Other rooms in the house: (i) Reiaan’s room door leads from the dining room which has the length and width . The length of the dining room extends from point to point . Sketch the dining room and mark the coordinates of its corners. (ii) Place a rectangular dining table precisely in the centre of the dining room. Write down the coordinates of the feet of the table.

Orienting Yourself: The Use of Coordinates | Exercise 1.2

Question 4

Other rooms in the house:

(i) Reiaan’s room door leads from the dining room which has the length $18\text{ ft}$ and width $15\text{ ft}$ . The length of the dining room extends from point $P$ to point $A$ . Sketch the dining room and mark the coordinates of its corners.

(ii) Place a rectangular $5\text{ ft} \times 3\text{ ft}$ dining table precisely in the centre of the dining room. Write down the coordinates of the feet of the table.

Solution

Understand the Question

The dining room is a rectangle of dimensions $18\text{ ft} \times 15\text{ ft}$ , with its top edge extending from point $P(-6, 0)$ to point $A(12, 0)$ on the $x$ -axis and spanning $15\text{ ft}$ downwards along the negative $y$ -axis.
The corner coordinates are determined by offsetting the $x$ and $y$ positions by the length and width.
The center of the room is found using the midpoint formula, and the coordinates of the table's feet are obtained by offsetting half the table's length and width from the center.

Step 1 · Find the Coordinates of the Dining Room Corners

From the given layout, point $P$ is at $(-6, 0)$ and point $A$ is at $(12, 0)$ . Diagram 1

Length along the $x$ -axis

\begin{aligned} \text{PA length} &= x_A - x_P \\ &= 12 - (-6) \\ &= 12 + 6 \\ &= 18\text{ ft} \end{aligned}

The room has a width of $15\text{ ft}$ extending downwards below $PA$ , so the $y$ -coordinate of the bottom corners is $y = 0 - 15 = -15$

Let the bottom corners below $A$ and $P$ be $Q$ and $S$ respectively:

$Q = (12, -15)$
$S = (-6, -15)$

Answer

(i) The coordinates of the corners are $(-6, 0)$ , $(12, 0)$ , $(12, -15)$ , and $(-6, -15)$ .

(ii) Place a rectangular $5\text{ ft} \times 3\text{ ft}$ dining table precisely in the centre of the dining room. Write down the coordinates of the feet of the table.

Step 1 · Find the Center of the Dining Room

The coordinates of the room's center are given by the midpoints of the $x$ and $y$ bounds

\begin{aligned} x\text{-coordinate of center} &= \dfrac{-6 + 12}{2} = \dfrac{6}{2} = 3 \\[0.6em] y\text{-coordinate of center} &= \dfrac{0 + (-15)}{2} = -7.5 \end{aligned}

Thus, the center of the dining room is $(3, -7.5)$ .

Step 2 · Calculate the Coordinates of the Table's Feet

The table dimensions are $5\text{ ft} \times 3\text{ ft}$ .

Half of the length $= \dfrac{5}{2} = 2.5\text{ ft}$
Half of the width $= \dfrac{3}{2} = 1.5\text{ ft}$

Calculating the boundaries of the table from the center $(3, -7.5)$

\begin{aligned} x_{\text{left}} &= 3 - 2.5 = 0.5 \\ x_{\text{right}} &= 3 + 2.5 = 5.5 \\ y_{\text{bottom}} &= -7.5 - 1.5 = -9 \\ y_{\text{top}} &= -7.5 + 1.5 = -6 \end{aligned}

The coordinates of the four corners (feet) are $(0.5, -9), \quad (5.5, -9), \quad (5.5, -6), \quad (0.5, -6)$

Answer

(ii) $(0.5, -9)$ , $(5.5, -9)$ , $(5.5, -6)$ , and $(0.5, -6)$

Common Mistakes

Sign Error in $y$ -Coordinates: Since the room extends below the $x$ -axis, the $y$ -coordinates must be negative ( $-15$ , not $+15$ ).
Offsetting Full Dimensions Instead of Half: Adding/subtracting the full dimensions ( $5\text{ ft}$ and $3\text{ ft}$ ) from the center rather than half-dimensions ( $2.5\text{ ft}$ and $1.5\text{ ft}$ ).
Incorrect Center Calculation: Forgetting to account for negative coordinates when finding the midpoint of the $y$ -values.

Question 4

More questions in Exercise 1.2