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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # Three parallel lines r, s and t pass, respectively, through three cons

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GMATH Teacher P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Three parallel lines r, s and t pass, respectively, through three cons  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 30% (03:44) correct 70% (02:45) wrong based on 34 sessions

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Three parallel lines r, s and t pass, respectively, through three consecutive vertices A, B and C of square ABCD. If the distance between r and s is 5 and the distance between s and t is 7, the numerical value of the area of ABCD is:

(A) prime
(B) divisible by 7
(C) divisible by 11
(D) divisible by 23
(E) divisible by 37

Source: http://www.GMATH.net

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Posts: 8201
Re: Three parallel lines r, s and t pass, respectively, through three cons  [#permalink]

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fskilnik wrote:
Three parallel lines r, s and t pass, respectively, through three consecutive vertices A, B and C of square ABCD. If the distance between r and s is 5 and the distance between s and t is 7, the numerical value of the area of ABCD is:

(A) prime
(B) divisible by 7
(C) divisible by 11
(D) divisible by 23
(E) divisible by 37

Source: http://www.GMATH.net

Draw the figure as attached...

Now since the three lines are parallel, ∆AEB and ∆BFC are similar and congruent triangles as they have similar angles X,90-x and 90 . Also the hypotenuse is same - 'a'
So CF =EB=5.. therefore side of square =a=√(7^2+5^2)=√74

Area =a^2=(√74)^2=74
So area is divisible by 37

E
Attachments Screenshot_2018-09-27-08-17-20-898_com.viettran.INKredible.png [ 102.31 KiB | Viewed 547 times ]

_________________
GMATH Teacher P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Three parallel lines r, s and t pass, respectively, through three cons  [#permalink]

### Show Tags

1
fskilnik wrote:
Three parallel lines r, s and t pass, respectively, through three consecutive vertices A, B and C of square ABCD. If the distance between r and s is 5 and the distance between s and t is 7, the numerical value of the area of ABCD is:

(A) prime
(B) divisible by 7
(C) divisible by 11
(D) divisible by 23
(E) divisible by 37

Source: http://www.GMATH.net

Perfect, chetan2u. Thanks for the contribution!

There are two possible configurations. As we will see, any one of them will go to the same answer! (Otherwise the question would be ill posed.)

$$?\,\,\,:\,\,\,{L^{\,2}}\,\,{\text{divisib}}{\text{.}}\,\,{\text{property}}$$

From the images below, please note that (in both cases): 1. Right triangles AEB and BFC are similar. (All corresponding internal angles are equal.)
2. The ratio of similarity is 1:1 (AB and BC are homologous sides and AB=BC)

Conclusion: Right triangles AEB and BFC are congruent ("equal"), hence (in both cases):

3. Each triangle has hypotenuse (length) L, and legs (with lengths) 5 and 7.

Finally:

$$?\,\,\,:\,\,\,\,{L^{\,2}} = {5^{\,2}} + {7^{\,2}}\, = 74\,\,\left( { = 2 \cdot 37} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( E \right)\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net Three parallel lines r, s and t pass, respectively, through three cons   [#permalink] 27 Sep 2018, 13:27
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