What is the area of right triangle XYZ?

Question

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

littlewarthog · Accepted Answer

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.

BrentGMATPrepNow · Answer

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: 
 https://i.imgur.com/TSVshdl.png 
Here, the  area = (1/2)(9)(12) =  54

case b: 
 https://i.imgur.com/1qvTmU5.png 
Here, the  area = (1/2)(9)(15)=  67.5  
Since we cannot answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer = E

Cheers,
Brent

minwoswoh · Answer

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?

minwoswoh · Answer

Thanks for the explanation!
+1 for you

Gonga · Answer

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?

Arpitkumar · Answer

Such a sly Question , got it wrong !!!

Kimberly77 · Answer

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

BrentGMATPrepNow · Answer

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

ScottTargetTestPrep · Answer

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.

Eliminate answer choice B.

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.

Answer: E

Kimberly77 · Answer

Thanks Brent  BrentGMATPrepNow

Kimberly77 · Answer

Great explanation. Thanks Scott  ScottTargetTestPrep

ScottTargetTestPrep · Answer

Sure thing!

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the area of right triangle XYZ?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

What is the area of right triangle XYZ?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards