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In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt

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In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt [#permalink]

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New post Updated on: 14 Aug 2017, 06:02
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Difficulty:

  95% (hard)

Question Stats:

39% (01:23) correct 61% (01:26) wrong based on 172 sessions

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In triangle ABC, the measure of angle ABC is 90 degrees, and the lengths of two sides of triangle ABC are shown. Triangle DEF is similar to ABC and has integer side lengths. Which of the following could be the area of triangle DEF ?

A)15
B)48
C)90
D)150
E)204


Source => Kaplan.
Any laconic way to solve this up ?

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Originally posted by stonecold on 14 Aug 2017, 04:48.
Last edited by Bunuel on 14 Aug 2017, 06:02, edited 2 times in total.
Edited the question.
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In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt [#permalink]

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New post 14 Aug 2017, 05:52
stonecold wrote:
Image

In triangle ABC, the measure of angle ABC is 90 degrees, and the lengths of two sides of triangle ABC are shown. Triangle DEF is similar to ABC and has integer side lengths. Which of the following could be the area of triangle DEF ?

A)15
B)48
C)90
D)150
E)204


Source => Kaplan.
Any laconic way to solve this up ?



Hi..
Sides of ∆DEF will be in similar ratio as sides of ∆ABC..
So EF=8x and DE=6x...
Since the sides 6 and 8 have 2 as Common factor, x will be an integer or a fraction with 2 in denominator.
Area of ∆ABC = 1/2 *6*8=24..
Area of∆DEF = 1/2 *6x*8x=24x^2..
Now this x should come out as a fraction with 2 in denominator or INTEGER..

Check with choices..
A)15
24x^2=15.... x=√(5/8).no

B)48
24x^2=48.....X=√2..no

C)90.
24x^2=90...X=√(15/4)=√15/2...no

D)150
24x^2=150...x^2=150/24=25/4..
X=√(25/4)=5/2...yes

E)204..
24x^2=204...X=√34/2 No

D
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt [#permalink]

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New post 14 Aug 2017, 07:04
1
chetan2u wrote:
stonecold wrote:
Image

In triangle ABC, the measure of angle ABC is 90 degrees, and the lengths of two sides of triangle ABC are shown. Triangle DEF is similar to ABC and has integer side lengths. Which of the following could be the area of triangle DEF ?

A)15
B)48
C)90
D)150
E)204


Source => Kaplan.
Any laconic way to solve this up ?



Hi..
Sides of ∆DEF will be in similar ratio as sides of ∆ABC..
So EF=8x and DE=6x...
Area of ∆ABC = 1/2 *6*8=24..
Area of∆DEF = 1/2 *6x*8x=24x^2..
Now this x should come out as a fraction or INTEGER..
Check with choices..
A)15
24x^2=15...no
B)48..no
C)90..no
D)150
24x^2=150...x^2=150/24=25/4..
X=√(25/4)=5/2...yes
E)204..No

D

Hi Chetan,
The way you explain are real awesome and u really deserve a Kudo from me . And I Have given also...[GRINNING FACE WITH SMILING EYES]

But 1 thing I don't understand why They value of x^2 should be a fraction ??
Tough for me to interpret.
Pls help.

Thanks in advance.

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app
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Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt [#permalink]

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New post 14 Aug 2017, 07:20
kumarparitosh123 wrote:
chetan2u wrote:
stonecold wrote:
Image

In triangle ABC, the measure of angle ABC is 90 degrees, and the lengths of two sides of triangle ABC are shown. Triangle DEF is similar to ABC and has integer side lengths. Which of the following could be the area of triangle DEF ?

A)15
B)48
C)90
D)150
E)204


Source => Kaplan.
Any laconic way to solve this up ?



Hi..
Sides of ∆DEF will be in similar ratio as sides of ∆ABC..
So EF=8x and DE=6x...
Area of ∆ABC = 1/2 *6*8=24..
Area of∆DEF = 1/2 *6x*8x=24x^2..
Now this x should come out as a fraction or INTEGER..
Check with choices..
A)15
24x^2=15...no
B)48..no
C)90..no
D)150
24x^2=150...x^2=150/24=25/4..
X=√(25/4)=5/2...yes
E)204..No

D

Hi Chetan,
The way you explain are real awesome and u really deserve a Kudo from me . And I Have given also...[GRINNING FACE WITH SMILING EYES]

But 1 thing I don't understand why They value of x^2 should be a fraction ??
Tough for me to interpret.
Pls help.

Thanks in advance.

Sent from my Lenovo TAB S8-50LC using GMAT Club Forum mobile app



Hi..
The sides of TWO similar triangle will have same ratio with corresponding sides.

Say here sides are 6 and 8...
If corresponding side of 6 of similar triangle is 6*1/2=3, so side corresponding to 8 will be 8*1/2=4..

Had it not been given that sides are integer than ofcourse x could be anything...

Yes if sides were 3 and 4 or co-prime, fraction would not have been possible.
Here 6 and 8 have 2 as Common factor so a fraction with 2 in denominator can also be the ratio ..
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt [#permalink]

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New post 27 Aug 2017, 20:25
Triangle DEF is similar to ABC, so \(\frac{DE}{6}\) = \(\frac{EF}{8}\) => DE = \(\frac{4EF}{3}\)

Let area of triangle DEF is S, S = \(\frac{(DE*EF)}{2} =\frac{4EF}{3} * \frac{EF}{2} = \frac{2EF^2}{3}\)

So \(\frac{(S *3)}{2}\) = \(EF^2\) we can conclude:
1- S is an even number, eliminate answer A
2-\(\frac{(S * 3)}{2}\) must be a perfect square of an integer.

B- S = 48, \(\frac{(S * 3)}{2} = 144/2\) = 72. No.
C- S = 90, \(\frac{(S * 3)}{2} = 270/2\) = 135. No.
D- S = 150,\(\frac{(S * 3)}{2} = 450/2 = 225\)= \(15^2\). Yes.
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Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt [#permalink]

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New post 11 Sep 2017, 02:53
1
Not worth of 95% difficulty:-

Answer is D 150

AB/BC = 6/8 = 3/4 = DE/EF
Area DEF = 1/2 DE xEF
=> 6
6 is the area when 3/4 is the least ratio of integers => if the ratio is increased then it will grow in squares of numbers from 1 ,2,3 and so on
because
3/4 = 3/4
6/8 = 3x2/4x2
9/12 = 3x3/4x3

so we can see both sides are being multiplied by 1,2,3 two times

so possible values of area can be 6 x( 1,4,9,16,25,36)
6x25 is the value => 150 is the answer D
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Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt   [#permalink] 11 Sep 2017, 02:53
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