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Is the perimeter of triangle ABC greater than 20? (1)

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Re: Perimeter of triangle ABC. [#permalink] New post 17 Dec 2010, 08:01
Bunuel wrote:
sriharimurthy wrote:
Thanks Bunuel.

Though I just realized something. I considered the area given to be 40 by mistake. It is actually 20.

Although the logic will be same, the calculations will be harder since we will have to use the real value of \(\sqrt{3}\). (no margin for approximations!)

Sorry for the mistake guys. However, as long as you understand the logic, it shouldn't matter. At least you'll know how to go about approaching such questions in the future!

Cheers.

Ps. Any trick for the calculations Bunuel?


I would go backward.

Let's assume the perimeter is 20. The largest area with given perimeter will have the equilateral triangle, so side=20/3. Let's calculate the area and if the area will be less than 20 it'll mean that perimeter must be more than 20.

\(Area=s^2*\frac{\sqrt{3}}{4}=(\frac{20}{3})^2*\frac{\sqrt{3}}{4}=\frac{100*\sqrt{3}}{9}=~\frac{173}{9}<20\)

Think this way is easier. \(\sqrt{3}\approx{1.73}\).


Bunuel,

is it necessary to solve it? what if you just recognized that you could solve it and the answer is either > 20 or < 20? I'm just thinking in terms of timing strategy...
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Re: Perimeter of triangle ABC. [#permalink] New post 17 Dec 2010, 08:51
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psirus wrote:
Bunuel wrote:
sriharimurthy wrote:
Thanks Bunuel.

Though I just realized something. I considered the area given to be 40 by mistake. It is actually 20.

Although the logic will be same, the calculations will be harder since we will have to use the real value of \(\sqrt{3}\). (no margin for approximations!)

Sorry for the mistake guys. However, as long as you understand the logic, it shouldn't matter. At least you'll know how to go about approaching such questions in the future!

Cheers.

Ps. Any trick for the calculations Bunuel?


I would go backward.

Let's assume the perimeter is 20. The largest area with given perimeter will have the equilateral triangle, so side=20/3. Let's calculate the area and if the area will be less than 20 it'll mean that perimeter must be more than 20.

\(Area=s^2*\frac{\sqrt{3}}{4}=(\frac{20}{3})^2*\frac{\sqrt{3}}{4}=\frac{100*\sqrt{3}}{9}=~\frac{173}{9}<20\)

Think this way is easier. \(\sqrt{3}\approx{1.73}\).


Bunuel,

is it necessary to solve it? what if you just recognized that you could solve it and the answer is either > 20 or < 20? I'm just thinking in terms of timing strategy...


You have to solve it. If we get that the minimum perimeter possible for a triangle with an area of 20 is less than 20 then we won't be able to answer the question. Similarly if we get that the maximum area possible for a triangle with a perimeter of 20 is more than 20 (for example 25) then knowing that area is 20 won't mean that perimeter must be more than 20.

Hope it's clear.
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Re: Perimeter of triangle ABC. [#permalink] New post 14 Jun 2011, 21:07
A bc-ac < ab < bc + ac

ab > 10 and bc > 10 thus adding ab+bc = 20 already.Hence sufficient.

B base * altitude = 40
base = bc
altitude = ad

now assuming values bc = 0.1,1 and 4 , ad = 400,40 and 10. all giving values of perimeter > 20.

sufficient.

D
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 13 Jan 2012, 02:08
Hi Bunuel , Is there any way to find out the other two wxtreme in those two statement . Like :
A. For a given perimeter equilateral triangle has the largest area: What is the lowest area ?

B. For a given area equilateral triangle has the smallest perimeter : What is the largest area ?
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 13 Jan 2012, 11:06
Good one:

TRIANGLE RULES:
1. Third side < Sum of other two sides
2. Third side > Difference of other two sides

1. Starting with BC=11, AC=1, we get AB=9. Rule 1 is violated because BC > AB + AC. Keep on going BC = 12, 13... it gets worst. Therefore, the perimeter of the triangle has to be greater than 20. Suff.
2. This one I had to semi-guess. I picked 3-4-5 triangle and found that area < perimeter. So, perimeter has to be > 20. Suff.

D.
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 07 May 2012, 20:56
While solving statement-2 in this problem, I realized that –
For a right triangle with a given area, perimeter will be minimum when that right triangle is an "isosceles" right triangle.

I feel this property makes sense. Still, can someone confirm it please?

Using this property the problem can be solved as follows:
Assume that the given right triangle with area 40, is an isosceles right triangle and then find the perimeter. If the perimeter of isosceles right triangle is greater than 20, that will mean any perimeter for that right triangle is definitely greater than 20.

