In the figure above, ABC is a right triangle with AC as its hypotenuse

Question

https://gmatclub.com/forum/download/file.php?id=26493 In the figure above, ABC is a right triangle with AC as its hypotenuse, and PQRS is a square. What is the area of the square? (1) AC is 70 units long. (2) The product of the length of AS and the length of RC is 396. Kudos for a correct solution . conundrum.gif

Zhenek · Answer

#1 is clearly insufficient. 
#2 - since PQ and AC are parallel, we can figure out that triangles APS and QRC are similar. Going by angles we get the relation:QR/AS = RC/PS 
QR*PS = AS*RC = 396. QR*PS is pretty much the area of the square, sufficient.

B

Shree9975 · Answer

Hi  Bunuel :

Can you please explain how are the two triangles similar??
And can you give a detailed solution to the problem.

Zhenek · Answer

The only reason they are similar is coz angle B is 90. 
Say angle BPQ is q, then angle BQP = 90 - q, Since PQRS is a square, PQ is parallel to AC which means that angle A = angle BPQ = q and angle C = angle BQP = 90 - q. Now if you look closely at these 2 triangles which are right triangles by the way (courtesy of PQRS being a square), you can infer that angle A = angle RQC = q and angle C = angle APS = 90 - q which makes these 2 triangles similar.

EgmatQuantExpert · Answer

https://gmatclub.com/forum/download/file.php?id=26493 (1) AC is 70 units long. That is, AS + RS + RC = 70 We need to know the value of RS in order to find the area of the square. But the above equation also has 2 other unknowns. Even if we try to express AS and RC in terms of RS, we cannot do so without involving other dimensions of the triangles in this figure. Therefore, St. 1 is not sufficient to determine a unique value of RS. (2) The product of the length of AS and the length of RC is 396. That is, AS*RC = 396 In right triangle ABC, tanC = \frac{AB}{BC} . . . (1) In right triangle CRQ, tanC = \frac{QR}{RC} . . . (2) By equating (1) and (2), we get: \frac{AB}{BC} = \frac{QR}{RC} That is, RC = \frac{QR*BC}{AB} . . . (3) Now, in right triangle ABC, tanA = \frac{BC}{AB} . . . (1') In right triangle ASP, tanA = \frac{PS}{AS} . . . (2') By equating (1') and (2'), we get: \frac{BC}{AB} = \frac{PS}{AS} That is, AS = \frac{PS*AB}{BC} . . . (3') By substituting equations (3) and (3') in the red equation above, we get: \frac{PS*AB}{BC}*\frac{QR*BC}{AB} = 396 That is, PS*QR = 396 (Side of square)^2 = 396 So, the area of square = 396 sq. units. Thus, St. 2 is sufficient to find the area of the square. Hope this helped!

Best Regards Japinder

GMATinsight · Answer

I never suggest students to take help of any Trigonometry therefore the important way to learn to solve this question is identifying the similar triangles

Look at Triangle PAS and Triangle QRC

in Triangle PAS
Let Angle PAS = x
then Angle APS = (90-x)
and angle PSA = 90

Also look at triangle ABC where if Angle A = x then angle C = (90-x) because angle B=90

Now in Triangle QRC

Since Angle QCR = (90-x)
therefore, Angle RQC = x
and Angle QRC = 90

Since angles between Triangle Triangle PAS and Triangle QRC are equal hence they become Similar Triangle

i.e. PS/AS = RC/QR

i.e. AS x RC = PS x QR 
But PS and QR are equal as they are sides of Square therefore PS x QR represents area of Square
 hence, Area of Square = AS x RC

Statement 1: AC = 70 
 NOT SUFFICIENT

Statement 2: AS x RC = 396 
 SUFFICIENT

Answer: Option  B

UJs · Answer

triangle.png to get area of square we need one side of square or some relation with square sides and triangle stmt 1 : AC = 70 ; not sufficient one value and lots of unknowns stmt 2: AS x RC = 396 ; looks promising lets evaluate these sides now refer to these sides and triangles consisted with these side { AS } and { RC } , they have one side of square common, one right angle common, possibility of similar triangles and (ratio of sides can lead us to side^{2} ) lets see riangle APS ~ riangle CRQ \angle a = \angle x ; this is because a, b, c in bigger right riangle have eq (a + c =90) and similarly in smaller right riangle APS (a + y = 90) and CRQ (x + c = 90) \angle y = \angle c \angle S = \angle R ; both 90^{\circ} now in similar triangles riangle APS ~ riangle CRQ \frac{PS}{AS} = \frac{RC}{QR} side^{2} = AS x RC = 396 : sufficient Ans: B

streetking · Answer

Thanks for this detailed explanation.

1 question though, do we need to know trigonometry to this level for the GMAT. I would like a Q51

hellosanthosh2k2 · Answer

Statement 1 : clearly insufficient

Statement 2: AS * RC = 396

All the triangles are similiar

AS/PS = QR/RC

PS = QR = side of square

AS * RC = PS * QR = 396 = area of square => sufficient (B)

bumpbot · Answer

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In the figure above, ABC is a right triangle with AC as its hypotenuse

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