Question

A regular hexagon A has the midpoints of its edges joined to form a smaller hexagon B. This process is repeated by joining the midpoints of the edges of the hexagon B to get a third hexagon C. What is the ratio of the area of hexagon C to the area of hexagon A?

Answer 1

What's the ratio of one hexagon's area to the other is the what you're looking for. Since it's a regular hexagon, we know that it's really 6 equilateral triangles like this:

Answer 2

If you can find the area of one of those triangles, you can multiply that area by 6 to find the whole area. With a little bit of playing around, you can see that you can find what the area of the hexagon is based on one of the little lines.

Answer 3

That's all the hints I'm giving right now, see how far you can get, or if you have specific questions like, "What is a ratio?" or something, I'll answer them. =)

Answer 4

yeah, please give me the ratio of the hexagon A to hexagon B. ^_^

Answer 5

No.

Answer 6

the correct answer is 9/11. how come?

Answer 7

yup it is 9/11

Answer 8

please explain

Answer 9

It's a bunch of triangles...

Answer 10

but how it was derived to 9/11?

Answer 11

some solutions please :<

Answer 12

You can prove that the area of a hexagon is proportional to the square of its side. The constant of proportionality is not important here because it will cancel out when taking ratios. So the area of first hexagon of side s is k * s^2

The next step: Join the midpoints of any two adjacent sides of the hexagon and figure the length of the side of the second hexagon in terms of the side s of the first hexagon. The area of the second hexagon will be k * (side of second hexagon)^2

Take the ratio of areas of the second hexagon to the first hexagon and simply square it to get the ratio of the area of the third hexagon to the first one.

Answer 13

A hexagon is made up of six equilateral triangles. The area of an equilateral triangle of side s = 1/2 * s * (s * sin(60)). Thus area of a hexagon = k * s^2.

Next step, join the midpoints and figure out length of new side in terms of s.

Answer 14

AC and AB are two sides of the first hexagon. DE joining their midpoints is the side of the second hexagon. Due to similar triangles ABC and ADE, DE = 1/2*BC
BC = s*cos(30) + s*cos(30) = 2scos(30) = sqrt(3) * s
Therefore, DE = 1/2 * BC = 1/2 * sqrt(3) * s

Much like we did for the first hexagon, the area of the second hexagon = k * (side)^2
= k * {1/2 * sqrt(3) * s}^2 = k * 3/4 * s^2

Now I will let you finish the rest.