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

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

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26 Apr 2019, 03:13
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If x and y are the lengths of the legs of a right triangle, what is the value of xy ?

(1) The hypotenuse of the triangle is $$10\sqrt{2}$$.
(2) The area of the triangular region is 50.

DS94602.01
OG2020 NEW QUESTION

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Re: If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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26 Apr 2019, 06:39
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.
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If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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Updated on: 11 May 2019, 09:53
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.

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Originally posted by DavidTutorexamPAL on 27 Apr 2019, 07:40.
Last edited by DavidTutorexamPAL on 11 May 2019, 09:53, edited 1 time in total.
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Re: If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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27 Apr 2019, 08:14
Bunuel wrote:
If x and y are the lengths of the legs of a right triangle, what is the value of xy ?

(1) The hypotenuse of the triangle is $$10\sqrt{2}$$.
(2) The area of the triangular region is 50.

DS94602.01
OG2020 NEW QUESTION

#1
200= x^2+y^2
insufficient
#2
x*y= 100
sufficient
IMO B
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Re: If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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27 Apr 2019, 23:18
1
1
DavidTutorexamPAL wrote:
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 neither x not y).
Statement (2) is just what we were looking for: xy, as mentioned above, is half the triangular area.

Posted from my mobile device

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

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Re: If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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11 May 2019, 08:34
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Top Contributor
Bunuel wrote:
If x and y are the lengths of the legs of a right triangle, what is the value of xy ?

(1) The hypotenuse of the triangle is $$10\sqrt{2}$$.
(2) The area of the triangular region is 50.

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

Cheers,
Brent
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Re: If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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11 May 2019, 09:52
m1033512 wrote:
DavidTutorexamPAL wrote:
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 neither x not y).
Statement (2) is just what we were looking for: xy, as mentioned above, is half the triangular area.

Posted from my mobile device

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

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
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Re: If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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11 May 2019, 14:32
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

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Re: If x and y are the lengths of the legs of a right triangle, what is th  [#permalink]

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14 May 2019, 14:02
m1033512 wrote:
DavidTutorexamPAL wrote:
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 neither x not y).
Statement (2) is just what we were looking for: xy, as mentioned above, is half the triangular area.

Posted from my mobile device

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

made the same mistake... i thought any of the two sides can be the legs
Re: If x and y are the lengths of the legs of a right triangle, what is th   [#permalink] 14 May 2019, 14:02
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