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

Question

https://gmatclub.com/forum/download/file.php?id=36942 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 ? Untitled.png

chetan2u · Accepted Answer

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

kumarparitosh123 · Answer

Hi Chetan,
The way you explain are real awesome and u really deserve a Kudo from me . And I Have given also...

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

chetan2u · Answer

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 ..

NamVu1990 · Answer

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.

sahilvijay · Answer

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

firas92 · Answer

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) ------------------

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)

Fdambro294 · Answer

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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ScottTargetTestPrep · Answer

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

carouselambra · Answer

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

ansab · Answer

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..

unraveled · Answer

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.

\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.

hadimadi · Answer

Hi,

Area of initial triangle=1/2*g*h=24

For similar triangles, sides g' and h' of the new similar triangle will ways be in the same ratio as g and h:
g':h' <-> 6:8 <-> 3:4

Area_similar_triangle= 1/2*g'*h' =1/2*(3x*4x)=1/2*(12x^2)=6x^2

Plugging in numbers for x we quickly see that for x=5, we get for the area 6*5^2=6*25=150 -> (D)

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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

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