Chapter 1

Chapter 1: Orienting Yourself: The Use of Coordinates — Solutions Guide

1. EXERCISE SET 1.1

Referring to Fig. 1.3, answer the following questions:
(i) If $D_1R_1$ 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 $D_1$?
(iii) If $R_1$ 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 $B_1(0, 1.5)$ and $B_2(0, 4)$ represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?

Solution:

(i) Distance of the door:
Point $D_1$ is at $x = 8$ on the x-axis.
Distance from the left wall (y-axis) = 8 units (feet).
Since $D_1$ lies directly on the x-axis ($y = 0$), the distance from the x-axis = 0 units.

(ii) Coordinates of $D_1$:
$D_1 = (8, 0)$

(iii) Width and Accessibility of Room Door:
Width of door $D_1R_1 = 11.5 - 8 = 3.5$ feet.
Standard wheelchair accessibility requires a minimum width of around 2.5 to 3 feet. A 3.5 ft wide door is comfortable and easily accessible for a wheelchair user.

(iv) Comparison with Bathroom Door:
Width of bathroom door $B_1B_2 = 4 - 1.5 = 2.5$ feet.
Width of room door = 3.5 feet.
The bathroom door (2.5 ft) is narrower than the room door (3.5 ft).

2. EXERCISE SET 1.2

Question 1: Place Reiaan's rectangular study table with three of its feet at the points $(8, 9)$, $(11, 9)$ and $(11, 7)$.
(i) Where will the fourth foot of the table be?
(ii) Is this a good spot for the table?
(iii) What is the width of the table? The length? Can you make out the height of the table?

Solution:

(i) Fourth foot position:
The table forms a rectangle in the 2D plane. The x-coordinates pair as $(8, 11)$ and y-coordinates as $(7, 9)$.
Fourth foot = $(8, 7)$

(ii) Practical evaluation:
Yes, this spot is in the upper right quadrant of the room near the right wall, keeping the center and doorway clear.

(iii) Dimensions:
Length = $11 - 8 = 3$ units (feet).
Width = $9 - 7 = 2$ units (feet).
Height cannot be determined because a 2-D floor map only provides $x$ and $y$ dimensions (length and width), not $z$ (height).

Question 3: Look at Reiaan's bathroom (Fig 1.5).
(i) What are the coordinates of the four corners O, F, R, and P of the bathroom?
(ii) What is the shape of the showering area SHWR in Reiaan's bathroom? Write the coordinates of the four corners.
(iii) Mark off a $3\text{ ft} \times 2\text{ ft}$ space for the washbasin and a $2\text{ ft} \times 3\text{ ft}$ space for the toilet. Write the coordinates of the corners.

Solution:

(i) Bathroom corners:
From Fig 1.5, the bathroom extends left from the origin $O(0,0)$ to $x = -6$, and vertically up to $y = 9$.
$O(0,0), F(0,9), R(-6,9), P(-6,0)$

(ii) Showering area shape and corners:
The showering area is bounded by vertices $S(-6,6), H(-2,6), W(-2,9), R(-6,9)$.
Since $SR = 3$, $HW = 3$, $SH = 4$, $RW = 4$, it forms a rectangle.
Shape: Rectangle | Vertices: $S(-6,6), H(-2,6), W(-2,9), R(-6,9)$

3. END-OF-CHAPTER EXERCISES

Question 1: What are the x-coordinate and y-coordinate of the point of intersection of the two axes?

The intersection of the coordinate axes is the origin $O$.

x-coordinate = 0, y-coordinate = 0 $\implies (0, 0)$

Question 2: Point W has x-coordinate equal to 5. Can you predict the coordinates of point H which is on the line through W parallel to the y-axis? Which quadrants can H lie in?

A line parallel to the y-axis passing through $x = 5$ has the equation $x = 5$. Thus, any point H on this line must have an x-coordinate of 5, giving coordinates $(5, y)$.

Since $x > 0$, point H can lie in Quadrant I (where $y > 0$) or Quadrant IV (where $y < 0$).

Question 3: Consider the points $R(3, 0)$, $A(0, -2)$, $M(-5, -2)$ and $P(-5, 2)$. If they are joined in the same order, predict:
(i) Two sides of RAMP that are perpendicular to each other.
(ii) One side of RAMP that is parallel to one of the axes.
(iii) Two points that are mirror images of each other in one axis. Which axis will this be?

