Orienting Yourself: The Use of Coordinates | Exercise 1.1

Question 1

Fig. 1.3 shows Reiaan’s room with points OABC marking its corners. The x- and y-axes are marked in the figure. Point O is the origin.

Referring to Fig. 1.3, answer the following questions:

(i) If $\text{D}_1\text{R}_1$ represents the door to Reiaan’s room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis?

(ii) What are the coordinates of $\text{D}_1$ ?

(iii) If $\text{R}_1$ is the point $(11.5, 0)$ , how wide is the door? Do you think this is a comfortable width for the room door? If a person in a wheelchair wants to enter the room, will he/she be able to do so easily?

(iv) If $\text{B}_1 (0, 1.5)$ and $\text{B}_2 (0, 4)$ represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?

Solution

Understand the Question

The room is set on a Cartesian coordinate plane with corner $O$ at the origin $(0,0)$ .
The left wall lies along the $y$ -axis ( $x = 0$ ), so the distance of any point from the left wall is its $x$ -coordinate.
The bottom wall lies along the $x$ -axis ( $y = 0$ ), so the distance of any point from the $x$ -axis is its $y$ -coordinate.
The width of a door lying along an axis is found by calculating the difference between the coordinates of its two endpoints.

(i) If $\text{D}_1\text{R}_1$ represents the door to Reiaan’s room, how far is the door from the left wall (the y-axis) of the room? How far is the door from the x-axis?

Step 1 · Find Distance from y-axis and x-axis

Diagram 1

The door starts at point $\text{D}_1$ , where $x_{\text{D}_1} = 8$ and $y_{\text{D}_1} = 0$ .

\begin{aligned} \text{Distance from y-axis} &= x_{\text{D}_1} \\[0.6em] &= 8\text{ units} \end{aligned}

\begin{aligned} \text{Distance from x-axis} &= y_{\text{D}_1} \\[0.6em] &= 0\text{ units} \end{aligned}

Answer

(i) Distance from $y$ -axis: $8\text{ units}$ ; Distance from $x$ -axis: $0\text{ units}$

(ii) What are the coordinates of $\text{D}_1$ ?

Step 1 · Find Coordinates of $\text{D}_1$

Diagram 2

Point $\text{D}_1$ lies on the $x$ -axis with $x = 8$ and $y = 0$ :

\begin{aligned} \text{Coordinates of D}_1 &= (x_{\text{D}_1}, y_{\text{D}_1}) \\[0.6em] &= (8, 0) \end{aligned}

Answer

(ii) $(8, 0)$

Step 1 · Calculate Room Door Width and Accessibility

Diagram 3

Door width is the difference between the $x$ -coordinates of $\text{R}_1(11.5, 0)$ and $\text{D}_1(8, 0)$ :

\begin{aligned} \text{Width of room door} &= x_{\text{R}_1} - x_{\text{D}_1} \\[0.6em] &= 11.5 - 8 \\[0.6em] &= 3.5\text{ units} \end{aligned}

Assuming $1\text{ unit} = 1\text{ foot} = 12\text{ inches}$ :

Door width $= 3.5\text{ feet} = 42\text{ inches}$ .
Standard room doors are $30\text{--}36\text{ inches}$ , so $42\text{ inches}$ is very comfortable.
Standard wheelchair accessibility requires at least $32\text{ inches}$ . Since $42 > 32$ , a person in a wheelchair can enter easily.

Answer

(iii) Width is $3.5\text{ units}$ ( $42\text{ inches}$ ). Yes, it is a comfortable width and easily accessible for a wheelchair.

(iv) If $\text{B}_1 (0, 1.5)$ and $\text{B}_2 (0, 4)$ represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?

Step 1 · Calculate Bathroom Door Width and Compare

Diagram 4

Bathroom door width is the difference between the $y$ -coordinates of $\text{B}_2(0, 4)$ and $\text{B}_1(0, 1.5)$ :

\begin{aligned} \text{Width of bathroom door} &= y_{\text{B}_2} - y_{\text{B}_1} \\[0.6em] &= 4 - 1.5 \\[0.6em] &= 2.5\text{ units} \end{aligned}

Comparing door widths: $\text{Bathroom door } (2.5\text{ units}) < \text{Room door } (3.5\text{ units})$

Therefore, the bathroom door is narrower.

Answer

(iv) The bathroom door is narrower than the room door ( $2.5\text{ units} < 3.5\text{ units}$ ).

Common Mistakes

Axis Inversion: Confusing distance from the $y$ -axis with the $y$ -coordinate. The distance from the $y$ -axis is given by the $x$ -coordinate, and the distance from the $x$ -axis is given by the $y$ -coordinate.
Points on Axes: Forgetting that any point on the $x$ -axis has $y = 0$ , so $\text{D}_1$ is $(8, 0)$ and not $(0, 8)$ .