Reiaan's Whole House: Exercise Set 1.2 (Part 1) — Class 9 Mathematics Chapter 1

🖼 Image PendingA top-down architect's view of Reiaan's whole apartment — bedroom on the right, bathroom on the left in the negative-x region, with a faint Cartesian grid overlaid and dawn light pouring through invisible windows

AI Generation Prompt

Ultra-wide cinematic banner (16:5 ratio). A top-down architectural view of Reiaan's whole apartment — bedroom on the right side of the frame, bathroom (with showering area, toilet, washbasin) on the left side, both rendered in warm wood and cool tile colours. A faint Cartesian coordinate grid is overlaid on the entire space, with the origin at the meeting corner of the two rooms. Soft dawn light streams across the floor. The image conveys: an architect's plan is a coordinate plane in disguise. Painterly cinematic illustration with subtle technical overlay. Dark background. No text, no labels.

✦Think About It

Real architects and designers do their work on coordinate paper. Every door, every window, every piece of furniture is a set of numbers before it is anything else.

Now that you have learned about all four quadrants, a whole house opens up. Reiaan's bathroom is to the left of the bedroom — in the negative-x region. The same room, the same coordinates, just a wider Cartesian plane.

If the bathroom door swings into the bedroom, how do you check whether it will hit the wardrobe?

A door is a line segment with one fixed end (the hinge) and one moving end (the free end). As it opens, the free end traces a circular arc. The question becomes: does that arc cross the wardrobe's rectangle?

✦सूत्रात्मक सिद्धांत

From the Indian Mathematical Tradition — On Careful Surveying

प्रत्येकं स्थानं संख्यया वेदितव्यम्।

एकस्य अपि क्षेपः सर्वम् असत्यं करोति॥

(pratyekaṃ sthānaṃ saṃkhyayā veditavyam / ekasya api kṣepaḥ sarvam asatyaṃ karoti)

'हर जगह को संख्या से जानना चाहिए। एक भी ग़लती सब कुछ बिगाड़ देती है।'

'Every position must be known by its number. Even one error throws everything off.'

Indian survey manuals from the Sulba-sutra tradition warned that one miscounted unit could ruin an entire layout. The same is true today: an architect who mis-places one corner of a wardrobe by 30 cm may design a door that does not open. Coordinate geometry is the discipline of being right about every single number.

Reiaan's apartment — the whole grid

On the previous pages we worked only with the bedroom — the rectangle from $(0, 0)$ to $(12, 10)$ . The full apartment, shown below, adds a bathroom in the negative-x region: the rectangle from $(-6, 0)$ to $(0, 9)$ , attached to the bedroom along the y-axis.

Key labelled points (units: 1 grid step = 1 foot):

$O = (0, 0)$ — corner where bedroom meets bathroom (origin)
$A = (12, 0)$ , $B = (12, 10)$ , $C = (0, 10)$ — three other bedroom corners
$W_1, W_2, W_3, W_4$ — corners of the wardrobe
$S_1, S_2, S_3, S_4$ — corners of the bed
$B_1 = (0, 1.5)$ and $B_2 = (0, 4)$ — endpoints of the bathroom door (a vertical segment on the y-axis)
$D_1, R_1$ — endpoints of the room (entrance) door, on the bottom wall
$F$ — top-left corner of the bathroom; $H$ — left wall corner; $P, R$ — far corners of the bathroom in the negative-x region

Using this single coordinate plane, we can now answer NCERT Exercise Set 1.2 Questions 1 and 2 with no measuring tape — just arithmetic.

🖼 Image PendingA top-down floor plan of an apartment drawn on a Cartesian grid. The bedroom is a 12 ft × 10 ft rectangle in the right half (positive x), with the y-axis as its left wall. Inside the bedroom: a bed labelled with corners S1, S2, S3, S4 against the upper-left wall; a wardrobe (4 ft × 2 ft) on the lower-middle wall with corners W1, W2, W3, W4. The room door has endpoints D1 and R1 on the bottom wall. The bathroom is a 6 ft × 9 ft rectangle in the left half (negative x), with the y-axis as its right wall. Inside the bathroom: a showering area, a toilet, a washbasin, and corner labels F, H, R, P. Two bathroom door endpoints B1 = (0, 1.5) and B2 = (0, 4) are marked on the y-axis itself. A potted plant sits in the upper-right of the bedroom.

AI Generation Prompt

Top-down architectural floor plan of an apartment drawn on a Cartesian coordinate grid. Two rooms share the y-axis as a wall: the bedroom occupies the rectangle from (0, 0) to (12, 10) on the right (warm beige); the bathroom occupies the rectangle from (-6, 0) to (0, 9) on the left (cool blue tile). Origin O = (0, 0) labelled at the bedroom-bathroom corner. Other corners labelled: A = (12, 0) bottom-right, B = (12, 10) top-right, C = (0, 10) top-left of bedroom; F = (0, 9) top of shared wall; bathroom corners labelled R, P. Bedroom interior: a bed in the upper-left with corners S1, S2, S3, S4; a wardrobe (4 ft × 2 ft) on the lower wall with corners W1, W2, W3, W4. Bathroom interior: a showering area in the upper-left (with shower head and SHWR label); a toilet in the lower-left; a washbasin labelled. Two bathroom door endpoints labelled B1 (lower) and B2 (upper) on the y-axis itself. Tick marks at every integer from −6 to +12 on x-axis, +1 to +10 on y-axis. Style: clean architectural illustration. Dark background, orange accent labels, clean technical illustration style.

