If x and y are the lengths of the legs of a right triangle, what is th

Given: x and y are the lengths of the legs of a right triangle
We have something like this:


Target question: What is the value of xy?

Statement 1: The hypotenuse of the triangle is \(10\sqrt{2}\).
There are infinitely-many different right triangles that meet this condition. Here are two:
Case a: x = 10 and y = 10

CHECK: If h = the hypotenuse, then we get 10² + 10² = h²
Solve: 200 = h²
So, h = √200 = 10√2
In this case, the answer to the target question is xy = (10)(10) = 100

Case b: x = √50 and y = √150

CHECK: If h = the hypotenuse, then we get (√50)² + (√150)² = h²
Solve: 200 = h²
So, h = √200 = 10√2
In this case, the answer to the target question is xy = (√50)(√150) = √7500 = 50√3

Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT


Statement 2: The area of the triangular region is 50
Area of triangle = (base)(height)/2
So, we can write: (x)(y)/2 = 50
Multiply both sides by 2 to get: xy = 100
So, the answer to the target question is xy = 100
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

The area of the triangle is xy/2, so Statement 2 is immediately sufficient.

Statement 1 is not sufficient - if we just draw a hypotenuse of length 10√2, we can make triangles of any small area at all. For example, we might have one leg of length 0.0000001, and another of length just less than 10√2. That triangle will have a tiny area, so xy will be very small. Or this might be a 45-45-90 triangle with legs of length 10, and might have quite a large area.

The Logical approach to this question will start by pointing out that since the area of a right triangle is half the product of its legs, then xy is twice the triangle's area.
Statement (1) gives us just the hypotenuse. Since the pythagorean theorem is an equation with three variables we must know the value of two of the three in order to solve it (i.e. x² + y² = 200 wouldn't give us the value of x or the value of y).
Statement (2) is just what we were looking for: xy, as mentioned above, is half the triangular area.
The correct answer is (B).

Posted from my mobile device

#1
200= x^2+y^2
insufficient
#2
x*y= 100
sufficient
IMO B

legs of a right angle triangle = two sides other than the Hypo ?

the question did not mention which side was Hypo .

So I chose C because Statement 1 gave the HYPO and confirmed that the other two sides are base and height .
and so from statement 2 , we could get va;ue of xy


Please let me know your opinion .

Hey m1033512,
Yes, a right-angle triangle has 3 sides: two 'legs', which form an angle of 90 degrees between them, and one hypotenuse.
So in the equation a^2 + b^2 = c^2 'a' and 'b' are the legs whereas 'c' is the hypotenuse.
It isn't necessary to think of a 'base' and 'height' of a right-angle triangle because it doesn't really matter which is which: both legs can be either the base or the height.

Best,
David

Hi All,

We're told that X and Y are the lengths of the legs of a RIGHT triangle. We're asked for the value of (X)(Y). This question can be solved with a mix of Geometry rules and TESTing VALUES.

(1) The hypotenuse of the triangle is 10√2.

With the information in Fact 1 - and the Pythagorean Theorem - we can create the following equation:

X^2 + Y^2 = (10√2)^2
X^2 + Y^2 = 200

Since this one equation has two variables, it's likely that there are LOTS of different possible values for X and Y - and as a result, there are probably lots of different values for (X)(Y). Here are two examples:

IF....
X=10 and Y= 10, then the answer to the question is (10)(10) = 100
X = 9 and Y = √119, then the answer to the question is 9√119
Fact 1 is INSUFFICIENT

(2) The area of the triangular region is 50.

With the information in Fact 2 - and the area formula - we can create the following equation:

Area = (1/2)(Base)(Height)
50 = (1/2)(X)(Y)
100 = (X)(Y)
This is the exact answer to the question that is asked. Since there's only one possible answer here, there's no more work needed.
Fact 2 is SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

made the same mistake... i thought any of the two sides can be the legs

Solution:

Question Stem Analysis:

We need to determine the value of xy given that x and y are the lengths of the legs of a right triangle.

