If the area of triangular region RST is 25, what is the perimeter

VERY IMPORTANT: For geometry Data Sufficiency questions, we are typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in our free video: https://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1103

This technique can save a lot of time.

Target question: What is the perimeter of RST?

Given: The area of triangular region RST is 25.

Statement 1: The length of one of the sides is 5√2
There are several possible triangles such that the length of one side is 5√2. Here are two:

Notice that the perimeter for each triangle is DIFFERENT. In other words, statement 1 does not lock our shape into having just one perimeter.
As such, statement 1 is NOT SUFFICIENT

Statement 2: The triangle is a right isosceles triangle
This fact alone forces the triangle into having a 90-degree angle, 2 equal angles and 2 equal sides. Of course there still many different triangles (with different perimeters) that meet these conditions:

HOWEVER, it is given that the area of the triangle is 25. Among the infinite number of isosceles right triangles, ONLY ONE has an area of 25.
So, statement 2 (along with the given information) "locks" our triangle into ONE and ONLY ONE shape, which means there's only one possible perimeter.
As such, statement 2 is SUFFICIENT.
IMPORTANT: Need we actually find the perimeter of this triangle? No. We need only recognize that we COULD find the perimeter (if we so inclined to do so)

Answer:

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B

1/2*a^2 = 25 --> a =?
then b =a*sqare root(2)

and perimeter = b+ 2a, suff

I think C is right..how do we know from B which side is base?

i mean i could have a 90-degree-45-45 isso triangle and it will have perimeter of 10+5+5=20..

i could have a different isso triangle..

hi fresinha12,
is it required to know whats the base???
in any case let what ever be the base the area will be same

I agree with Rohit, you can change the position but the relative sides will be the same hence you can trace a perpendicular for the height wherever you want and you would still have the same area and same perimeter.

Hence for me (B) is the correct choce
Let us know if this is the OA will you?
Thanks
Cheers
J

The correct answer is B. From (2) we know that 1/2*leg^2=25 --> we can get the length of the legs, and since it's a 45-45-90 right isosceles triangle, we can get the length of the hypotenuse too.
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I don't understand why (1) is not sufficient. Doesn't (1) imply that the triangle is an isosceles right triangle since the height = 5*sqrt(2)?

No. Any triangle with the base (side) of \(5\sqrt{2}\) and height of \(\frac{10}{\sqrt{2}}\) would have the area of 25.
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So the prompt tells us that the area of a triangle is 25 or that b*h=50
we are looking for the perimeter

(I) doesn't really tell us anything. Giving us one side is useless because we have no idea what the other sides could be
(II) actually tells us quite a lot. The fact that it is an isosceles right triangle tells us that b=h, so we know that those sides are rt(50) long and we can use that information to figure out the hypotenuse and add them up to get the perimeter.

Bunuel egmat

It is stated in the question thread that we are dealing with a rst(right-side triangle), if i understand it right.

if the given triangle is indeed a right angle triangle, then :

statement1: one side =\(5\sqrt{2}\). if the hypotenuse is this side, then the area can't be 25 as the other sides come out to be 5 & 5, so one of the other sides has to be this side.
we already know that b*h=50, so the second side is also equal to \(5\sqrt{2}\). We can know find the hypotenuse (10) and thus the perimeter.

This, this option is sufficient.

Statement2: I agree that this statement is sufficient.

Thus the correct option should be C.

Let me know if i am missing something.

Hi

1) In the main question (before the two statements), Nowhere is it stated that RST is a right angled triangle. It just means that the three vertices of this triangle are named as R, S, T respectively. RST doesn't mean right-side triangle, there is no such official term as 'right side triangle'. So we cannot assume it to be a right angled triangle.

2) When each statement alone is sufficient to answer a question in GMAT, then the answer to be marked is D, not C. (but of course here answer is neither C nor D, its B).

First of all, if you claim that each statement is sufficient, then the answer should be D, not C. Next, when considering (1) alone, we don't know whether the triangle is right angled nor that it's isosceles. All we know is the area and the lengths of one side, which is not enough to get other two sides.

Hope it's clear.
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Assume b is the base and h is the height of the triangle.
(1/2)bh = 25.
Since we have 2 variables and 1 equation, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
Since there two cases, we can't determine the perimeter of RST.
We don't know which side is 5√2.
Those cases are b is 5√2 or another side length is 5√2.
This is not sufficient.

Condition 2)
Assume that the length of two legs is a.
The area of the right isosceles triangle is (1/2)a^2 = 25.
Then a = 5 and the hypotenuse is 5√2 .
The perimeter is 10 + 5√2.
Condition 2) is sufficient.

Therefore, the answer is B.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

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