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Geometry
Source: Beat The GMAT — Problem Solving |
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.
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.
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.
Rey Fernandez
Instructor
Manhattan GMAT
Instructor
Manhattan GMAT
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
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
I am in agreement with pepeprepa here. There is no need to even write any calculations down!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
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.
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