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# What is the maximum possible area of triangle?

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What is the maximum possible area of triangle? [#permalink]

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21 May 2012, 20:33
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What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.
[Reveal] Spoiler: OA

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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 00:17
Smita04 wrote:
What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.

1. Two sides are 7 and 14 but we don't know which is base and which is height. Third side is 7< x < 21.
2. We are only given one side and no info whether this is B or H.

Together not sufficient either.

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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 07:18

Two sides are 7 an 14 so by the triangle sides property third side can be in between of 7 and 21. We need to maximise the area of triangle which is possible with these two sides and another sides so we will choose the third side as 20 (Assumption is sides are integer).
Triangle with sides 7 ,14 and 20 will have the max area and this can be calculate using the below formula.

s= a+b+c /2 , Area = Sqrt { s(s-a)(s-b)(s-c)}

Please correct me if something is worng.

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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 07:27
IMO E,
we need to find an answer and nowhere we are given that 3rd side is an integer.

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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 07:37
Yes that can be true. however this question does not looks likea Gmat question.
GMAT question generally do not have these confusions.

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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 08:29
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Smita04 wrote:
What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.

Because using 1 we can know the max area of triangle.
As for any triangle area is 0.5*AB*BC*sin B = 0.5*BC*CA*sin C = 0.5*CA*AB*sin A

since max value of a sine of any angle is 1
max area of given triangle is 49.

But by using statement 2 we can't come to any conclusion like that.

Hope that helps.
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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 21:39
Bunuel, can you please comment on this? Is it not a GMAT type question? If that is the case, then there is no point discussing it.

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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 21:59
To maximize the area of a triangle and if you know the length of two sides make them perpendicular to maximize the area. Hence A is sufficient.
where are the experts?

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Re: What is the maximum possible area of triangle? [#permalink]

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22 May 2012, 22:19
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Smita04 wrote:
Bunuel, can you please comment on this? Is it not a GMAT type question? If that is the case, then there is no point discussing it.

Hi Smita04,

This is very much a GMAT query.

Evaluating statement 1 only:
Here, we know that the length of the two sides are 7 cms and 14 cms respectively. Now just picture this. Let us try to a triangle with base = 14 cm and then try to put the 7 cm side such that ther area is the maximum.

Let the side AB = 7 cm and BC = 14 cm. The figure shows 3 possibilities for AB that would result in the maximum possible area for triangle ABC.
Now, we k now that area
= 1/2 * base * altitude = 1/2 * BC * altitude.
Now, the area will be maximum for the maximum value of the altitude. This is only possible with AB as the altitude, as in the other two cases the length of the altutide goes down.
Hence, ABC is right angled and the maximum area = 1/2 * 14 * 7
We can eliminate options B, C and E.

Evaluating statement 2 only:
Let us picture this.

Let side AB = 7 cm. Now depending on the size of the circle, the area of the triangle can keep increasing. Hence, statement 2 alone is insufficient.
Hence D is eliminated.

Hope this helps.

Regards,

Shouvik.
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Re: What is the maximum possible area of triangle? [#permalink]

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23 May 2012, 21:32
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Smita04 wrote:
What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.

For the greatest area, the triangle should be a right angles triangle.
Also, Area = 1/2 * base * height.

If you make 14 the greatest side (hypotenuse), and 7 as base, the height would be smaller than 14.

To maximise area, consider 14 as height, in that case Area = 1/2 * 7 * 14.

A is sufficient.

B is clearly not sufficient, because we have to know atleast one more side, or the radius of the triangle.

Hope it helps.

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Re: What is the maximum possible area of triangle? [#permalink]

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24 May 2012, 02:56
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For the greatest area, the triangle should be a right angles triangle.

whats the reason ?

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Re: What is the maximum possible area of triangle? [#permalink]

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24 May 2012, 05:52
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For any triangle ABC its area = 1/2*b*c*sinA=1/2*c*b*sinC = 1/2*c*a*sinB.

since two sides are known the area would be 1/2*7*14*sin(angle bw sides 7 & 14)

maximum value of sine of angle is 1 when angle is 90 degrees..
so maximum area is 49.

So statement A is itself sufficient.

Since triangle is inscribed as of statement B the side 7 can be a chord or diameter.So not much data.

Hope its clear to you all.

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Re: What is the maximum possible area of triangle? [#permalink]

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24 May 2012, 06:13
vikram4689 wrote:
Quote:
For the greatest area, the triangle should be a right angles triangle.

whats the reason ?

Area of triangle = 1/2 * ab*sin(C)
sin(c) is max when angle b/w A & B = C = 90.

