Parent Request

I am trying to help my son solve this word problem on his geometry review sheet. Any help you can provide would be greatly appreciated.

The sum of the areas of two regular decagons is 39 square inches, and their radii are in the ratio of 2:3. Find the area of the larger decagon.

Thanks!

since the radii are in the ratio 2:3, the areas are in the ratio 4:9 (if it doesn't feel intuitively true that you square both the 2 and 3, think about what happens to area when you double and triple the lengths of the sides of a square

sum of areas is 39 ratio of areas is 4:9

Thankfully, this scenario has whole number solutions, so I can avoid some algebraic calculation. With a little trial and error we find that multiplying both numbers in the ratio 4:9 by three gives the answer

ratio of areas is 4:9, which is also 12:27 And 12+27 is 39

Therefore, the larger decagon has an area of 27 square inches.

Note that the shape of the fiqure is unneeded information!