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

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 ..

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.

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

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)
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Please let me know if I’m missing something.....

In triangle ABC, Angle ABC is 90 degrees.

This means the 2 Legs are 6 and 8

This is a Multiple of a 3-4-5 Pythagorean Triplet.

We are told that the 2nd Triangle is a Similar Right Triangle. Further the Side Lengths must be Integers. Thus, the sides must be in the Ratio of: 3x - 4x - 5x

Taking the Areas of Increasing Multiples of the Pythagorean Triplet 3-4-5

1/2 * 3 * 4 = 6

1/2 * 9 * 12 = 54

1/2 * 12 * 16 = 96

1/2 * 15 * 20 = 150 ——- -an Answer Choice Match

D

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Solution:

We see that triangle ABC is a 3-4-5 right triangle. Since triangle DEF is similar to triangle ABC, it’s also a 3-4-5 right triangle. Therefore, the two legs of triangle DEF could be one of the following pairs: {3, 4}, {6, 8}, {9, 12}, {12, 16}, {15, 20}, etc. If it’s the first pair, the area of the triangle is ½ x 3 x 4 = 6 (recall that the area of a right triangle is half the product of its two legs). If it’s any larger pair, the area will be 6 multiplied by a perfect square. That is, if it’s the second pair, the area will be 6 x 4 = 24; third pair, 6 x 9 = 54; fourth pair, 6 x 16 = 96, and fifth pair, 6 x 25 = 150. We see that 150 is given as one of the choices. So 150 is the correct answer.

Answer: D
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Yeah I agree with some of the solutions here too.
Triangle A,B,C are in the ratios 3x:4x:5x
Hence, we will have a similar ratio in the triangle DEF since they are similar.

Now, area of triangle DEF= 1/2*3x*4x = 6x^2
Equate 6x^2 with the options given.
Only 150 gives you the side in the ratio of 5x.

Hence IMO D

For similar triangles: (where A is Area and L is length)

(L1/L2)^2=(A1/A2)

The simplest ratio and area for ABC is 3:4:5 and 6 units sq respectively

(4X/4)^2=(A1/6)

X^2=(A1/6)-----Try trial and error for each solution replacing A1 to get the integer value.

Only 150 (Option D) gives an integer value..

Any right triangle has area = \(\frac{1}{2} * 3x * 4x = 6 * x^2\) (for this question only)
Since DEF is similar to ABC, we just need to check for multiple of 6 here.
Possible answers are
6 * 1^2 = 6
6 * 2^2 = 24
6 * 3^2 = 54
6 * 4^2 = 96
6 * 5^2 = 150
6 * 6^2 = 216
....

Answer D.

[Note: Area of right triangle = \(\frac{1}{2} * 3x * 4x = 6 * x^2 OR \frac{1}{2} * 12x * 5x = 30 * x^2 OR \frac{1}{2} * 7x * 24x = 84 * x^2 OR \frac{1}{2} * 9x * 40x = 180 * x^2 \) OR any set of number that satisfy the pythagoras theorem.
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