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Re: What is the maximum possible area of triangle? [#permalink]

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06 Jun 2012, 13:36
what is OA smita?
i got E as i thought it is not possible to find are untill we know which side is given what?
can someone explain what is regarding maximising sides?

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Re: What is the maximum possible area of triangle? [#permalink]

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10 Oct 2012, 03:08
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Smita04 wrote:
What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.

Good question. I feel this can be perfect GMAT question.

Q ->What is the maximum possible area of triangle ?
1) Let the third side is x which ranges from 7<x<21
where x can take any value within this range.

Now the three sides of the triangle are 7, 14, 7<x<21. Using these 3 sides numerous triangles can be made but only 1 triangle will have the maximum area , which we can find out (but it is not required to find the max area)
Sufficient

2) Outright insufficient

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Re: What is the maximum possible area of triangle? [#permalink]

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10 Oct 2012, 09:28
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fameatop wrote:
Smita04 wrote:
What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.

Good question. I feel this can be perfect GMAT question.

Q ->What is the maximum possible area of triangle ?
1) Let the third side is x which ranges from 7<x<21
where x can take any value within this range.

Now the three sides of the triangle are 7, 14, 7<x<21. Using these 3 sides numerous triangles can be made but only 1 triangle will have the maximum area , which we can find out (but it is not required to find the max area)
Sufficient

2) Outright insufficient

(1) You are absolutely right. Since only two sides of the triangle are given, the area varies depending on the third side. Since we can make the area as small as we wish and the third side must be between 7 and 21, the area must be finite for every one of these possible triangles. So, there must be a maximum, and because this is a DS question, we are not supposed to find that maximum. Trigonometry is out of question on the GMAT, but even without it, we can figure out when the area is maximum.

Since the area of a triangle is the same regardless which side we take as a base, we can consider 14 (denoted by BC in the attached drawing) as a constant base, and look at the various triangles that can be formed. Angle ABC varies between 0 and 180, with the side AC of constant length 7. Maximum height corresponding to BC is obtained when AB is perpendicular to BC, and in fact we now know that the maximum area will be 14*7/2 = 49.
Indeed (1) is sufficient.

(2) Any triangle can be inscribed in a circle and through two given points (apart at a distance of 7) there are infinitely many circles passing through.
Obviously not sufficient.

Attachments

TriangleMaxArea.jpg [ 9.9 KiB | Viewed 16808 times ]

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Re: What is the maximum possible area of triangle? [#permalink]

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31 May 2016, 23:02
EvaJager wrote:
fameatop wrote:
Smita04 wrote:
What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.

Good question. I feel this can be perfect GMAT question.

Q ->What is the maximum possible area of triangle ?
1) Let the third side is x which ranges from 7<x<21
where x can take any value within this range.

Now the three sides of the triangle are 7, 14, 7<x<21. Using these 3 sides numerous triangles can be made but only 1 triangle will have the maximum area , which we can find out (but it is not required to find the max area)
Sufficient

2) Outright insufficient

(1) You are absolutely right. Since only two sides of the triangle are given, the area varies depending on the third side. Since we can make the area as small as we wish and the third side must be between 7 and 21, the area must be finite for every one of these possible triangles. So, there must be a maximum, and because this is a DS question, we are not supposed to find that maximum. Trigonometry is out of question on the GMAT, but even without it, we can figure out when the area is maximum.

Since the area of a triangle is the same regardless which side we take as a base, we can consider 14 (denoted by BC in the attached drawing) as a constant base, and look at the various triangles that can be formed. Angle ABC varies between 0 and 180, with the side AC of constant length 7. Maximum height corresponding to BC is obtained when AB is perpendicular to BC, and in fact we now know that the maximum area will be 14*7/2 = 49.
Indeed (1) is sufficient.

(2) Any triangle can be inscribed in a circle and through two given points (apart at a distance of 7) there are infinitely many circles passing through.
Obviously not sufficient.

Statement 2 >
Among all triangles inscribed in a given circle, the equilateral one has the largest area.

So if we consider side as 7, we can get the maximum area. Note that, question is asking for maximum area, not exact area.
I think OA should Be D.

@bunnel: Pl help.

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Re: What is the maximum possible area of triangle? [#permalink]

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08 Sep 2017, 00:38
Smita04 wrote:
What is the maximum possible area of triangle?

(1) Two sides of the triangle are 7 cm and 14 cm.
(2) The triangle is inscribed in a circle and one of its sides is 7 cm.

ohhh okay I see what this question is doing

So to maximize the area of a triangle you would make it perpendicular- and although we don't know the third side of the triangle it we make it perpendicular it cannot be greater than 21 because 7 +14 must be greater than 21- so eitherway this is actually sufficient

A

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Re: What is the maximum possible area of triangle?   [#permalink] 08 Sep 2017, 00:38
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