Lets call base of right triangle = b
Lets call height of right triangle = h
Area given = 20. ==> (1/2)b.h = 20 ==> b.h = 40

If we assume this right triangle to be isosceles, then
b=h= sq.root(40)
=> hypotenuse = sq.root(80)
=> perimeter = 2. sq.root(40) + sq.root(80)
=> This is greater than 20.
=> So minimum perimeter is more than 20. So any perimeter for this right triangle with the area as 20, will be more than 20.
=> Statement#2 is sufficient.
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 29 Dec 2012, 07:21
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Great qs. Bunuel...!And very nice back calculations shown :)

Could you please share some more like it ?
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 03 Jan 2013, 21:46
very good question. thanks a lot for the explanation
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 07 Jul 2013, 23:54
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 09 Jul 2013, 08:55
(1). The Third side is greater than the difference betn the other two sides. Therefore atleast two sides are greater than 10. Therefore perimeter is >20.
Sufficient.

(2) 1/2 bh =20 , if the base corresponding to this height is 1 (a smaller value gives an even greater height) , height is 40 , thus other side is greater > 40 (since other side is the hypotenuse in the right triangle formed by this height and base). Thus, perimeter > 20.
Sufficient.

Answer is (D).
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 10 Dec 2013, 05:26
Is the perimeter of triangle ABC greater than 20?

(1) BC-AC=10.
This can be rewritten as BC=10+AC.
The sum of any two sides of a triangle must be greater than the third. So, for example, BC + AC must be greater than AB. Because BC is at least 10, AB must be greater than 10 as well. SUFFICIENT.

(2) The area of the triangle is 20.
This I solved with a bit of intuition and drawing out but it's best to recognize that for triangles of the same area, perimeter is minimized when the triangle is equilateral. Knowing the area of an equilateral triangle will allow us to figure out a side length and thus, the perimeter:
20 = s^2 (Sqrt3/4)
s^2 = 80/(Sqrt 3)
s = 6.8
6.8+ 6.8+ 6.8 = 20.4 > 20

So when we solve for a triangle with the minimal possible value for it's given area it's greater than 20. SUFFICIENT.

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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 10 May 2014, 11:30
Bunuel wrote:
Is the perimeter of triangle ABC greater than 20?

(1) BC-AC=10.
(2) The area of the triangle is 20.


Dear Bunnel

as this was a yes no qs.... the ans was D... but if they had asked value of the perimeter... then the answer would be E right??
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 11 May 2014, 05:10
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 20 Jun 2014, 02:15
An easier way to understand the calculation of the second statement is Following.
Since for a given area an equilateral triangle has the smallest perimeter, Hence for an area of 20 , each side of the equilateral comes out to be 6.8. Hence the minimum perimeter for a triangle with area 20 is 3a=20.4; hence the perimeter has to be greater than 20.
Hope it helps
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Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 20 Sep 2015, 06:43
sriharimurthy wrote:
Question Stem : Is AB + BC + AC > 20?

St. (1) : BC = AC + 10

Triangle Property : The sum of any two sides of a triangle is always greater than the third.

Since we are given that one of the sides is greater than 10, the sum of the other two sides must also be greater than 10.
Hence the perimeter will always be greater than 20.
Statement is sufficient.

St. (2) : A = 40

Triangle Property : For triangles with same area, the perimeter is smallest for an equilateral triangle.

Area of equilateral triangle with side x = \(\frac{\sqrt{3}}{4}x^2\)

Therefore, \(\frac{\sqrt{3}}{4}x^2\) = 40

\(x^2 = \frac{160}{\sqrt{3}}\)

Now, in order to speed up calculations, I will assume \(\sqrt{3}\) to be equal to 2.

If the condition is satisfied with \(\sqrt{3}\) equal to 2 then it will definitely be satisfied with the actual value of \(\sqrt{3}\) which is less than 2.

Therefore, \(x^2 = \frac{160}{2}\) = 80

This tells us that x is almost 9. More importantly, it tells us that x is greater than 8. Thus perimeter will be 3*x = 24.

Since this is the minimum perimeter possible (actually it is still less than what the actual minimum would be due to our approximations), we can conclude that the question stem will always be true.

Hence Sufficient.

Answer : D


Another interesting triangle property : For triangles with same perimeter, the area is maximum for an equilateral triangle. (If you think about it, this property goes hand in hand with the one we used in St. 2).



Dear Bunnel,
Thanks for your explanation. I think I almost understood the logic but cannot figure out clearly how the triangle property you mentioned "For triangles with same perimeter, the area is maximum for an equilateral triangle" is related to the Statement 2.
Can you please explain how I can link the property to the statement 2?

Thanks in advanace
regards
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Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 20 Sep 2015, 07:02
andy2whang wrote:
sriharimurthy wrote:
Question Stem : Is AB + BC + AC > 20?

St. (1) : BC = AC + 10

Triangle Property : The sum of any two sides of a triangle is always greater than the third.

Since we are given that one of the sides is greater than 10, the sum of the other two sides must also be greater than 10.
Hence the perimeter will always be greater than 20.
Statement is sufficient.