Statement One Alone:

Knowing the length of the hypotenuse of the triangle is not sufficient to answer the question. Obviously, the triangle can be an isosceles right triangle and having a hypotenuse = 10√2 means both legs are 10. So xy = (10)(10) = 100. However, the triangle can also be a non-isosceles right triangle. For example, x can be 2 and y can be 14 (notice that 2^2 + 14^2 = 4 + 196 = 200 = (10√2)^2). In this case, xy = (2)(14) = 28. Statement one alone is not sufficient.

Statement Two Alone:

Recall that the area of a right triangle is 1/2 the product of the lengths of its legs. Here, the area of the triangle will be 1/2 * xy. Since we are given that the area is 50, we have:

1/2 * xy = 50

xy = 100

Statement two alone is sufficient.

Answer: B
_________________

Answer: Option B

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What if we take 45-45-90 triangle for statement 1 and this gives us that side opposite to 90° is 10√2 and thus the legs of the triangle come out to be 10 each. This gives us 10*10 = 100.

Why is this approach wrong.

Posted from my mobile device

Hi Shrihari12,

You've come up with one possible example that fits the given information; however, there are LOTS of potential examples that fit.

We're told that X and Y are the lengths of the legs of a RIGHT triangle. We're asked for the value of (X)(Y). Fact 1 tells us that the hypotenuse of the triangle is 10√2. Using the Pythagorean Theorem, we can create the following equation:

X^2 + Y^2 = (10√2)^2
X^2 + Y^2 = 200

Since this one equation has two variables, it's likely that there are LOTS of different possible values for X and Y - and as a result, there are probably lots of different values for (X)(Y). Here are two examples:

IF....
X=10 and Y= 10, then the answer to the question is (10)(10) = 100
X = 9 and Y = √119, then the answer to the question is 9√119
Fact 1 is INSUFFICIENT

GMAT assassins aren't born, they're made,
Rich

The two perpendicular sides of a right triangle are called the legs of the triangle; therefore, x and y are the two perpendicular sides of the right triangle.

We need to find the value of xy i.e. the product of the perpendicular sides.

For a right triangle, Area = ½ * (Product of perpendicular sides)
Therefore, product of perpendicular sides = 2 * Area

From statement I alone, hypotenuse = 10√2.

Using Pythagoras theorem, \(x^2 + y^2 = (10√2)^2\).
We have 2 unknowns in 1 independent equation. We cannot find out unique values for x and y and hence cannot find a unique answer.

Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, area of triangle = 50.

As highlighted in the analysis above, product of perpendicular sides = xy = 2 * Area.
Therefore, xy = 2 * 50 = 100
Statement II alone is sufficient. Answer options C and E can be eliminated.

The correct answer option is B.

Hope that helps!
Aravind B T
_________________

Hi Shrihari12

Apologies for late reply!

IMPORTANT SUGGESTION: Never make any assumptions in any Quant question

As per your comment, you are making an assumption that it's a 45-45-90 triangle whereas it could also be a 30-60-90 triangle hence the statement is NOT SUFFICIENT

I hope that's clear to you now.

Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1

Why can't i say that if its a right triangle i know that the lengths must follow the ratios x:x√3:2x and i know the hypothenuse is 10√2 so 2x=10√2 , hence i know the value of x and y?

The sides of a right triangle are in the ratio \(1:√3:2\) if and only it's a 30°-60°-90° triangle. When evaluating (1) we don't know that.




Hope it helps.
_________________

Hi user_00607,

You've come up with ONE possible example that fits the given information; however, there are LOTS of potential examples that fit.

We're told that X and Y are the lengths of the legs of a RIGHT triangle. We're asked for the value of (X)(Y). Fact 1 tells us that the hypotenuse of the triangle is 10√2. Using the Pythagorean Theorem, we can create the following equation:

X^2 + Y^2 = (10√2)^2
X^2 + Y^2 = 200

Since this one equation has two variables, it's likely that there are LOTS of different possible values for X and Y - and as a result, there are probably lots of different values for (X)(Y). Here are two examples:

IF....
X=10 and Y= 10, then the answer to the question is (10)(10) = 100
X = 9 and Y = √119, then the answer to the question is 9√119
Fact 1 is INSUFFICIENT

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: Rich.C@empowergmat.com
www.empowergmat.com

Can't we use 45:45:90 rule here?