Question 1

We will use the properties of a rectangle to find the fourth foot and then analyze the table's placement and dimensions.

Step 1 — Find the fourth foot

Let the three given feet of the rectangular table be A($\textbf{8}$, $\textbf{9}$), B($\textbf{11}$, $\textbf{9}$), and C($\textbf{11}$, $\textbf{7}$). In a rectangle, opposite sides are parallel and equal. Points A and B have the same y-coordinate, so AB is a horizontal side. Points B and C have the same x-coordinate, so BC is a vertical side. The fourth foot, D, must complete the rectangle. It will share the x-coordinate with A and the y-coordinate with C.

The x-coordinate of D is $\textbf{8}$. The y-coordinate of D is $\textbf{7}$.

$\boxed{\text{The fourth foot is at (8, 7)}}$

Step 2 — Evaluate the spot

Let's check the table's position relative to the room. The room extends from x=$\textbf{0}$ to x=$\textbf{12}$ and y=$\textbf{0}$ to y=$\textbf{10}$. The table's x-coordinates range from $\textbf{8}$ to $\textbf{11}$. The table's y-coordinates range from $\textbf{7}$ to $\textbf{9}$. All table coordinates are within the room's boundaries. The table is placed against the right wall (x=$\textbf{12}$) and near the top wall (y=$\textbf{10}$). This placement uses wall space efficiently. It leaves the central area of the room clear.

$\boxed{\text{Yes, this is a good spot for the table}}$

Step 3 — Calculate dimensions and height

Let's find the lengths of the sides of the table. The distance between ($\textbf{8}$, $\textbf{9}$) and ($\textbf{11}$, $\textbf{9}$) is the absolute difference in x-coordinates.

$= |11 - 8|$

$\boxed{3 \text{ units (length)}}$

The distance between ($\textbf{11}$, $\textbf{9}$) and ($\textbf{11}$, $\textbf{7}$) is the absolute difference in y-coordinates.

$= |9 - 7|$

$\boxed{2 \text{ units (width)}}$

The diagram is a 2D floor plan. It only shows x and y coordinates. There is no information about the third dimension, height.

$\boxed{\text{The height of the table cannot be determined}}$

Answer

(i) The fourth foot of the table will be at (8, 7). (ii) Yes, this is a good spot for the table as it is placed against two walls, leaving the main room area clear. (iii) The length of the table is 3 units, and the width is 2 units. The height of the table cannot be made out from this 2D floor plan.