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

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14 Aug 2017, 04:48

6

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

95% (hard)

Question Stats:

38% (01:21) correct 62% (01:27) wrong based on 161 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 ?

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

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

Show Tags

14 Aug 2017, 07:04

1

This post received KUDOS

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.

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.

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 ..
_________________

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.

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

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11 Sep 2017, 02:53

1

This post received KUDOS

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 _________________

Give Kudos for correct answer and/or if you like the solution.