Question

A $33$-gon $P_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $33$ vertices equals $99$. The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$. Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$. Find the sum of the $x$-coordinates of the vertices of $P_3$.

Answer 1

1.
The midpoint MPQ of PQ is given by  (a + c / 2, b + d / 2)

2.
Let the x coordinates of the vertices of P_1 be : 

x1, x2, x3,…x33

the x coordinates of P_2 be :

z1, x2, x3,…z33

and the x coordinates of P_3 be:


w1, w2, w3,…w33

3.
We are given with: 

X1 + x2 + x3… + x33 = 99

We also want to find the value of w1 + w2 + w3… + w33.

4.

Now, based from the midpoint formula:

 

Z1 = (x1 + x2) / 2

Z2 = (x2 + x3) / 2

Z3 = (x3 + x4) / 2

Z33 = (x33 + x1) / 2

and 

W1 = (z1 + z1) / 2


W2 = (z2 + z3) / 2

W3 = (z3 + z4) / 2

W13 = (z33 + z1) / 2

.
.

5.

W1 + w1 + w3… + w33 = (z1 + z1) / 2 +  (z2 + z3) / 2 + (z33 + z1) / 2 = 2 (z1 + z2 + z3… + z33) / 2

Z1 + z1 + z3… + z33 = (x1 + x2) / 2 + (x2 + x3) / 2 + (x33 + x1) / 2

2 (x1 + x2 + x3… + x33) / 2 = (x1 + x2 + x3… + x33 = 99


Answer: 99