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!

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