Author 
Message 
TAGS:

Hide Tags

Current Student
Joined: 12 Aug 2015
Posts: 2548

In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt
[#permalink]
Show Tags
Updated on: 14 Aug 2017, 06:02
Question Stats:
34% (02:40) correct 66% (02:05) wrong based on 161 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 13903 times ]
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
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
Joined: 02 Aug 2009
Posts: 8284

In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt
[#permalink]
Show Tags
14 Aug 2017, 05:52
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
_________________



Manager
Joined: 02 Nov 2015
Posts: 162

Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt
[#permalink]
Show Tags
14 Aug 2017, 07:04
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 S850LC using GMAT Club Forum mobile app



Math Expert
Joined: 02 Aug 2009
Posts: 8284

Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt
[#permalink]
Show Tags
14 Aug 2017, 07:20
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 S850LC using GMAT Club Forum mobile appHi.. 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 coprime, 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
Joined: 18 Aug 2017
Posts: 29

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



Senior Manager
Joined: 29 Jun 2017
Posts: 417
GPA: 4
WE: Engineering (Transportation)

Re: In triangle ABC, the measure of angle ABC is 90 degrees, and the lengt
[#permalink]
Show Tags
11 Sep 2017, 02:53
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



Director
Joined: 16 Jan 2019
Posts: 507
Location: India
Concentration: General Management
WE: Sales (Other)

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






