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QOTD #6 Two sides of a triangle have length 6 and 8

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QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 20 Jul 2016, 13:07
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Two sides of a triangle have length 6 and 8. Which of the following can be the possible areas of the triangle?

I. 2
II. 12
III. 24

Indicate all possible values



[Reveal] Spoiler:
I, II and III
[Reveal] Spoiler: OA

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Last edited by GreenlightTestPrep on 25 Jan 2017, 12:25, edited 2 times in total.
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Re: Two sides of a triangle have length 6 and 8 [#permalink] New post 20 Jul 2016, 19:32
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The largest triangle will be right angle triangle with sides 6 and 8, so the maximum area will be 1/2*6*8 = 24. And the minimum area will be when one of the angle approaches 180 degree. When when of the angle approaches 180 degree, the area will approach 0. Hence all three choice are valid.
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Re: QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 02 Jan 2017, 15:55
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Hi-

so is the answer not B and C?

I thought that the third side must be at least 2+ (based on third side rule), if we multiply 2 *6/2, the smallest area I computed was 6.

Are we just supposed to assume that this third side is not the same as the height of the triangle? and we can make the area very small if the angle between the two larger sides is approaching zero?

Thanks,


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Re: QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 25 Jan 2017, 12:48
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soumya1989 wrote:
Two sides of a triangle have length 6 and 8. Which of the following can be the possible areas of the triangle?

I. 2
II. 12
III. 24

Indicate all possible values



[Reveal] Spoiler:
I, II and III


Image

As we can see from the diagrams above, all 3 areas are possible.

Area = (base)(height)/2

The area of triangle A = (8)(6)/2 = 24
The area of triangle B = (8)(3)/2 = 12
The area of triangle C = (8)(0.5)/2 = 2

Once we prove that, by positioning the two lengths at right angles, we can create a triangle with area 24, we can rotate the side with length 6 so that the triangle's height gets smaller and smaller, which allows us to create triangles with area 12 and 2.

Answer: I, II and III

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Re: QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 06 Feb 2018, 20:28
Please, someone help me...

If two sides of a triangle have length 6 and 8 third side (x) has to between 2 < x < 14, which means answer A cannot be correct.
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Re: QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 06 Feb 2018, 20:55
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yell2012prime wrote:
Please, someone help me...

If two sides of a triangle have length 6 and 8 third side (x) has to between 2 < x < 14, which means answer A cannot be correct.



When the lengths of two sides of a triangle are given, there is no description on type of triangle you are dealing with. It may be a right triangle, isosceles or a very small triangle but not a straight line

and as such, you have to find the maximum and minimum value for the area

the maximum area will occur if the two sides form the legs of a right triangle.

Therefore Area \(= (\frac{1}{2})*(6)*(8) = 24 =\) maximum possible area

Now the range of possible areas is \(0 < Area\leq{24}\)

So all three values are possible.
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Re: QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 24 Feb 2018, 05:32
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Holy Cow! This problem is so devious! I also had B and C as my answers, but then realized that the height of the triangle can be chosen arbitrarily to make the area of the triangle equal to 2.

Last edited by gremather on 25 Feb 2018, 10:34, edited 4 times in total.
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Re: QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 25 Feb 2018, 10:18
Expert's post
Please avoid expressions such as Holy Christ. Please. For a stupid question, I do think is not necessary

Crap is already more than enough. Or similar to this.

Regards
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Re: QOTD #6 Two sides of a triangle have length 6 and 8 [#permalink] New post 25 Feb 2018, 10:24
Just relax there, dude. Not meant to offend or anything. Anyway, I've changed it.

Carcass wrote:
Please avoid expressions such as Holy Christ. Please. For a stupid question, I do think is not necessary

Crap is already more than enough. Or similar to this.

Regards
Re: QOTD #6 Two sides of a triangle have length 6 and 8   [#permalink] 25 Feb 2018, 10:24
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