Two sides of a triangle have length 6 and 8. Which of the following ar

Question

Two sides of a triangle have length 6 and 8. Which of the following are possible areas of the triangle?

I. 2 
II. 12 
III. 24

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II, and III

Kudos for a correct  solution .

EMPOWERgmatRichC · Accepted Answer

Hi All,

When you're given the lengths of two sides of a triangle, you have no way of knowing what type of triangle you're dealing with. It MIGHT be a right triangle. It MIGHT be isosceles. It MIGHT be a really long, thin triangle that's ALMOST a straight line (but still has 3 distinct sides).

As such, you have to find the maximum value for the area; with that, you know that every SMALLER area is also possible. In this type of situation, the maximum area will occur if the two sides form the legs of a right triangle. With these side lengths (6 and 8), we'd have...

Area = (1/2)(6)(8) = 24 = maximum possible area

Now we know that the range of possible areas is 0 < Area <= 24.

So all three values are possible.

Final Answer:  E

GMAT assassins aren't born, they're made,
Rich

Bunuel · Answer

MAGOOSH OFFICIAL SOLUTION: This requires a picture explanation. Consider the leg of 6 and the leg of 8 attached at a "hinged" joint at B. originals.jpg Here, the angle is made very narrow, only 1/100 of a degree. The area of this triangle would be 0.00419 --- you don't need to be able to calculate something like this for the test. The point is -- if the angle were one millionth, or one billionth, the area could be really really small --- greater than zero, but a really tiny decimal. So, clearly, the area can be less than two. Here, I moved the legs of 6 and 8 apart a little, and produced a triangle with an area of exactly two: originals (1).jpg Again, you don't have to know how to build a triangle like this: this is just to show you that it is, indeed possible. As we increase the angle, we get triangles with more and more area. You do not need to know how to find these areas --- this is just to demonstrate that the area would increase. originals (2).jpg originals (3).jpg originals (4).jpg Now, there's a geometry theorem that states, when the angle is 90°, the area of the triangle will be a maximum. At that point, the length of 8 will be the base, and the length of 6 will be the height, so the area = 0.5*bh = 24, the maximum possible area. Thus, the triangle can have all three areas: I, II, and III. Answer = E.

Plastelin · Answer

I. 2 
IMHO this area is impossible, because if this would be possible than the height to the largest side with length of 8 would have to be 2*2/8 = 1/2
Third side would be larger than 2 (because of sides 6 and 8), and i think the height to the largest side will be larger than 1/2

III. 24
This area is possible, 6*8*1/2 = 24

II. 12 
Upon analysing answer options without I and with III option there is only one answer: C

Answer is C, II and III are possible areas

p.s. solved very quickly, in less than 20 sec

chetan2u · Answer

hi why cant be the height be less than 2...
yes the third side has to be more than>2,otherwise it will become a straight line...
say the side is just above 2.. then the height can be .1,.2,1/3,1/4 etc 
so we have base 8 and height 1/8... area =1/2*1/8*8=1/2 .. so 2 is possible

chetan2u · Answer

hi,
lets see the minimum value and max value..
1)min value...
two sides are 6,8.. third side has to be more than 2, otherwise we will have a straight line...
with height slightly above 2, we can have height as 1/10,1/8,1/2 etc.. so min value can be any positive number, fraction decimal etc...
2) max valve ..
lets take these as two sides of a right angle triangle, then area=1/2*8*6=24..

all three values fit in so ans E..

PareshGmat · Answer

Answer = E. I, II, and III

With dimensions of only 2 sides given (for a triangle), this becomes an open ended question

Any area is possible out of the three

chetan2u · Answer

hi paresh,
there is a limitation on the upper end of area, which is area of a right angle triangle with these two sides..

PareshGmat · Answer

Fully agree Chetan... that's why mentioned out of the three given

Area is maximum when height is maximum Height is maximum when its a right triangle angle.png

Bunuel · Answer

Similar questions to practice: if-two-sides-of-a-triangle-are-12-and-8-which-of-the-141227.html if-two-sides-of-a-triangle-have-lengths-2-and-5-which-of-th-163409.html

Tmoni26 · Answer

Hi Rich, Going with your logic on the areas, and looking at this example- if-two-sides-of-a-triangle-are-12-and-8-which-of-the-141227.html Why cant we find possible outcomes for the third leg and then try and see if we get an area Eg, from the link, let third side be x then 12-8

EMPOWERgmatRichC · Answer

Hi Tmoni26,

You can actually catch the error in this logic yourself. Try drawing the triangle that you describe. What are the 3 sides (and how does this NOT match up with the information in the prompt)?

GMAT assassins aren't born, they're made,
Rich

ravindra88 · Answer

Well, there is another way to solve such questions using trignometry. Area of a triangle is  1/2absin ( \alpha ), where  a  and  b  are any two sides of the triangle and  \alpha  is the angle contained by these two sides. Now area is maximum when sine function becomes  1  at  \alpha   =  90^{\circ} . Also area tends to  0  as  \alpha  approaches  0^{\circ} .

EMPOWERgmatRichC · Answer

Hi ravindra88,

You're going to find that most GMAT questions can be solved in a variety of ways, so you should look for whichever method is easiest/fastest for you. It is worth noting, however, that Trigonometry is NOT tested on the GMAT, so choosing to use such an approach is up to you (but is not expected of you).

GMAT assassins aren't born, they're made,
Rich

mvictor · Answer

2 ways to solve it:
1 - plug in numbers:
h=1/2. 1/2*8 /2 = 2. 1 yes.
h=1. 1*8/2 = 4. 2 yes.
h=6. 6*8/2 = 24. 3 yes.

other method:
third side must be greater than 2, less than 14.
greatest possible area thus can be almost 14*8/2 or slightly less than 56.
everything below - is possible, so long as it is not 0 or negative.

brs1cob · Answer

24 is the one possibility if it is a right angled triangle. once we get this, the only thing we need to do is to change the angle between the two given sides and thus changing the height before it becomes 0 degree or 180 degree (both excluded). so area can be anything based on the height of the triangle.

Shiv2016 · Answer

I did it this way.

Area= 0.5 *base*height

We have no clue about the third side. It may not be an integer.
We just know that: 2<third side<14

But there are so many values that c can take even in this range.

Let's work with the options:

1) Area= 2= 0.5 *b*h
b*h= 4

2) b*h= 24

3) b*h= 48

There are many possible values that can result in these products.
So I selected all three.

Does this make sense?

Buttercup3 · Answer

Thanks RIch

EMPOWERgmatRichC · Answer

Hi Buttercup3,

The answer to your question can be found in a number of posts earlier on in the thread. Here's one of them:

When you're given the lengths of two sides of a triangle, you have no way of knowing what type of triangle you're dealing with. It MIGHT be a right triangle. It MIGHT be isosceles. It MIGHT be a really long, thin triangle that's ALMOST a straight line (but still has 3 distinct sides).

As such, you have to find the maximum value for the area; with that, you know that every SMALLER area is also possible. In this type of situation, the maximum area will occur if the two sides form the legs of a right triangle. With these side lengths (6 and 8), we'd have...

Area = (1/2)(6)(8) = 24 = maximum possible area

Now we know that the range of possible areas is 0 < Area <= 24.

So all three areas (including "2") are possible.

GMAT assassins aren't born, they're made,
Rich

Buttercup3 · Answer

Thank You Sir for the reply :-D

Two sides of a triangle have length 6 and 8. Which of the following ar

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Two sides of a triangle have length 6 and 8. Which of the following ar

Prep Toolkit

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My Rewards