Other rooms in the house:
- (i) Reiaan’s room door leads from the dining room which has the length 18 ft and width 15 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 ft × 3 ft dining table precisely in the centre of the dining room. Write down the coordinates of the feet of the table.
Step-by-step solution
Idea: The room door is in the wall along the x-axis, so the dining room is on the other side of that wall, below the x-axis. The centre of a rectangle is halfway across and halfway down; the table is placed symmetrically about it.
(i) Reiaan’s room door leads from the dining room which has the length 18 ft and width 15 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.
- P = (−6, 0) and A = (12, 0), so PA = 12 − (−6) = 18 ft: exactly the length of the dining room. So PA is one long wall.½ mark
- The door D1R1 is in the wall PA (the x-axis), and it leads from the dining room into Reiaan’s room. So the dining room is on the other side of this wall: below the x-axis.½ mark
- Its width is 15 ft, so the far wall is at y = 0 − 15 = −15. The other two corners are straight below A and P: (12, −15) and (−6, −15).1 mark
(ii) Place a rectangular 5 ft × 3 ft dining table precisely in the centre of the dining room. Write down the coordinates of the feet of the table.
- Centre of the dining room: halfway across, x = (−6 + 12) ÷ 2 = 3; halfway down, y = (0 + (−15)) ÷ 2 = −7.5. Centre = (3, −7.5).1 mark
- Put the 5 ft side along the room’s length: the table reaches 2.5 ft either side of the centre, x from 3 − 2.5 = 0.5 to 3 + 2.5 = 5.5. The 3 ft side reaches 1.5 ft either side: y from −7.5 + 1.5 = −6 to −7.5 − 1.5 = −9.½ mark
- Taking the feet at the four corners of the table top: (0.5, −6), (5.5, −6), (5.5, −9), (0.5, −9).½ mark
- (If the table is turned the other way, with the 5 ft side across the room, the feet are (1.5, −5), (4.5, −5), (4.5, −10), (1.5, −10). Both are correct.)
Check: Table: 5.5 − 0.5 = 5 ft and −6 − (−9) = 3 ft ✓. Its centre: ((0.5 + 5.5) ÷ 2, (−6 + (−9)) ÷ 2) = (3, −7.5), the room’s centre ✓.
Answer to write in the exam
(i)
PA = 12 − (−6) = 18 ft = length of dining room
Door D1R1 is on the x-axis ⇒ dining room lies below the x-axis
Width 15 ft ⇒ far wall at y = −15
∴ Corners: P (−6, 0), A (12, 0), (12, −15), (−6, −15)
(ii)
Centre = ((−6 + 12)/2, (0 − 15)/2) = (3, −7.5)
5 ft side along the length: x = 3 ± 2.5 ⇒ 0.5 to 5.5
3 ft side: y = −7.5 ± 1.5 ⇒ −9 to −6
∴ Feet: (0.5, −6), (5.5, −6), (5.5, −9), (0.5, −9)
Common mistakes that cost marks
- Drawing the dining room above the x-axis, on top of the bedroom. The door in the x-axis wall leads out of the bedroom, so the dining room is below it.
- Taking the centre as (3, 7.5). Below the x-axis the y-coordinate is negative: (3, −7.5).
- Adding the full 5 ft on each side of the centre (making a 10 ft table). Each side of the centre gets half: 2.5 ft and 1.5 ft.
How this can come in the exam
The centre of the rectangle with corners (−4, −2), (10, −2), (10, −8) and (−4, −8) is
- (3, −5)
- (7, −5)
- (3, −3)
- (14, −10)
Show answer
(A) (3, −5)
x = (−4 + 10) ÷ 2 = 3 and y = (−2 + (−8)) ÷ 2 = −5.
A hall has corners (0, 0), (20, 0), (20, −12) and (0, −12) (1 unit = 1 ft). A 6 ft × 4 ft stage is placed exactly at its centre with the 6 ft side along the x-axis direction. Find the corners of the stage.
Show answer
Centre (10, −6) (½ mark). x: 10 ± 3 → 7 to 13; y: −6 ± 2 → −8 to −4 (½ mark). Corners (7, −4), (13, −4), (13, −8), (7, −8) (1 mark).Try one yourself
A garden 16 m × 10 m has corners (−8, 0), (8, 0), (8, −10) and (−8, −10). A 4 m × 2 m pond is dug at its centre, with the 4 m side along the length. Find the corners of the pond.
Show answer
Centre (0, −5). Corners (−2, −4), (2, −4), (2, −6), (−2, −6).
More questions like this
- Look at triangle ADM in the figure. Triangle ADM is an acute angled triangle in the first quadrant. How do we find the lengths of its sides AD, DM and MA?
- 1. In moving from A (3, 4) to D (7, 1), what distance has been covered along the x-axis? What about the distance along the y-axis?
2. Can these distances help you find the distance AD? - What if, x1, x2, y1, y2 take negative values? In the figure, triangle AMD is reflected in the y-axis. What are the coordinates of the images of points A, M, and D?
- 1. What has remained the same and what has changed with this reflection?
2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)? - What are the x-coordinate and y-coordinate of the point of intersection of the two axes?