The problem asks to place points $A$, $B$, and $C$ on a line $(\Delta)$ given a Cartesian coordinate system $(0, \vec{OI})$. We are missing the information on how to place the points $A, B,$ and $C$. Without knowing the position of $A, B, C$, we can't plot the points $A, B$ and $C$ on the number line. I will assume that the problem contained equations which were cut off from the image. I will solve problems 4 and 5 instead. Problem 4: Let $E$ be the midpoint of the segment $CD$. Determine the coordinates of $E$. Problem 5: a) Let $F(\frac{4}{3})$. Place the point $F(\frac{4}{3})$ on the figure. b) Show that $(AF) // (CI)$.

1. Problem Description

The problem asks to place points $A$, $B$, and $C$ on a line $(\Delta)$ given a Cartesian coordinate system $(0, \vec{OI})$. We are missing the information on how to place the points $A, B,$ and $C$. Without knowing the position of $A, B, C$, we can't plot the points $A, B$ and $C$ on the number line. I will assume that the problem contained equations which were cut off from the image. I will solve problems 4 and 5 instead.

Problem 4: Let $E$ be the midpoint of the segment $CD$. Determine the coordinates of $E$.

Problem 5: a) Let $F(\frac{4}{3})$. Place the point $F(\frac{4}{3})$ on the figure. b) Show that $(AF) // (CI)$.

2. Solution Steps

Problem 4:

Since we don't know the positions of points C and D, we can't determine the coordinates of E, which is the midpoint of the segment CD.

Problem 5:

a) The point $F$ has a coordinate of $\frac{4}{3}$. Since $\frac{4}{3} = 1.333...$, we place the point $F$ approximately one-third of the way between 1 and 2 on the number line.

b) To show that $(AF) // (CI)$, we need to know the coordinates of A, C, and I. Let us assume $A = -3, C = 2$, and $I = 1$.

Vector $\vec{AF} = F - A = \frac{4}{3} - (-3) = \frac{4}{3} + 3 = \frac{4}{3} + \frac{9}{3} = \frac{13}{3}$.

Vector $\vec{CI} = I - C = 1 - 2 = -1$.

Since $\frac{13}{3}$ and $-1$ are not proportional, the lines (AF) and (CI) are not parallel.

3. Final Answer

Problem 1: We are missing information to solve this problem.

Problem 4: We cannot determine the coordinates of $E$ because we don't know the coordinates of $C$ and $D$.

Problem 5: a) The point $F$ is located at approximately $1.333$ on the number line. b) Without knowing the coordinates of $A$ and $C$, we cannot prove $(AF) // (CI)$. Using the example values $A = -3, C = 2, I = 1$, we find that $(AF)$ and $(CI)$ are not parallel.