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Triangle PQR has a right angle at Q and a perimeter of 60.

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Triangle PQR has a right angle at Q and a perimeter of 60. [#permalink] New post 30 Mar 2008, 11:54
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Triangle PQR has a right angle at Q and a perimeter of 60. Line segment QS is perpendicular to PR and has a length of 12. PQ > QR. What is the ratio of the area of triangle PQS to the area of triangle RQS?


A. 3/2

B. 7/4

C. 15/8

D. 16/9

E. 2
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Re: PS: Nasty triangle [#permalink] New post 30 Mar 2008, 12:17
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D.
1-method)
Euclid theorem:h^2=m*n=144
m+n=c(hypothenus), where m>n.
PQS/RQS=(m/n)^2

only 16 and 9 gives 144 (where m=16x and 9x)

2-method) PQ=a, QR=b and PR=c (hypothenus)
a+b+c=60 so a+b=60-c
ab/2=12c/2 (triangle area formula)
a^2+b^2=c^2
(a+b)^2=a^2+b^2+2ab
so (60-c)^2=c^2+24c this gives c=25
now Euclid theorem: 12^2=x(25-x)
x=16 or 9 so D.
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Re: PS: Nasty triangle [#permalink] New post 30 Mar 2008, 12:19
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D

:? Only back-solving helped me to solve it under 2 min....
Does anybody know a fast direct approach?
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Re: PS: Nasty triangle [#permalink] New post 30 Mar 2008, 12:23
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UMB, I like your second method :)
+1
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Re: PS: Nasty triangle [#permalink] New post 30 Mar 2008, 14:41
Remember that the GMAT loves the 3-4-5 right triangle, which has a perimeter of 12, a factor of 60 (12*5)

15-20-25

QS is indeed 12

Ratio of areas is 12(16) to 12(9) i.e. 16:9
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Re: PS: Nasty triangle [#permalink] New post 30 Mar 2008, 17:23
I kinda went about Deriving what Euclid already put out there I guess.

Drawing the triangle out, you realize that what you really need is the ratio of PS:SR (the height is the same for both triangles and area is 1/2 b*h)

Writing out what Mr.Pythagoras says..

PR^2 = PQ^2 + QR^2 ------1
PQ^2 = 144 + PS^2 ------2
QR^2 = 144 + SR^2 -------3

And we know PR = PS+SR -----4

Substitute 2 and 3 on the right side of equation 1 and 4 on the left side

=> (PS+SR)^2=288 + PS^2 + SR^2

Solving this,
PS*SR = 144

Looking at the answers 16:9 jumps out. (You should solve for n^2 * x*y = 144, where x:y is the ratio and n is an integer <= 1)

Last edited by daszero on 30 Mar 2008, 21:10, edited 1 time in total.
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Re: PS: Nasty triangle [#permalink] New post 30 Mar 2008, 19:22
kevincan wrote:
Remember that the GMAT loves the 3-4-5 right triangle, which has a perimeter of 12, a factor of 60 (12*5)

15-20-25

QS is indeed 12

Ratio of areas is 12(16) to 12(9) i.e. 16:9


Kevincan,
I like, no, I love this. It definitely saves time. But long time to be expert like you! :o
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Re: PS: Nasty triangle   [#permalink] 30 Mar 2008, 19:22
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Triangle PQR has a right angle at Q and a perimeter of 60.

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