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Current Student D
Joined: 12 Aug 2015
Posts: 2604
Schools: Boston U '20 (M)
GRE 1: Q169 V154 In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt  [#permalink]

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12 00:00

Difficulty:   95% (hard)

Question Stats: 33% (02:39) correct 67% (02:03) wrong based on 154 sessions

### HideShow timer Statistics 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 ?

Attachment: Untitled.png [ 1.77 KiB | Viewed 13471 times ]

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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.
Math Expert V
Joined: 02 Aug 2009
Posts: 7754
In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt  [#permalink]

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1
1
stonecold wrote: 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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Manager  B
Joined: 02 Nov 2015
Posts: 163
GMAT 1: 640 Q49 V29 Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt  [#permalink]

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1
chetan2u wrote:
stonecold wrote: 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
Math Expert V
Joined: 02 Aug 2009
Posts: 7754
Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt  [#permalink]

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kumarparitosh123 wrote:
chetan2u wrote:
stonecold wrote: 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 ..
_________________
Intern  B
Joined: 18 Aug 2017
Posts: 30
GMAT 1: 670 Q49 V33 In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt  [#permalink]

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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.
Senior Manager  P
Joined: 29 Jun 2017
Posts: 438
GPA: 4
WE: Engineering (Transportation)
Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt  [#permalink]

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2
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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Give Kudos for correct answer and/or if you like the solution.
Senior Manager  P
Joined: 16 Jan 2019
Posts: 425
Location: India
Concentration: General Management
WE: Sales (Other)
In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt  [#permalink]

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By property, if two similar triangles have side lengths in the ratio a:b, then their areas will be in the ratio a^2 : b^2

Let DE=x

Then (Area of ABC)/(Area of DEF) = 6^2/x^2

=> x^2 = 6^2*(Area of DEF)/(Area of ABC)

=> x^2 = 3/2*(Area of DEF) ------------------ [Area of ABC=24]

We know that x^2 must be a perfect square since x is an integer.

Plugging in the options we get Area of DEF = 150

(D) In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt   [#permalink] 09 Apr 2019, 01:31
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