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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