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# WHat is the length of QS

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Director
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WHat is the length of QS [#permalink]  06 Oct 2004, 13:19
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WHat is the length of QS?
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[#permalink]  06 Oct 2004, 14:12
12/5
1) 3^2 - (5-SR)^2 = QS^2
2) 4^2 - SR^2 = QS^2

Solve for SR and you get SR = 16/5
QS^2 = 4^2 - (16/5)^2
QS = 12/5

Quite calculation intensive. Took me about 3 min.
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[#permalink]  06 Oct 2004, 14:32
12/5 is 2.4. Is there any smarter way to do this which is being tested?
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[#permalink]  06 Oct 2004, 14:37
It is an homothetic right angled triangle in another one :

so QS = 3*4/5 = 12/5
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[#permalink]  06 Oct 2004, 14:43
Twixt, can you plz explain the concept mate?
Director
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[#permalink]  06 Oct 2004, 15:00
Saurya,

PQR and PQS have same lengths regardless the homothetic (al ?) ratio.

PS is similar to PQ rgdless ratio
QS QR ..........
PQ PR .............

so QS = QR * ratio

ratio = PQ/PR = 3 / 5
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[#permalink]  06 Oct 2004, 17:58
I used area formula to get it

(1/2)*3*4 = (1/2)*5*x

you will get 12/5 =>2.4
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[#permalink]  06 Oct 2004, 18:01
crackgmat3 wrote:
I used area formula to get it

(1/2)*3*4 = (1/2)*5*x

you will get 12/5 =>2.4

Nice crackgmat3!
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[#permalink]  06 Oct 2004, 22:51
There is a formula from grade 9 of how to find it.

1/QS^2 = 1/QP^2 + 1/QR^2 (I call it the umbrella formula)

applies for a right triangle with an altitude relative to hypotenuse

evrything can solve it .
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[#permalink]  10 Oct 2004, 22:15
I found a fastest and easier way and I am quite surprised because usually I am not so strong in Quant

Area of the triangle is (B*h)/2 and you can use different "h", there are 3 in every triangle. In this triangle there is QS but also QR and another one that is not drawn (from point P to somewhere in QR where it will be perpendicular) but we don't really care about the last one so it's ok

Area of PQR is (3*4)/2 = 6 (using QR as "h")

The area has to be the same so you can find QS by replacing the data : (5* QS)/2 = 12/5

Sounds ok to me, is there any objection ?

Ps : sorry, just realized crackgmat3 used the same way...I don't delete my post, the explanations are worth it
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[#permalink]  11 Oct 2004, 13:04
it will be helpful to remember the following:

we can say 2 triangles are similar if two angles of one triangle are equal to two angles of another triangle.

Having said that,
triangle pqr and qsr(or qsp) are similar

reason:
ang pqr = 90 = ang qsr
ang prq = ang qrs (they represent the same angle in the figure)

now because of that since corresponding sides are proportional,
qs/4 = 3/5

so qs = 12/5

-anish

saurya_s wrote:
Twixt, can you plz explain the concept mate?
[#permalink] 11 Oct 2004, 13:04
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# WHat is the length of QS

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