Please help me on this!

1) Jina is not a member of the Music Club.

2) Three students are a member of the Music Club but not the Math Club.

3 Mary and Dan are in all three clubs.

Step-by-step explanation:

Look at the diagram and notice how each club has its own circle. If a student is in that circle, then they are a member of that club. When circles overlap, that overlap space means that a student is a member of two or all three clubs, depending on how many and which circles overlap.

1) Looking at Jina's location on the diagram, she is in the overlap between the math and chess club circles. However, her position does NOT overlap with the Music Club circle, therefore she is not a member of the Music Club.

2) Let's look at the students of the Music Club circle and its overlaps with the Chess Club circle, but NOT the overlaps with the Math Club circle. We can see that Lucy, Josh, and Juan are in these locations, therefore there are three students who are members of the Music Club but not the Math Club.

3) To find which students are in all three clubs, you would look at the middle of the diagram where all three circles overlap. The students listed in this location are Mary and Dan, therefore they are in all three clubs.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!