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What is the area of right triangle XYZ?

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What is the area of right triangle XYZ? [#permalink]
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Hi there,

I found this question really tricky.

You can see that if the right answer is E, the reasoning obviously implies that even combining both statements you don´t have sufficient information to answer the question. As for this problem, this means that:
- EITHER XZ is the hypotenuse and therefore XY is 12 (the magic Pythagorean triple we all learnt!)
- OR XZ is not the hypotenuse and therefore XY is actually the hypotenuse. This would be written as 9^2 + 15^2 = XY^2. Therefore, XY = $$\sqrt{306}$$ = 17.5 approx.

Both options yield to different areas for the triangle. Hence, E.

My confusion is the following:
If the question stem states that we have a right triangle XYZ, doesn´t that imply that Y = 90º (the central letter = 90º)? In other words, doesn´t that imply that YZ and YX MUST BE the legs?
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Re: What is the area of right triangle XYZ? [#permalink]
littlewarthog wrote:
When the GMAT is describing a right triangle using the names of the vertices (ABC, XYZ, etc.), there is no convention on where the right angle is, so it could be at vertex A, B or C.

However, when the GMAT is describing angles by using the adjacent vertices, then the middle letter is the vertex where the angle is located. E.g. angle ABC is the angle inscribed by the vertices A, B and C.

Thanks for the explanation!
+1 for you
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Re: What is the area of right triangle XYZ? [#permalink]
Can we also think of it like, they said it was a right angle triangle, but we don't know if it's a 30-60-90 or a 45-45-90 triangle. Would that also be a factor we can consider in this question?
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Re: What is the area of right triangle XYZ? [#permalink]
Such a sly Question , got it wrong !!!
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Re: What is the area of right triangle XYZ? [#permalink]
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BrentGMATPrepNow wrote:
minwoswoh wrote:
What is the area of right triangle XYZ?

(1) Side YZ is 9 inches long.
(2) Side XZ is 15 inches long.

Target question: What is the area of right triangle XYZ?

KEY CONCEPT: Area of triangle = (1/2)(base)(height)
So, to answer the target question, we need lengths of BOTH the base AND the height.

Statement 1: Side YZ is 9 inches long.
We have only 1 measurement. So, there's no way to determine the area of the triangle.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Side XZ is 15 inches long.
We have only 1 measurement. So, there's no way to determine the area of the triangle.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
We still don't have enough information.
Consider the following 2 cases (which both satisfy the given conditions).

case a:

Here, the area = (1/2)(9)(12) = 54

case b:

Here, the area = (1/2)(9)(15)= 67.5
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Cheers,
Brent

Hi BrentGMATPrepNow, a bit confused with this one that given XZ is 15 inches long but in case b diagram is 15 = YZ and not XZ?
Both statements combined we can say that third side is 12. Therefore shouldn't this fixed the longest length is 15 therefore make it C? Not sure what did I miss here? Thanks Brent
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Re: What is the area of right triangle XYZ? [#permalink]
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Kimberly77 wrote:
Hi BrentGMATPrepNow, a bit confused with this one that given XZ is 15 inches long but in case b diagram is 15 = YZ and not XZ?
Both statements combined we can say that third side is 12. Therefore shouldn't this fixed the longest length is 15 therefore make it C? Not sure what did I miss here? Thanks Brent

I made a mistake with the second graphic.
I have not replaced it.
Please check out my new response.

Thanks for the heads up! ! Kudos for you!!!!

Cheers,
Brent
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Re: What is the area of right triangle XYZ? [#permalink]
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minwoswoh wrote:
What is the area of right triangle XYZ?

(1) Side YZ is 9 inches long.
(2) Side XZ is 15 inches long.

Question Stem Analysis:

We need to determine the area of right triangle XYZ. We are not provided with any other information in the question stem.

Statement One Alone:

$$\Rightarrow$$ Side YZ is 9 inches long.

Without knowing anything about the remaining two sides of the triangle, we cannot determine its area. Statement one alone is not sufficient.

Eliminate answer choices A and D.

Statement Two Alone:

$$\Rightarrow$$ Side XZ is 15 inches long.

