SOLUTION: A regular hexagon A has the midpoints of its edges joined to form a smaller hexagon B and this process is repeated by joining the midpoints of the edges of B to get a third hexagon

Question 1112136: A regular hexagon A has the midpoints of its edges joined to form a smaller hexagon B and this process is repeated by joining the midpoints of the edges of B to get a third hexagon C. What is the ratio of the area of C to the area of A? Please Explain.
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

The ratio of the area of hexagon A to the area of hexagon B is going to be the same as the ratio of the area of hexagon B to the area of hexagon C, because the same process is used to create each smaller hexagon.

For hexagon A and hexagon B, choose one side of hexagon A and draw segments from the center of the hexagon to the midpoint of that side and to one endpoint of that side.

Those two segments, along with the half side of hexagon A, form a 30-60-90 right triangle, so the ratio of the lengths of the two segments is sqrt(3):2.

But those two segments are corresponding parts of hexagons A and B. Since the ratio of corresponding linear measurements between hexagons B and A is sqrt(3):2, the ratio of the areas of hexagons B and A is the square of that ratio, which is 3:4.

So the area of hexagon B is 3/4 the area of hexagon A; and similarly the area of hexagon C is 3/4 the area of hexagon B. So the area of hexagon C is (3/4)^2 = 9/16 the area of hexagon A.

Answer: The ratio of the area of hexagon C to the area of hexagon A is 9:16.
RELATED QUESTIONS
A regular hexagon A has the midpoints of its edge joined to form a smaller hexagon B, and (answered by greenestamps)

In a regular hexagon, the midpoints of the sides are joined to form a shaded regular... (answered by ikleyn,greenestamps)

The midpoints of a hexagon are connected to form a smaller hexagon. What is the ratio of... (answered by CubeyThePenguin)

A regular hexagon is inscribed in a circle. another regular hexagon is circumscribed... (answered by solver91311)

A regular hexagon ABCDEF has vertices at A(4,4√3), B(8,4√3), C(10,2√3), (answered by Alan3354)

A regular hexagon has an apothem of 7root3 cm. What is the area of this... (answered by Earlsdon)

A regular hexagon has a perimeter of 54cm. What is the area of this hexagon to the... (answered by robertb)

What is the difference between a regular hexagon, equiangular hexagon and equilateral... (answered by solver91311)

A regular hexagon (with all sides equal and all angles equal) has perimeter 60. Another... (answered by Alan3354)