Geometry

is the answer A) 3/4K?

Yes.....Can you please solve it in Detail

Sure.

We know that all the triangles are equilateral and that the sides of the larger triangle are 2 times bigger. I basically plugged in numbers to figure out the area.

So, for the large triangle, lets make the side length=4. To find the area, we apply the (bh)/2 formula. B/c its equilateral, we know that the height will break about the equilateral into two identical 30-60-90 right triangles. So, the side=4 is our hypotenuse, and the height will be 4/2 (sqrt 3) or 2 (sqrt 3). Thus, the area of the large triangle is [4* 2(sqrt 3)]/2 = 4(sqrt 3).

Lets now look at the smaller triangles. We know that the length of the smaller equilateral is half of that of the larger. So, given that I used 4 for the larger side, here we'll use half of that which is 2. Again, same concept applies, we come up with two identical 30-60-90 right triangles and our height will be (sqrt 3). The area then of one small equilateral is [2*(sqrt 3)]/2 = (sqrt 3).

Compare the area of the larger to that of the smaller. Smaller triangle area is 1/4 that of the larger triangle. So if we take out that one small unshaded triangle, we are left with 3(sqrt 3) which is 3/4K where K=area of the large triangle.

Hope that helps.

I find the formula for the area of an equilateral triangle is a good one to memorize.

If the length of the side of an equilateral triangle is s, its area is s^2root(3)/4.

For this problem, let's call the smaller triangle's side s. Therefore, the larger triangle's side is 2s. It follows that:

A(small) = s^2root(3)/4
A(large) = (2s)^2root(3)/4 = 4s^2root(3)/4 = s^2root(3)

This shows that the larger triangle's area is 4 times the area of the smaller triangle. If the larger triangle's area is K, then the smaller triangle's area is K/4.

Therefore, the shaded region's area is K - K/4 or 3K/4.

You can use the formula of two similar triangles:
ratio of the areas is equal to the square of the ratio.
For example, if one equilateral side is 1 and the other has side of 2, the ratio is 2 times bigger so the area of the bigger one will be 4 times more than the area of the smaller one.
So the small is 1/4 of the big and we subtract to have 3/4

Another way is to draw the triangles, you can see that you can arrange 4 small triangles in the big one, so the small one is 1/4 and the gray area is 3/4

pepeprepa wrote:Another way is to draw the triangles, you can see that you can arrange 4 small triangles in the big one, so the small one is 1/4 and the gray area is 3/4

I am in agreement with pepeprepa here. There is no need to even write any calculations down!

If you know that all triangles are similar triangles, then the largest triangle is made up of 4 smaller ones. 3 out of 4 of them are shaded.
Bingo.