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.
Given any 2 points P(a, b) and Q(c, d), the midpoint
`M_P_Q` of PQ is given by
`( (a+c)/(2), (b+d)/(2) )`.

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


`{x_1, x_2, x_3....x_3_3}`

the x coordinates of P_2 be :


`{z_1, z_2, z_3....z_3_3}`

and the x coordinates of P_3 be:


`{w_1, w_2, w_3....w_3_3}`

3.
we are given that


`x_1+ x_2+ x_3....+x_3_3=99`

and we want to find the value of
`w_1+ w_2+ w_3....+w_3_3`.

4.

According to the midpoint formula:


`z_1= (x_1+x_2)/(2)`


`z_2= (x_2+x_3)/(2)`


`z_3= (x_3+x_4)/(2)`
.
.

`z_3_3= (x_3_3+x_1)/(2)`

and



`w_1= (z_1+z_2)/(2)`


`w_2= (z_2+z_3)/(2)`


`w_3= (z_3+z_4)/(2)`
.
.

`w_3_3= (z_3_3+z_1)/(2)`

5.


`w_1+ w_2+ w_3....+w_3_3=(z_1+z_2)/(2)+(z_2+z_3)/(2)+...(z_3_3+z_1)/(2)= (2(z_1+z_2+ z_3....+z_3_3))/(2)`


`=(z_1+z_2+ z_3....+z_3_3)=(x_1+x_2)/(2)+(x_2+x_3)/(2)+...(x_3_3+x_1)/(2)`


`=(2(x_1+x_2+ x_3....+x_3_3))/(2)=(x_1+x_2+ x_3....+x_3_3)=99`


Answer: 99