The figure 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 the figure, answer the following questions:
- (i) If D1R1 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 D1?
- (iii) If R1 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 B1 (0, 1.5) and B2 (0, 4) represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?
Step-by-step solution
Idea: The x-coordinate of a point is its distance from the y-axis and the y-coordinate is its distance from the x-axis. The width of a door lying along an axis is the difference of the coordinates of its two ends.
(i) If D1R1 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 end of the door nearer the left wall is D1. In the figure D1 is at the mark 8 on the x-axis, so the door starts 8 ft from the left wall.½ mark
- Both ends D1 and R1 lie on the x-axis (the bottom wall). So the door is 0 ft from the x-axis: it is in the wall along the x-axis.½ mark
(ii) What are the coordinates of D1?
- x-coordinate = distance of D1 from the y-axis = 8.½ mark
- y-coordinate = distance of D1 from the x-axis = 0, since D1 is on the x-axis. So D1 = (8, 0).½ mark
(iii) If R1 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?
- Both ends are on the x-axis, so the width is the difference of the x-coordinates: 11.5 − 8 = 3.5 ft.½ mark
- 3.5 ft is about 107 cm (1 ft ≈ 30.5 cm). Room doors in homes are usually about 2½ to 3 ft (75 to 90 cm) wide, so 3.5 ft is a comfortable, even generous, width.
- A wheelchair is usually about 2 to 2½ ft wide and needs a clear opening of about 3 ft (90 cm). The door is 3.5 ft wide, so yes, a person in a wheelchair can enter easily.½ mark
(iv) If B1 (0, 1.5) and B2 (0, 4) represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door?
- Both ends are on the y-axis, so the width is the difference of the y-coordinates: 4 − 1.5 = 2.5 ft.½ mark
- 2.5 ft < 3.5 ft, so the bathroom door is narrower than the room door, by 3.5 − 2.5 = 1 ft.½ mark
Check: Count grid squares along the bottom wall from D1 to R1: 3 full squares and a half, 3.5 ✓. From B1 up to B2 on the left wall: 2 and a half squares, 2.5 ✓.
Answer to write in the exam
(i)
Scale: 1 unit = 1 ft
D1 is at 8 on the x-axis.
∴ Distance of the door from the left wall = 8 ft
D1, R1 lie on the x-axis ∴ distance of the door from the x-axis = 0 ft
(ii)
x-coordinate of D1 = distance from y-axis = 8
y-coordinate of D1 = distance from x-axis = 0
∴ D1 = (8, 0)
(iii)
Width of door = D1R1 = 11.5 − 8 = 3.5 ft
Usual room door ≈ 2½ to 3 ft; a wheelchair needs ≈ 3 ft clear opening.
∴ 3.5 ft is a comfortable width, and a wheelchair user can enter easily.
(iv)
Width of bathroom door = B1B2 = 4 − 1.5 = 2.5 ft
2.5 ft < 3.5 ft
∴ The bathroom door is narrower than the room door (by 1 ft).
Common mistakes that cost marks
- Writing D1 as (0, 8). The first number is always the distance along the x-axis; D1 is 8 along and 0 up, so (8, 0).
- Giving the door width as 11.5 ft (the coordinate of R1) instead of 11.5 − 8 = 3.5 ft. The door starts at D1, not at O.
- For the bathroom door, subtracting the x-coordinates (0 − 0 = 0). The door runs up the y-axis, so subtract the y-coordinates.
How this can come in the exam
On a room plan drawn to the scale 1 unit = 1 ft, a door in the wall along the x-axis runs from (4.5, 0) to (7.5, 0). The width of the door is
- 12 ft
- 3 ft
- 4.5 ft
- 7.5 ft
Show answer
(B) 3 ft
Width = 7.5 − 4.5 = 3 ft.
On a plan (1 unit = 1 ft), a cupboard has corners (2, 0), (6, 0), (6, 1.5) and (2, 1.5). Find its length and depth, and the floor area it covers.
Show answer
Length = 6 − 2 = 4 ft (½ mark). Depth = 1.5 − 0 = 1.5 ft (½ mark). Area = 4 × 1.5 = 6 sq ft (1 mark).Try one yourself
On the same kind of plan, a door has ends E (0, 6) and G (0, 9.5) on the y-axis. How wide is the door, and how far is its lower end from the x-axis?
Show answer
Width = 9.5 − 6 = 3.5 ft; the lower end E is 6 ft from the x-axis.
More questions like this
- 1. What are the standard widths for a room door? Look around your home and in school.
2. Are the doors in your school suitable for people in wheelchairs? - So far, we have only considered points on the two coordinate axes. What can you say about the coordinates of points that are not on either axes?
- Copy the figure and mark S and Q in your diagram. Mark any point P in Quadrant I and any point R in Quadrant III, and write down their coordinates.
- 1. What is the x-coordinate of a point on the y-axis?
2. Is there a similar generalisation for a point on the x-axis?
3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.
4. If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true? - Place Reiaan’s rectangular study table with three of its feet at the points (8, 9), (11, 9) and (11, 7).