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Two 10-sided convex polygons are similar. The side of the larger polygon is 3 1/3 times longer than the corresponding side of the smaller polygon. What is the ratio of the area of the larger polygon to the area of the smaller polygon? angel

 Dec 29, 2015

Best Answer 

 #1
avatar+128089 
+10

Similar polygons are similar in all respects.....

 

If we divide each polygon into 10 equal triangles...the base of each triangle will be formed by a side.....and the side [base] of a triangle in the larger polygon will be  3+1/3  = 10/3   that of the base of a triangle in the smaller one.......and the height of each corresponding triangle in both figures will have the same relationship....[10/3].......so the ratio of the area of a triangle in the larger polygon to a corresponding triangle in the smaller polygon will just be the square of the ratio of the sides = [10/3]^2 = 100/9

 

And......the ratio of the area of one triangle in the larger polygon to the area of a triangle in the smaller polygon will have the same relationship as the area of the larger polygon to the area of smaller polygon  = [10/3]^2  = 100/9

 

 

cool cool cool

 Dec 29, 2015
edited by CPhill  Dec 29, 2015
 #1
avatar+128089 
+10
Best Answer

Similar polygons are similar in all respects.....

 

If we divide each polygon into 10 equal triangles...the base of each triangle will be formed by a side.....and the side [base] of a triangle in the larger polygon will be  3+1/3  = 10/3   that of the base of a triangle in the smaller one.......and the height of each corresponding triangle in both figures will have the same relationship....[10/3].......so the ratio of the area of a triangle in the larger polygon to a corresponding triangle in the smaller polygon will just be the square of the ratio of the sides = [10/3]^2 = 100/9

 

And......the ratio of the area of one triangle in the larger polygon to the area of a triangle in the smaller polygon will have the same relationship as the area of the larger polygon to the area of smaller polygon  = [10/3]^2  = 100/9

 

 

cool cool cool

CPhill Dec 29, 2015
edited by CPhill  Dec 29, 2015
 #2
avatar+466 
+5

That's what I thought... thank you for confirming that in my mind! cool

 Dec 29, 2015
 #3
avatar+118587 
+10

Hi Shades and CPhill.

 

Huh, that sounded like a lot of work chris.

 

If the sides of two similar figures have the ratio of  \(1:\frac{10}{3}\)

 

Then the area of the 2 figures have the ratio of

    \(1^2:\frac{10^2}{3^2}\\ 1:\frac{100}{9}\)

 

 

That is of course the small one to the large one, you asked for it the other way around so the sides need to be swapped.

 Dec 29, 2015

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