St. (2) : A = 40

Triangle Property : For triangles with same area, the perimeter is smallest for an equilateral triangle.

Area of equilateral triangle with side x = \(\frac{\sqrt{3}}{4}x^2\)

Therefore, \(\frac{\sqrt{3}}{4}x^2\) = 40

\(x^2 = \frac{160}{\sqrt{3}}\)

Now, in order to speed up calculations, I will assume \(\sqrt{3}\) to be equal to 2.

If the condition is satisfied with \(\sqrt{3}\) equal to 2 then it will definitely be satisfied with the actual value of \(\sqrt{3}\) which is less than 2.

Therefore, \(x^2 = \frac{160}{2}\) = 80

This tells us that x is almost 9. More importantly, it tells us that x is greater than 8. Thus perimeter will be 3*x = 24.

Since this is the minimum perimeter possible (actually it is still less than what the actual minimum would be due to our approximations), we can conclude that the question stem will always be true.

Hence Sufficient.

Answer : D


Another interesting triangle property : For triangles with same perimeter, the area is maximum for an equilateral triangle. (If you think about it, this property goes hand in hand with the one we used in St. 2).



Dear Bunnel,
Thanks for your explanation. I think I almost understood the logic but cannot figure out clearly how the triangle property you mentioned "For triangles with same perimeter, the area is maximum for an equilateral triangle" is related to the Statement 2.
Can you please explain how I can link the property to the statement 2?

Thanks in advanace
regards
Andy


You are not going to get a reply as the post you are quoting is from 2009.

Let me try to answer your question. You are given a particular area in statement 2. Now, based on this value of area is there a property of triangles that you can apply to see whether you do get triangles with perimter >20 and area =20?

As per the property mentioned, of all triangles with EQUAL areas, equilateral triangle will have the smallest perimeter. Thus, side of an equilateral triangle with area of 20

---> \(\frac{\sqrt{3}*a^2}{4} = 20\), where a = side of the equilateral triangle.

----> a = (approx.) 6.79 ---> Perimeter (minimum of the triangles possible with area = 20 ) = 3*a=20.3 > 20.

Thus, when the minimum perimeter is 20.3, then all the other possible traingles will have the perimeter > 20. Thus this statement is sufficient.

Alternately, as mentioned by Bunuel at is-the-perimeter-of-triangle-abc-greater-than-87112.html#p836517, based on the property mentioned above, you know that for a given perimeter, an equilateral triangle will have the smallest area.

Thus if the perimeter of the equilateral triangle is 20, then each side of the triangle = 20/3.

Thus, the area of such an equilateral triangle = \(\frac{\sqrt{3}*a^2}{4} = 20\) = \(\frac{\sqrt{3}*(20/3)^2}{4} = =173/9 < 180/9 =20\). Thus we see that with perimeter 20 , the smallest area is <20.

Thus, if we are given an area of 20 , then the perimeter of the smallest triangle (=an equilateral triangle) MUST be >20.
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 27 Sep 2015, 03:33
For triangles with same area, the perimeter is smallest for an equilateral triangle.
please Bunuel clarify this property. I do not grasp this concept so far. I memorize it but I can not apply it my self in similar question and when I should. thank you in advance for your great help.
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Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 27 Sep 2015, 03:44
hatemnag wrote:
For triangles with same area, the perimeter is smallest for an equilateral triangle.
please Bunuel clarify this property. I do not grasp this concept so far. I memorize it but I can not apply it my self in similar question and when I should. thank you in advance for your great help.


The concept IS what you mentioned.

For all the triangles given to you with a fixed area, the equilateral triangle will have the smallest perimeter.

This is a very unique property of equilateral triangles.

Let's say in a PS question, you are given that area of a triangle is 20 square units, what is the smallest perimeter of this triangle ?

You will be able to use the property above to see that for a triangle with a given area to have the smallest perimeter, it must be an equilateral triangle. Based on this you can use the formula

\(\frac{\sqrt{3}*a^2}{4} = 20\) ----> calculate 'a' and then for perimeter= 3a

This is how you will be able to apply this property.

Hope this helps
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Re: Is the perimeter of triangle ABC greater than 20? (1) [#permalink] New post 30 Sep 2015, 07:52
i got answer using another approach which may not be correct always.
statement 2 :

1/2 * b * h = 20
b*h =40
possible combinations 10*4,20*2, 8*5 etc. in these combinations let's consider base as 10,20,8. and we know the angle property that sum of 2 sides is greater than the 3rd side, so in the above combinations sum of other 2 sides must be greater than 10 and 20. in 3rd case since height is 5, using common sense we can conclude that sum of other two sides must be greater than 12.
Re: Is the perimeter of triangle ABC greater than 20? (1)   [#permalink] 30 Sep 2015, 07:52

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