SOLUTION: Two 12-sided polygons as similar. A side of the larger polygon is 9 times as long as the corresponding side of the smaller polygon. What is the ratio of the area of the smaller pol

Algebra ->  Polygons -> SOLUTION: Two 12-sided polygons as similar. A side of the larger polygon is 9 times as long as the corresponding side of the smaller polygon. What is the ratio of the area of the smaller pol      Log On


   

Question 550978: Two 12-sided polygons as similar. A side of the larger polygon is 9 times as long as the corresponding side of the smaller polygon. What is the ratio of the area of the smaller polygon to the area of the larger polygon?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the area of the larger polygon will be 9^2 = 81 times the area of the smaller polygon.
this makes the area of the smaller polygon equal to 1 / 81 times the area of the larger polygon.
every regular polygon can be thought of as being composed of a set number of regular triangles equal to the number of sides of the polygon starting with a rectangle. the base of each of those triangle is the side of the polygon and the vertex opposite the base of those triangles is the center of the polygon.
a triangle is composed of 3.
a rectangle is composed of 4
a pentagon is composed of 5.
a hexagon is composed of 6, etc.
the area of each of those polygons is equal to the sum of the area of each of the constituent triangles of each of those polygons.
if the polygons are similar, then their constituent triangles are similar.
since the area of proportional triangle is equal to the square of the length of the corresponding sides of each of those triangles, it follows that the area of the polygon will also have the same ratio based on the ratio of the length of the sides of those triangles.
an easy example is a square.
the length of each side of square 1 is equal to 5.
the length of each side of square 2 is equal to 45.
the area of square 1 is equal to 5^2 and the area of square 2 is equal to 45^2.
the ratio of the length of square 2 to square 1 is equal to 45/5 = 9.
the ratio of the area of square 2 to square 1 is equal to 45^2 / 5^2 which is equal to (45*45) / (9*9) which is equal to 9*9 which is equal to 81.
if you divide each square into their constituent triangles, you will find that square 1 has 4 triangles, each with a base of 5 and each with an altitude of 2.5.
the area of each of these triangles is equal to 1/2 * b * h = 1/2 * 5 * 2.5 which is equal to 2.5 * 2.5 which is equal to 6.25.
multiply that by 4 triangles and you get an area of 4 * 6.25 = 25.
you can do the same with square 2 to find that it's area is going to be equal to 4 * 1/2 * 45 * 22.5 which is equal to 2025.
if you divide 2025 by 25 you get 81.
the ratio is the area of each similar polygon is equal to the square of the ratio of the length of each correspondding side of each similar polygon.
the following diagram shows what your polygon looks like after being broken up into its constituent triangles.
$$$$$