Answer:12: 90 diagonals13: heptagon14: 11.413 feet15: 7.416 feet16: 423.194 square feetStep-by-step explanation:12: So, in a shape every vertex will have (n-3) diagonals, where n is the number of sides. Look at simpler shapes like a triangle, square and pentagon just to verify. Now, if you just multiply that by the number of sides you would be counting some twice. Again, with a square there are 4 vertexes. If you count the number of diagonals each vertex has you would get 4, but the diagonal from the upper left to lower right would be the same as the lower right to upper left one. So each diagonal is counted twice in this method, which is the same for any shape. so the formula to find the number of diagonals is (n(n-3))/213: Here we just have to use the formula, set it equal to 14 and then solve for n. so this gets us (n(n-3))/2=14. We can turn this into a quadratic. [tex]n^2-3n=28[/tex] Then just subtract 28 from both sides and solve for n however you like . However you do it the two answers for the qudratic are -4 and 7. We can't have a shape with -4 sides so 7 it is, so a shape with 14 diagonals is a heptagon.14: This is a long one but a lot of what we do in this answer will be used for 15 and 16. I would actually try and find a video showing this or something so you could see it worked out. Let me know if I can clear anything up though.So when given a radius to find the apotham of a polygon, or vice versa the trick is to use triangles. So, we have a radius. If we make a second radius so the two radii are on either side of a side of the decagon, we now have a triangle with two equal sides, and in effect two equal angles. It is an isosceles triangle. We want to start finding angles so we can do some trig. The two angles that are the same are the ones at the base of the triangle, I will be finding the angle of the third, the one at the top. To do that we need to realize that we can do this all around the decagon to get 10 triangles like this. One for each side. Well, all of them have the same angles, and the sum of these "top angles" of each triangle will equal 360. so 360=10a. So for each triangle these angles near the center of the decagon will be 36 degrees. That means for the other two that are the same we just need to use that all three angles add up to 180 in a triangle. So the final two angles are 72 degrees.Now we know all angles and two of the sides of this triangle. We are going to solve for the area of this triangle and multiply it by 10, since we can split this decagon into 10 of these isosceles triangles. To find the area we need the base, which as mentioned before is one of the sides of the polygon. To find the base we need the height of the triangle, which hey, is the apotham. If we split the triangle in half we get two identical right triangles with the same base, which is half of the side of the decagon which we are trying to find, the same heing which we are tring to find so we can get the base and the same hypotenuse, which are the radii, so 12. Also the same angles, 72, 90 and 18 degrees.Now, we could actually use trig here to find both the height and base of the right triangles, or find one and use the pythagorean theorem to find the other. I'll just use trig for both. First the apotham of the deagon. That is the height of the right triangle or the distance from the center fo the decagon to the base. sin(72)=h/12 or 12sin(72) = h = 11.413. This brings us to...15: Now we want the side of the decagon, which is half the base of the right triangle in question. so 2b. Now the base itself can be found with trig. cos(72)=b/12 or 12cos(72)=b=3.708. So then the side of the decagon is 2b so 7.416. Now we have the base and height of the original isosceles triangle and can answer...16: Like I mentioned in 14, the area of the whole shape is the sum of each of these triangle's areas. there are going to be 10 triangles so that's 10A, where A is the area of one triangle. Now the area of a triangle is 1/2*base*height. plugging in gets us (1/2)7.416*11.413=42.3194. Multiplying by 10 just moves the decimal place so 423.194.Sorry this took so long, hopefully it's an adequet explanation. Do let me know if I can explain anything more clearly though. Perhaps I can upload a drawing.