(i) Perpendicular sides: $AM$ (horizontal line $y = -2$) and $MP$ (vertical line $x = -5$). Thus, $AM \perp MP$.

(ii) Parallel side: Side $MP$ lies along $x = -5$, which is parallel to the y-axis. Side $AM$ lies along $y = -2$, which is parallel to the x-axis.

(iii) Mirror images: $M(-5, -2)$ and $P(-5, 2)$ have the same x-coordinate and opposite y-coordinates. They are reflection images across the x-axis.

Question 6*: Are the points $M(-3, -4)$, $A(0, 0)$ and $G(6, 8)$ on the same straight line? Suggest a method to check this without plotting and joining the points.

Method 1: Distance Formula (Collinearity check)
$MA = \sqrt{(0 - (-3))^2 + (0 - (-4))^2} = \sqrt{3^2 + 4^2} = 5$
$AG = \sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{6^2 + 8^2} = 10$
$MG = \sqrt{(6 - (-3))^2 + (8 - (-4))^2} = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$

Since $MA + AG = 5 + 10 = 15 = MG$, the points lie on the same straight line.

Yes, M, A, and G are collinear.

Question 7*: Use your method (from Problem 6) to check if the points $R(-5, -1)$, $B(-2, -5)$ and $C(4, -12)$ are on the same straight line.

$RB = \sqrt{(-2 - (-5))^2 + (-5 - (-1))^2} = \sqrt{3^2 + (-4)^2} = 5$
$BC = \sqrt{(4 - (-2))^2 + (-12 - (-5))^2} = \sqrt{6^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85} \approx 9.22$
$RC = \sqrt{(4 - (-5))^2 + (-12 - (-1))^2} = \sqrt{9^2 + (-11)^2} = \sqrt{81 + 121} = \sqrt{202} \approx 14.21$

Since $RB + BC = 5 + 9.22 = 14.22 \neq \sqrt{202}$, the points do not satisfy collinearity exactly.

Points R, B, and C are NOT on the same straight line.

Question 9*: Check whether M is the midpoint of segment ST using $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$:

S	M	T	Is M Midpoint?	Reason
(-3, 0)	(0, 0)	(3, 0)	Yes	$\left(\frac{-3+3}{2}, \frac{0+0}{2}\right) = (0, 0)$
(2, 3)	(3, 4)	(4, 5)	Yes	$\left(\frac{2+4}{2}, \frac{3+5}{2}\right) = (3, 4)$
(0, 0)	(0, 5)	(0, -10)	No	$\left(\frac{0+0}{2}, \frac{0+(-10)}{2}\right) = (0, -5) \neq (0, 5)$
(-8, 7)	(0, -2)	(6, -3)	No	$\left(\frac{-8+6}{2}, \frac{7+(-3)}{2}\right) = (-1, 2) \neq (0, -2)$

Question 10*: Use the connection found to find the coordinates of B given that $M(-7, 1)$ is the midpoint of $A(3, -4)$ and $B(x, y)$.

Using Midpoint Formula:
$\frac{3 + x}{2} = -7 \implies 3 + x = -14 \implies x = -17$
$\frac{-4 + y}{2} = 1 \implies -4 + y = 2 \implies y = 6$

Coordinates of B = (-17, 6)

Question 12*: (i) Given points $A(1, -8)$, $B(-4, 7)$ and $C(-7, -4)$, show that they lie on a circle K centered at origin $O(0, 0)$. What is the radius?
(ii) Check whether $D(-5, 6)$ and $E(0, 9)$ lie inside, on, or outside circle K.

(i) Distance from Origin $O(0,0)$:
$OA = \sqrt{1^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}$
$OB = \sqrt{(-4)^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}$
$OC = \sqrt{(-7)^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}$
Since $OA = OB = OC = \sqrt{65}$, all three points lie on circle K.

Radius = $\sqrt{65} \approx 8.06$ units

(ii) Testing D and E:
$OD = \sqrt{(-5)^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}$. Since $\sqrt{61} < \sqrt{65}$, D lies inside the circle.
$OE = \sqrt{0^2 + 9^2} = 9 = \sqrt{81}$. Since $\sqrt{81} > \sqrt{65}$, E lies outside the circle.