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Statement (1) by itself is insufficient. Let's pick numbers: if the sides of \(ABC\) are 1, 1 and \(\sqrt{2}\) (a half of a square with sides equal to 1), the area equals \(0.5\) and t perimeter is \(2+\sqrt{2}\). The perimeter is much greater than the area of a triangle with these values. However if the sides of \(ABC\) are 10, 10, and \(10\sqrt{2}\); then the perimeter is \(20+10\sqrt{2}\) and the area is 50. The perimeter is much smaller than the area.
Statement (2) by itself is insufficient. All it tells us is that both triangles are similar or proportionate to each other, but nothing about their size.
Statements (1) and (2) combined are insufficient. Combining the two statements, we still cannot determine whether the triangles have small values of their sides that yield greater perimeters or large values that yield greater area measurements.
Hi, Bunuel. I answered correctly but I just have a question. Is the solution below a right application of the properties of a triangle? or was I just lucky to arrive at the correct answer?
My approach:
Statement 1: Area ABC < Perimeter of DEF To maximize the area of ABC, let's assume that it's an equilateral triangle with side x. The area becomes [x^2(√3)]/4
To maximize the area given a perimeter, let's assume that the triangle DEF is also an equilateral triangle with side, y. The area DEF becomes [y^2(√3)]/4
Since we don't know the length of the sides, x and y, the statement is insufficient.
Statement 2: Angles of ABC = Angles of DEF This only tells us that the triangles are similar.
Combining both: Since we already assumed that both are equilateral triangles, and therefore similar, statement 2 doesn't add any additional information. We still don't know the length of the sides and therefore, the both statements taken together are insufficient.
Statement (1) by itself is insufficient. Let's pick numbers: if the sides of \(ABC\) are 1, 1 and \(\sqrt{2}\) (a half of a square with sides equal to 1), the area equals \(0.5\) and t perimeter is \(2+\sqrt{2}\). The perimeter is much greater than the area of a triangle with these values. However if the sides of \(ABC\) are 10, 10, and \(10\sqrt{2}\); then the perimeter is \(20+10\sqrt{2}\) and the area is 50. The perimeter is much smaller than the area.
Statement (2) by itself is insufficient. All it tells us is that both triangles are similar or proportionate to each other, but nothing about their size.
Statements (1) and (2) combined are insufficient. Combining the two statements, we still cannot determine whether the triangles have small values of their sides that yield greater perimeters or large values that yield greater area measurements.
Answer: E
Hi Bunuel ,
I am little confused about explanation for stmnt 1. In your explanation, you are talking about ABC only but the given stmnt talks about comparison of triangles. for stmnt 1 to be insufficient, it shd lead to 2 contradictory results. can you pls elaborate a little.
Since we know that a triangle can have an area > perimeter or perimeter > area, we can infer two scenarios: while, in the first, the area of DEF is bigger than perimeter of DEF, so area of DEF is bigger than area ABC, in the second, the perimeter of DEF is bigger than area of DEF, so we can have area of DEF > area of ABC and area of ABC > area of DEF. Therefore, statement 1 is insufficient.
Can someone please explain the solution since the official solution is a bit unclear as it only talks about the ABC triangle and does not mention DEF triangle. I am clear with the statement 2 but not sure about statement 1.
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. i could'nt find the solution clear.please provide a better one
Consider for ABC, A1 is the area and P1 is the perimeter.
Similarly consider A2 and P2 for DEF.
Now statement 1 gives A1< P2
And asks us if A1> A2?
Is is not hard to imagine A1< P2< A2 and therefor A1< A2 (if area of one triangle is smaller than the perimeter of another, we can easily imagine the latter to be infinitely big).
But can we imagine A2< A1< P2 (or A2< A1)?
The official solutions says that for triangles with small sides, perimeter> area. So we take smaller triangles to see if we can find an example.
Try with smaller sets (1, 1, √2/ 2, 2, 2√2 and 1, 1, √2 / 0.5, 0.5, 0.5√2) and you will find that statement goes both ways. Hence, insufficient.
This question can be solved easily by using only 2 cases :
Fact 1 - Area of ABC < Perimeter of DEF Consider both ABC and DEF as right angled triangles with 3, 4, 5 combination Area of ABC (3,4,5) = 1/2 * 3 * 4 = 6 Perimeter of DEF (3,4,5) = 3+4+5 = 12
So Is Ar. of ABC > Ar. of DEF ? [Is 6 > 6?] ........ NO
Consider both ABC and DEF as equi triangles ABC sides 3,3,3 and DEF sides 2,2,2. Area of ABC (3,3,3) = \(x^2\) \(\sqrt{3}/4\) = 9\(\sqrt{3}/4\) ~ 4.25 Perimeter of DEF (2,2,2) = 2+2+2 = 6
So Is Ar. of ABC > Ar. of DEF ? ........ Clearly YES
Fact 2 - Angles of ABC = Angles of DEF We already proved in Fact 1 that its INSUFFICIENT.
Combining Fact 1 and Fact 2 , we have nothing new. Hence E _________________
The question can also be solved by applying the property of similar triangles, i.e, When two triangles (lets say Triangle ABC and Triangle DEF) are similar, the the ratio of the area of the triangles is equal to the squares of their respective perimeter, sides, medians, heights etc. So, Area(ABC)/Area(DEF)=[Perimeter(ABC)]^2/[Perimeter(DEF)]^2 Now, from St.(1) The value of area of ABC is less than that of perimeter of DEF---->We cannot conclude anything-->insufficient. from St.(2) Angles of ABC = Angles of DEF----->Can only conclude that the triangles are similar(using the A-A-A property)--->Insufficient from St.1 and St.2: Since ABC and DEF are similar triangles, thus, Area(ABC)/Area(DEF)=[Perimeter(ABC)]^2/[Perimeter(DEF)]^2---eqn1 Now, St. 1 says area of ABC is less than that of perimeter of DEF.Applying this logic on eqn1 nd taking some integer values, we can see that the Perimeter(DEF)>Perimeter (ABC); thus Area (ABC)<Area(DEF). [No] On taking decimal values on eqn1, we get that the Perimeter(DEF)<Perimeter (ABC); thus Area (ABC)>Area(DEF)....[Yes] Thus insufficient.
_________________
Thanks, Rnk
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Drew two equal right triangles ABC and DEF with sides 3, 4 and 5. Area of each triangle is 6 and perimeter of each triangle is 12.
(1) says that value of area of ABC is less than value of perimeter of DEF.
If we change nothing, area of ABC will be equal to the area of DEF and (1) will be satisfied, i.e. 6 is less than 12. In this case answer to the question is no
However, if we multiply each side of DEF by 0.99999999 then, area of DEF will be less than area of ABC. After multiplying each side by 0.9999999 perimeter of DEF will be slightly less than 12. Therefore statement (1) will be satisfied, because value of perimeter will definitely be greater than 6. In this case answer to the question yes
insufficient
(2) above mentioned example is valid for statement 2 as well,
(1) and (2) together. Above mentioned example satisfies both statements, and we get both yes and no, therefore answer is E
close
57% (00:46) correct 
