NCERT Solutions Class 9 Maths Exercise 1.1 — Orienting Yourself: The Use of Coordinates

NCERT Class 9 Mathematics — Ganita Manjari Part I, Orienting Yourself: The Use of Coordinates, Exercise Set 1.1. All four parts solved.

This is the first exercise of the new Class 9 course, and it is not really about calculation. It is about reading a plan. Fig. 1.3 is a floor plan of Reiaan’s room with a coordinate grid laid over it, and every part of the question is answered by reading a coordinate off that grid and then doing one subtraction.

Two conventions carry the whole exercise:

• A point $P(x, y)$ has $x$ measured from the $y$-axis and $y$ measured from the $x$-axis. So a point’s distance from the $y$-axis is its $x$-coordinate, and its distance from the $x$-axis is its $y$-coordinate.
• Two points on the same horizontal line are $|x_2 – x_1|$ apart; two on the same vertical line are $|y_2 – y_1|$ apart.

The room is the rectangle $OABC$ with $O(0, 0)$, $A(12, 0)$, $B(12, 10)$ and $C(0, 10)$, so one unit on the grid is one foot, and the room is $12$ ft by $10$ ft.

Key insight. The door $D_1R_1$ lies along the bottom wall, which is the $x$-axis. Both its ends therefore have $y = 0$, and the only number that varies along it is $x$. That single observation answers every part: the door’s distance from the $x$-axis is zero, its distance from the $y$-axis is an $x$-coordinate, and its width is a difference of two $x$-coordinates.

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.

Solution. Read the marked points off the figure first. Along the bottom wall the figure marks $W_1(3, 0)$ and $W_2(7, 0)$ at the foot of the wardrobe, then $D_1$ on the $+8$ tick, then $R_1$, labelled $(11.5, 0)$, and finally the corner $A(12, 0)$. On the left wall it marks $B_1$ and $B_2$, the two ends of the bathroom door, and the question supplies their coordinates.

(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?

The left wall is the $y$-axis, so “how far from the left wall” is asking for an $x$-coordinate. The door stretches from $D_1(8, 0)$ to $R_1(11.5, 0)$, so its nearer edge is $8$ ft from the left wall and its far edge is $11.5$ ft from it.

The second half of the question is the one worth pausing on. Both ends of the door have $y = 0$, which means the door lies on the $x$-axis — it is part of the bottom wall. Its distance from the $x$-axis is therefore not a small number; it is exactly zero.

(ii) What are the coordinates of $D_1$?

$D_1$ is the end of the door nearer the wardrobe. It sits on the bottom wall, so its $y$-coordinate is $0$, and the figure places it on the $+8$ mark of the $x$-axis.

(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?

$D_1$ and $R_1$ both lie on the $x$-axis, so the distance between them is simply the difference of their $x$-coordinates:

$$D_1R_1 = 11.5 – 8 = 3.5 \text{ ft}$$

That is about $107$ cm. A door of that width is generous — most room doors are around $3$ ft — and it is comfortably wider than the $80$ to $90$ cm of clear opening a wheelchair needs. So yes, someone using a wheelchair can enter 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?

$B_1$ and $B_2$ both have $x = 0$, so they lie on the $y$-axis — the left wall — and the distance between them is the difference of their $y$-coordinates:

$$B_1B_2 = 4 – 1.5 = 2.5 \text{ ft}$$

Compare that with the $3.5$ ft found in part (iii).

Common mistakes

• Part (i), answering “$8$ ft” for the distance from the $x$-axis. This confuses the two questions being asked. The distance from the $y$-axis is an $x$-coordinate; the distance from the $x$-axis is a $y$-coordinate. Here every point of the door has $y = 0$, so the answer is $0$ — the door is part of the bottom wall, not set away from it.
• Part (ii), reading $D_1$ as $(8, 0.5)$ or similar. The label $D_1$ is printed below the axis in the figure, next to $W_2$, which makes it look as though the point sits off the line. It does not: the red dot is on the axis, at $x = 8$.
• Part (iii), measuring from the wrong end. The door is $D_1R_1$, not $OR_1$. Using $11.5 – 0$ gives $11.5$ ft, which is nearly the whole width of the room.
• Part (iii), subtracting in the wrong order. Writing $8 – 11.5 = -3.5$ and reporting a negative width. A distance is $|x_2 – x_1|$; always take the difference as a positive number.
• Part (iv), taking $B_1$ and $B_2$ off the figure instead of the question. In Fig. 1.3 the label $B_1$ sits between the $+1$ and $+2$ ticks and is easy to read as $(0, 1)$, which would give a width of $3$ ft. The question states $B_1(0, 1.5)$, and the printed number always beats an estimate from the drawing.
• Part (iv), answering “narrower” and stopping. The question invites a comparison, so say by how much: $3.5 – 2.5 = 1$ ft. A bare “narrower” leaves the arithmetic undone.

Practise next

• Exercise Set 1.2 — the same floor plan, extended to the left of the $y$-axis to take in the bathroom. It asks for coordinates in the second and third quadrants, where the signs start to matter, and it is the natural continuation of this exercise.
• The End-of-Chapter Exercises, Question 1 and Question 2 — both turn on the same idea used in part (i): what a coordinate of zero tells you about where a point lies.