In the figure shown, what is the area of triangular region PRT?

Question

image.png In the figure shown, what is the area of triangular region PRT? (1) The area of rectangular region PQST is 24. (2) The length of line segment RT is 5.

VeritasPrepBrandon · Answer

It statement 1, is it supposed to say "The area of the rectangular region PQST is 24? If so, statement 1 gives you enough information to prove that the area is 12. This is because the area of a quadrilateral is equal to base * height. Because the triangle is inscribed within the rectangle, we know that the base * height of the triangle must also equal 24. But the area of a triangle is equal to 1/2*base*height, so its area must equal 1/2*24 = 12.

Try some numbers to prove this to yourself. Say that the base of the rectangle is 24, and its height 1. That makes the base of the triangle 24 and its height 1, and 1/2*24*1 = 12. How about if the base is 6 and the height is 4? 1/2*6*4 = 12. The area of the triangle will always be 12.

Statement 2, on the other hand, provides neither the base nor the height, and is this insufficient.

This makes the correct answer A.

I hope this helps!!!

yuvrajsub · Answer

Thanks a lot for the crystal clear explanation! Never ever question the official answers. :oops:

meenakshi1 · Answer

@ bunuel - pls help. Is it always that area of any quadrilateral will be its bxh?

Bunuel · Answer

No, that's not true. Please check here: math-polygons-87336.html Hope it helps.

mau5 · Answer

Theory: The area of any # of trianlge(s) between 
A.2 fixed parallel lines, and 
B.With the same base is ALWAYS the same.

The first point makes sure that the height of the given triangles is always the same;the second point mandates the base lenght to be the same for all of them.

From the given image, try sliding the point R towards Q till the line segment RP co-incides with the side QP of the rectangle and thus, RT becomes the diagonal.
Now, if we were told to find the area of the newly formed triangle PRT, it would be nothing but  \frac{1}{2}*24

Hope this helps.

SweetiePi · Answer

I'm sorry but I have a bit of a question on this:

(1) The are of PQST is 24 (.5*B*H) so that's the area of the triangular region: sufficient

(2) The length of the line segment RT is 5.

RT is 5
If you drop a perpendicular line down from R to PT that forms a 90 degree triangle. Let's call that point X. 
Can we then assume a 3-4-5 triangle? I got 90 degrees and a 5 right?
Then triangle XRT is similar to XRP because they share the same side RX so you can set up proportions to find length PX and XT to get length PT and since you know RX is 4. b*h*.5 is the answer? thus sufficient???

Bunuel · Answer

No, the red part is not correct. Hypotenuse of 5 does not necessarily means that the legs must be 3 and 4.

Amit0507 · Answer

You may also prove this algebraically. Area of the rectangle is the sum of the all three embedded triangles:

(1/2*RQ*ST) + (1/2*RS*ST) + (1/2*PT*ST). But, you will notice that RQ +RS = QS= PT; hence the sum of area of triangle PQR and RST is equal to the area of PRT. Thus, the area of triangle PRT is half of the rectangle.

Cez005 · Answer

Area of triangle = 1/2bh.
Area of rectangle = bh

1) Gives bh and is therefore suff.

2) Cannot get b or h. Insuff.

A

Kudos if you agree!

jwang1191 · Answer

I picked E because I thought we couldn't assume R was on QS since it is a Data Sufficiency Question and therefore, we don't know the height of the triangle. Just as we couldn't assume PQST was a quadrilateral until statement 1. I'm confused on what we can and cannot assume in these questions. Does anybody have any good advice on this?

Bunuel · Answer

R is shown to be on QS so it is actually on QS. This will never be a trick on the GMAT.

Nunuboy1994 · Answer

Tricky question ahhh

What the trap in this question is getting you to think that you need a particular hypotenuse length in order to calculate the respective areas- HOWEVER if you think about the question from a formulaic perspective then you will see that the area of a right triangle

1/2 BH essentially equals the area of the square

It doesn't actually matter what the height and length are- 6 and 4 or 8 and 3 because they will multiply to 24 which can just be substituted for BH in

1/2 BH

And will result in the same answer

A

BrentGMATPrepNow · Answer

Target question :    What is the area of triangular region PRT?

Statement 1 : The area of rectangular region PQST is 24  
The area of ∆PRT = (base)(height)/2
Statement 1 tells us that  (base)(height) = 24 
Since the base and the height of the  rectangle  are the SAME as the base and the height of the  triangle  PRT, we can use this information to find the area of ∆PRT. 
The  area of ∆PRT =  (base)(height) /2 =  24 /2 = 12 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : The length of line segment RT is 5 
The area of ∆PRT = (base)(height)/2
Statement 2 tells us NOTHING about the base and height, so we can't use this to find the area of ∆PRT. 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

Gmatstudent2018 · Answer

Hi Bunuel,

If it is not always type that the area of a quadrilateral is base*height then how is that property being used on this question? I answered E to this question because i was using the formula for rectangle's area as length * width. I am not able to follow the solution of this problem as I'm unable to understand the concept that is applied in regards to the area of a quadrilateral being base*height. Isn't the formula for the area of a rectangle length*width? I apologize that i'm asking such a basic question, but I'm unclear on this problem.

Thank you!

MikeScarn · Answer

Gmatstudent2018 , We know that L*W = 24 for the rectangle. Let's say that PQ=ST=Length and QS=PT=Width

For the triangle, let's assume our Base is line PT. So our Base=Width. Cool?

Now we need to find out what our Height would be for the triangle.

Our height would start at vertex R and be a line that is perpendicular with our Base... maybe draw a dotted line on your scratch piece of paper.

Now, how does your dotted line compare with the PQ and ST?

Sparta_750 · Answer

When I saw this question, the figure that came to my mind was that of two different rectangles each of which were divided equally into triangles.

See attached image and solution. Not the most elegant one, but worked for me.

PelumiOgunlana · Answer

I know the GMAT can be literal, so how do we know that the triangle is inscribed on the rectangle?  Bunuel

Gmatboat · Answer

Hi Brent, for S2, why cant we draw a perpendicular line from VERTEX R downward to PT, thereby splitting the triangle into two 45-45-90 triangles. Once we have two 45-45-90 triangles, we can solve the question by using information from Statement 2? (EG: RT the "hypotenuse" will be 5, so height will be 5/ \sqrt{2}, etc)

Johnny1989 · Answer

Because it can be a 45-45-90 triangle or anything else..

In the figure shown, what is the area of triangular region PRT?

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