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Coordinates on a plan · 3 marks

Place Reiaan’s rectangular study table with three of its feet at the points (8, 9), (11, 9) and (11, 7).

  1. (i) Where will the fourth foot of the table be?
  2. (ii) Is this a good spot for the table?
  3. (iii) What is the width of the table? The length? Can you make out the height of the table?
ShowerareaBathroomBedWardrobexy−6−4−2246810122468S₁S₂S₃S₄W₃W₄B₁B₂RSHWFC (0, 10)B (12, 10)A (12, 0)O (0, 0)PW₁W₂D₁R₁
Answer: (i) (8, 7) (ii) Yes: it is clear of the bed, the wardrobe and the door’s swing, with open floor in front for a chair (iii) width 2 ft, length 3 ft; the height cannot be found from a floor plan

Step-by-step solution

Given: The plan of Reiaan’s bathroom and bedroom (see the figure), drawn to the scale 1 unit = 1 ft, with O (0, 0) at the foot of the wall between the two rooms; Three feet of the table at (8, 9), (11, 9) and (11, 7)

Idea: In a rectangle with sides along the grid lines, opposite corners share coordinates: each corner has the x-coordinate of one neighbour and the y-coordinate of the other. Lengths along the grid are differences of coordinates.

ShowerareaBathroomBedWardrobeTablexy−6−4−2246810122468S₁S₂S₃S₄W₃W₄B₁B₂RSHWFC (0, 10)B (12, 10)A (12, 0)O (0, 0)PW₁W₂D₁R₁(8, 9)(11, 7)(8, 7)

(i) Where will the fourth foot of the table be?

  1. (8, 9) and (11, 9) have the same y-coordinate, so they form one side along a horizontal grid line. (11, 9) and (11, 7) have the same x-coordinate, so they form the next side, a vertical one.
  2. The fourth corner is opposite (11, 9). It lies straight below (8, 9), so its x-coordinate is 8, and level with (11, 7), so its y-coordinate is 7.½ mark
  3. Fourth foot = (8, 7).½ mark
(8, 7)

(ii) Is this a good spot for the table?

  1. The table covers x from 8 to 11 and y from 7 to 9. The bed ends at x = 6.5, the wardrobe ends at y = 2 and the room door (in the bottom wall) swings only about 3.5 ft into the room. So the table blocks none of them.½ mark
  2. It is 1 ft from the right wall and 1 ft from the top wall, and below it (from y = 7 down to about y = 3.5) there is plenty of free floor for a chair. So yes, it is a good, quiet corner for a study table.½ mark
Yes. It does not block the bed, the wardrobe or the door, and there is room in front of it for a chair.

(iii) What is the width of the table? The length? Can you make out the height of the table?

  1. Width = 9 − 7 = 2 ft (the shorter side). Length = 11 − 8 = 3 ft (the longer side).½ mark
  2. The height cannot be found: the plan shows only the floor (length and breadth), and height is a third measurement it does not show.½ mark
Width 2 ft, length 3 ft. The height cannot be made out from the floor plan.
(i) The fourth foot is at (8, 7). (ii) Yes, it is a good spot: it is clear of the bed, wardrobe and door, with space for a chair. (iii) Width 2 ft, length 3 ft; the height cannot be found from a floor plan.

Check: Opposite sides: from (8, 7) to (11, 7) is 3 and from (8, 9) to (11, 9) is 3 ✓; from (8, 7) to (8, 9) is 2 and from (11, 7) to (11, 9) is 2 ✓. So the four feet form a 3 × 2 rectangle.

Answer to write in the exam

(i)

(8, 9), (11, 9): same y-coordinate; (11, 9), (11, 7): same x-coordinate

Fourth corner: x = 8 (below (8, 9)), y = 7 (level with (11, 7))

∴ Fourth foot = (8, 7)

(ii)

Table: 8 ≤ x ≤ 11, 7 ≤ y ≤ 9

Bed up to x = 6.5; wardrobe up to y = 2; door swing up to y ≈ 3.5

Free floor in front of the table for a chair.

∴ Yes, it is a good spot.

(iii)

Width = 9 − 7 = 2 ft

Length = 11 − 8 = 3 ft

Height: cannot be found (a floor plan shows only length and breadth).

Common mistakes that cost marks

  • Giving the fourth foot as (11, 9) or (8, 9), which are already given, or as (9, 8) by swapping the coordinates of (8, 9).
  • Saying the width is 11 − 7 = 4 by subtracting an x-coordinate from a y-coordinate. Subtract like from like.
  • Trying to read a height off the plan. A floor plan has no height at all.

How this can come in the exam

MCQ (1 mark)

Three corners of a rectangle are (−2, 1), (4, 1) and (4, 5). The fourth corner is

  1. (−2, 5)
  2. (5, −2)
  3. (4, −2)
  4. (1, 5)
Show answer

(A) (−2, 5)
It lies below/above (−2, 1), so x = −2, and level with (4, 5), so y = 5.

Short answer (2 marks)

A bookshelf on a floor plan (1 unit = 1 ft) has three corners at (0, 6), (2, 6) and (2, 8.5). Find the fourth corner and the floor area the shelf covers.

Show answerFourth corner (0, 8.5) (1 mark). Sides 2 ft and 2.5 ft, area = 2 × 2.5 = 5 sq ft (1 mark).

Try one yourself

A bed has three legs at (1, 5), (7, 5) and (7, 8) on a plan (1 unit = 1 ft). Where is the fourth leg, and what are the length and width of the bed?

Show answer

Fourth leg (1, 8); length 7 − 1 = 6 ft, width 8 − 5 = 3 ft.

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