The triangles in the figure above are equilateral and the ratio of the

Question

https://gmatclub.com/forum/download/file.php?id=6319 The triangles in the figure above are equilateral and the ratio of the length of a side of the larger triangle to the length of a side of the smaller triangle is 2/1. If the area of the larger triangular region is K, what is the area of the shaded region in terms of K? (A) 3/4*K (B) 2/3*K (C) 1/2*K (D) 1/3*K (E) 1/4*K 2triangles.GIF

Bunuel · Accepted Answer

Yes, the property given above is very useful. It states:  in two similar triangles, the ratio of their areas is the square of the ratio of their sides:  \frac{AREA}{area}=\frac{S^2}{s^2} .

As both big and inscribed triangles are equilateral then they are similar, so  \frac{AREA}{area}=\frac{S^2}{s^2}=\frac{2^2}{1^2}=4 , so if  AREA=K  then  area=\frac{K}{4}  --> the area of the shaded region equals to  area_{shaded}=K-\frac{K}{4}=\frac{3K}{4} .

Answer: A.

buffdaddy · Answer

an easy way to solve this would be to visualise the below triangle instead. it shows u the answer straight away without the need for any calculations

bkk145 · Answer

A would be my answer.
Area of shaded = Area of Large - Area of Small
Give: Area of Large = K
We need to find Area of Small in term of K.
Triangle area = 1/2 * base * height
and since a side of the large triangle is twice larger than a side of small triangle, we are dealing with (1/2) * (1/2) = 1/4 factor.
Therefore, Area of shaded = K - K/4 = 3K/4

sondenso · Answer

A.

the ratio tells you that the larger also is a equilateral triangle.
Length of small one is a
length of larger one is b
a =b/2

the larger arear = K= b^2*(square root(3)/4) I called square root(3)/4 S
the smaller area = a^2 *S = S*b^2 /4 = K/4

the shaded area = K -K/4 =3/4 K

sixsigma1978 · Answer

Another way to do this!
Theorem : If there are Two similar triangles with sides in Ratio : S1 : S2 - then their areas are in the ratio S1^2 : S2 ^2
=> Area of Larger : Area of Smaller = S1^2 : S2^2
=> K : As = 2^2 : 1
=> As = k/4

Therefore, Area of the shaded region : K-K/4 = 3k/4

gettinit · Answer

very useful ratio thanks guys.

sanjoo · Answer

m not geting this onne..how K/4??

large triangle have area 4...how k/4?

Bunuel · Answer

Check here: the-triangles-in-the-figure-above-are-equilateral-and-the-62201.html#p827422 We have that \frac{AREA}{area}=4 . Now, since AREA=K then \frac{K}{area}=4 --> area=\frac{K}{4} --> the area of the shaded region equals to area_{shaded}=K-\frac{K}{4}=\frac{3K}{4} . Hope it's clear.

pikachu · Answer

The easiest way to solve this one would be by picking numbers. Lets say each side of the larger triangle is 6 and given the ratio 2:1 each side of smaller triangle then is 3. Area of an equilateral triangle can be calculated using formula:  s^2(\sqrt{3})/4  where s = side. So area of large triangle = k =  9(\sqrt{3})  and area of small triangle =  9(\sqrt{3})/4  =  k/4 . So the area of the smaller triangle is 1/4 the area of the large triangle. Area of shaded region= k-k/4  =  3/4k

Bunuel · Answer

Bumping for review and further discussion.

GEOMETRY: Shaded Region Problems!

rohit8865 · Answer

Let side of larger eq. triangle=2X
area=sq.root3/4*(2X)^2------->sq.root3*x^2------(A)
therefore side of smaller eq. triangle=X(given ratio=2/1)
area of smaller- sq.root3*x^2/4
area of shaded part=larger-smaller triangle
=sq.root3*x^2-sq.root3*x^2/4------------>sq.root3*x^2(1-1/4)
substituting from eq. (A)
area of shaded part=3/4K
Ans A

KanishkM · Answer

KeyWord: Be careful of such questions

area of the  shaded   region in terms of K

Nothing much is to be done here

ratio of the length of a side of the larger triangle to the length of a side of the smaller triangle is 2/1 => 6/3

Area of bigger triangle => root3/4 * 36 = k -----(a)

Area of smaller triangle => root3/4 * 9 = k' --------(b)

divide (a) by (b)

k/k' = 4/1

k/4 = k'

Now what needs to be find ???

area of the shaded region in terms of K = 1 - k/4 = 3k/4

A

KarishmaB · Answer

All Equilateral triangles are similar by AA. And we know this about similar triangles - the ratio of their areas will be square of the ratio of their sides. This is discussed here: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2014/03 ... -the-gmat/ So if the ratio of their sides is 2/1, the ratio of their areas will be 4/1. So area of smaller triangle is 1/4 of larger triangle. The area of the rest of the larger triangle will be (3/4)K. Answer (A)

Kushchokhani · Answer

Bunuel

Link's post deleted?

Bunuel · Answer

Updated the link: GEOMETRY: Shaded Region Problems! Thank you very much!

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The triangles in the figure above are equilateral and the ratio of the

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The triangles in the figure above are equilateral and the ratio of the

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards