwise the sides will "overlap" each other and we would have a line => not a triangle.

Hope this clarifies, let me know if you have doubts

aceacharya · Answer

You're right. My Bad!! The maximum cant be infinity. The min however can tend to zero as the angle between the two tends to zero.

mau5 · Answer

OE :

The maximum area of a triangle for two given sides is when these two sides include a right angle.

From F.S 1, we know that GCD =2. All we know is that the sides are of the form 2a and 2b, where a and b are co-primes .Clearly Insufficient, as for different values of a and b, the area might/might not be more than 12 sq.units.

From F.S 2, we know that the LCM is 12. Thus, for sides as 1 and 12, we have the maximum area as \frac{1}{2}*1*12  = 6 sq units, and it is not more than 12 sq.units. However for sides as 6 and 12, the maximum area is \frac{1}{2}*6*12  = 36 sq. units, which is more than 12 sq.units. Insufficient.

Thus, combining both together, AB*BC = GCD*LCM = 2*12 = 24. Now, whatever values the sides take on, the product of the two sides will always be 24. Thus, the maximum area of this triangle = \frac{1}{2}*AB*BC  = 12. Thus, as the maximum area for the given triangle is not more than 12, we have a definitive answer for the question.

C.

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Is the area of the triangle ABC more than 12 sq. units?

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