If the bathroom door has a hinge at B1 and opens into the bedroom, will it hit the wardrobe? Are there any changes you would suggest if the door is made wider?
Step-by-step solution
Idea: A door turns about its hinge, so its free edge moves along a circle whose radius is the width of the door. The door hits an object only if the object is closer to the hinge than the door’s width.
- Width of the bathroom door = B1B2 = 4 − 1.5 = 2.5 ft.½ mark
- Hinged at B1 (0, 1.5), the free end starts at B2 (0, 4) and moves on a quarter circle of radius 2.5 ft. Fully open, the door lies along the line y = 1.5 and reaches the point (2.5, 1.5).½ mark
- The wardrobe’s left side W1W4 is the line x = 3 from y = 0 to y = 2. Its nearest point to B1 is (3, 1.5), which is 3 − 0 = 3 ft away.½ mark
- The door reaches only 2.5 ft from the hinge, and 2.5 < 3. So the door will not hit the wardrobe; there is a gap of 3 − 2.5 = 0.5 ft.½ mark
- If the door is made wider: once it is 3 ft or more, its edge reaches the line x = 3 and it will hit the wardrobe (a wheelchair-friendly door needs about 3 ft).½ mark
- Suggestions: hang the door on a hinge at its upper end so it swings away from the wardrobe; or make it open into the bathroom; or use a sliding door; or shift the wardrobe at least 1 ft to the right.½ mark
Check: Distance from B1 (0, 1.5) to the wardrobe corner W4 (3, 2) = √(32 + 0.52) = √9.25 ≈ 3.04 ft, also more than 2.5 ft ✓.
Answer to write in the exam
Width of door = B1B2 = 4 − 1.5 = 2.5 ft
Free end moves on a circle of radius 2.5 ft about B1 (0, 1.5); fully open it reaches (2.5, 1.5).
Nearest point of wardrobe = (3, 1.5); distance from B1 = 3 ft
2.5 ft < 3 ft
∴ The door will not hit the wardrobe (gap 0.5 ft).
A door 3 ft or wider would hit it. Suggest: hinge at the upper end, open into the bathroom, a sliding door, or move the wardrobe right.
Common mistakes that cost marks
- Comparing the door’s width with the wardrobe’s width (4 ft). What matters is the distance from the hinge to the wardrobe.
- Measuring from B2 instead of from the hinge B1. The door turns about the hinge.
- Saying “it will hit” from the picture alone, without comparing 2.5 ft with 3 ft.
How this can come in the exam
A door 3.5 ft wide is hinged at (12, 6) on a wall. A cupboard’s nearest point is (9, 6). When the door swings fully open towards the cupboard, it
- stops 0.5 ft short of the cupboard
- just touches the cupboard
- hits the cupboard
- stops 3 ft short of the cupboard
Show answer
(C) hits the cupboard
The cupboard is 12 − 9 = 3 ft from the hinge, but the door reaches 3.5 ft, so it hits.
In a flat, a kitchen door 2.5 ft wide is hinged at H (5, 0) on the wall along the x-axis (1 unit = 1 ft) and opens into the kitchen. A fridge stands with its nearest corner at F (5, 3).
(i) How far is the fridge from the hinge? (ii) Will the door hit the fridge? (iii) The owner wants a 3.5 ft door with the same hinge. What happens? (iv) Suggest one fix.
Show answer
(i) HF = 3 − 0 = 3 ft (1 mark). (ii) No: 2.5 < 3, gap 0.5 ft (1 mark). (iii) 3.5 > 3, so the wider door would hit the fridge (1 mark). (iv) Move the fridge at least 0.5 ft further, or hinge the door at its other end, or use a sliding door (1 mark).Try one yourself
A door 2 ft wide is hinged at (0, 5) and opens into a room. A table’s nearest point is (1.5, 5). Will the door hit the table? What is the widest door that would just clear it?
Show answer
The table is 1.5 ft from the hinge and the door is 2 ft wide, so yes, it hits. Any door narrower than 1.5 ft would clear it (in practice, move the table).
More questions like this
- Look at Reiaan’s bathroom.
- Other rooms in the house:
- 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?