Without knowing anything about the remaining two sides of the triangle, we cannot determine its area. Statement two alone is not sufficient.

Statements One and Two Together:

We should be careful not to assume that this is a 9 - 12 - 15 right triangle, because we don't know whether the side of length 15 is the hypotenuse.

If XZ is the hypotenuse, then we have a 9 - 12 - 15 right triangle, which means its area is 1/2 * 9 * 12 = 54.

If XZ is not the hypotenuse, then the area of the triangle is 1/2 * 9 * 15 = 67.5.

Since there is more than one possible value for the area of the triangle, statements one and two together is not sufficient.

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Re: What is the area of right triangle XYZ? [#permalink]
BrentGMATPrepNow wrote:
Kimberly77 wrote:
Hi BrentGMATPrepNow, a bit confused with this one that given XZ is 15 inches long but in case b diagram is 15 = YZ and not XZ?
Both statements combined we can say that third side is 12. Therefore shouldn't this fixed the longest length is 15 therefore make it C? Not sure what did I miss here? Thanks Brent

I made a mistake with the second graphic.
I have not replaced it.
Please check out my new response.

Thanks for the heads up! ! Kudos for you!!!!

Cheers,
Brent

Thanks Brent BrentGMATPrepNow
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Re: What is the area of right triangle XYZ? [#permalink]
ScottTargetTestPrep wrote:
minwoswoh wrote:
What is the area of right triangle XYZ?

(1) Side YZ is 9 inches long.
(2) Side XZ is 15 inches long.

Question Stem Analysis:

We need to determine the area of right triangle XYZ. We are not provided with any other information in the question stem.

Statement One Alone:

$$\Rightarrow$$ Side YZ is 9 inches long.

Without knowing anything about the remaining two sides of the triangle, we cannot determine its area. Statement one alone is not sufficient.

Eliminate answer choices A and D.

Statement Two Alone:

$$\Rightarrow$$ Side XZ is 15 inches long.

Without knowing anything about the remaining two sides of the triangle, we cannot determine its area. Statement two alone is not sufficient.

Statements One and Two Together:

We should be careful not to assume that this is a 9 - 12 - 15 right triangle, because we don't know whether the side of length 15 is the hypotenuse.

If XZ is the hypotenuse, then we have a 9 - 12 - 15 right triangle, which means its area is 1/2 * 9 * 12 = 54.

If XZ is not the hypotenuse, then the area of the triangle is 1/2 * 9 * 15 = 67.5.

Since there is more than one possible value for the area of the triangle, statements one and two together is not sufficient.

Great explanation. Thanks Scott ScottTargetTestPrep
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Re: What is the area of right triangle XYZ? [#permalink]
Kimberly77 wrote:
ScottTargetTestPrep wrote:
minwoswoh wrote:
What is the area of right triangle XYZ?

(1) Side YZ is 9 inches long.
(2) Side XZ is 15 inches long.

Question Stem Analysis:

We need to determine the area of right triangle XYZ. We are not provided with any other information in the question stem.

Statement One Alone:

$$\Rightarrow$$ Side YZ is 9 inches long.

Without knowing anything about the remaining two sides of the triangle, we cannot determine its area. Statement one alone is not sufficient.

Eliminate answer choices A and D.

Statement Two Alone:

$$\Rightarrow$$ Side XZ is 15 inches long.

Without knowing anything about the remaining two sides of the triangle, we cannot determine its area. Statement two alone is not sufficient.

Statements One and Two Together:

We should be careful not to assume that this is a 9 - 12 - 15 right triangle, because we don't know whether the side of length 15 is the hypotenuse.

If XZ is the hypotenuse, then we have a 9 - 12 - 15 right triangle, which means its area is 1/2 * 9 * 12 = 54.

If XZ is not the hypotenuse, then the area of the triangle is 1/2 * 9 * 15 = 67.5.

Since there is more than one possible value for the area of the triangle, statements one and two together is not sufficient.

Great explanation. Thanks Scott ScottTargetTestPrep

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Re: What is the area of right triangle XYZ? [#permalink]
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