📸 Fig. 1.5 — Reiaan's whole apartment. Bedroom in $x \geq 0$, bathroom in $x \leq 0$. Origin at the bedroom-bathroom corner. Each grid step is 1 foot.

Question 1: Where does the study table go?

NCERT Exercise Set 1.2, Q1. Place Reiaan's rectangular study table with three of its feet at the points $(8, 9)$ , $(11, 9)$ and $(11, 7)$ . Answer the three sub-questions below.

Q1 (i)Where will the fourth foot be?

NCERT

Three feet of a rectangular table are at $(8, 9)$ , $(11, 9)$ , $(11, 7)$ . Where is the fourth foot?

Solution

Step 1 — Identify the rectangle.

A rectangle has four corners; opposite sides are equal and parallel to each other. Let me name the three given corners $T_1(8, 9)$ , $T_2(11, 9)$ , $T_3(11, 7)$ .

Step 2 — Spot the right-angle.

$T_1 \to T_2$ : same y-coordinate (9), so this side is horizontal of length $|11 - 8| = 3$ .
$T_2 \to T_3$ : same x-coordinate (11), so this side is vertical of length $|9 - 7| = 2$ .

So the right angle of the rectangle is at $T_2(11, 9)$ . The remaining corner $T_4$ must be diagonally opposite to it.

Step 3 — Place the fourth corner.

The fourth corner sits below $T_1$ and to the left of $T_3$ . Its x-coordinate matches $T_1$ 's x ( $x = 8$ ); its y-coordinate matches $T_3$ 's y ( $y = 7$ ). So:

$T_4 = (8, 7)$

Answer: The fourth foot is at $(8, 7)$ .

Q1 (ii)Is this a good spot for the table?

NCERT

Is the table at $(8, 9), (11, 9), (11, 7), (8, 7)$ in a good spot inside Reiaan's bedroom?

Solution

Step 1 — Check the bounds.

The bedroom occupies $0 \le x \le 12$ and $0 \le y \le 10$ . All four feet are within these bounds, so the table fits.

Step 2 — Check for collisions with other furniture.

Bed corners (from Fig. 1.5): roughly $S_1(1, 5)$ to $S_3(7, 9)$ . The bed's region is $x \in [1, 7]$ , $y \in [5, 9]$ . Our table's region is $x \in [8, 11]$ , $y \in [7, 9]$ . The two regions do not overlap (the table starts at $x = 8$ , beyond the bed's right edge at $x = 7$ ). ✓
Wardrobe corners (from Fig. 1.5): $W_1(3, 1)$ to $W_3(7, 3)$ . The wardrobe is far below the table and far to its left. ✓

Step 3 — Practical reasoning.

The table sits in the upper-right corner of the bedroom, against the right wall and just below the back wall. This is a typical workspace placement — close to the window (top-right), away from the door (bottom-right), enough room behind the chair. Yes, this is a sensible spot.

Answer: Yes, the table fits inside the bedroom and does not collide with the bed or wardrobe — it is a reasonable spot.

Q1 (iii)Width? Length? Height?

NCERT

What is the width of the table? The length? Can you make out the height of the table from the floor plan?

Solution

Step 1 — Width and length from coordinates.

Using the corners $(8, 7), (11, 7), (11, 9), (8, 9)$ :

Horizontal sides (along x): length $= |11 - 8| = 3$ feet.
Vertical sides (along y): length $= |9 - 7| = 2$ feet.

By convention, the longer side is the length and the shorter side is the width. So:

Length = 3 feet (≈ 91 cm)
Width = 2 feet (≈ 61 cm)

Step 2 — Height?

The floor plan is a 2-D coordinate map. It captures only east-west and north-south position. Height is the third dimension (up from the floor) — it cannot be read off a 2-D plan. To know the table's height, we would need either a side-view elevation drawing or a 3-D coordinate $(x, y, z)$ for each foot.

Answer: Length = 3 ft, Width = 2 ft. The height cannot be determined from a 2-D floor plan.

Question 2: Will the bathroom door hit the wardrobe?

NCERT Exercise Set 1.2, Q2. The bathroom door is hinged at $B_1 = (0, 1.5)$ and opens into the bedroom. The door's free end (initially at $B_2 = (0, 4)$ when closed) sweeps through an arc as it opens. Will it hit the wardrobe? And what would you suggest if the door were made wider?

This is a real architectural question — and now you have the tools to settle it without ever stepping into the room.

Loading simulator…

Q2Door-swing analysis

NCERT

The bathroom door is a segment of length $|B_2 - B_1| = |4 - 1.5| = 2.5$ ft, hinged at $B_1(0, 1.5)$ . When fully open into the bedroom, the door points along the positive x-axis from $B_1$ . Will the door's swept arc hit the wardrobe (corners $W_1(3, 1), W_2(7, 1), W_3(7, 3), W_4(3, 3)$ )?

Solution

Step 1 — How far can the door reach?

The door is rigid and 2.5 ft long. As it pivots around $B_1(0, 1.5)$ , every point on the door stays within 2.5 ft of $B_1$ . So the door's swept region is a circular sector of radius 2.5 ft centred at $B_1$ .

Step 2 — How close is the wardrobe to $B_1$ ?

The wardrobe occupies $x \in [3, 7]$ and $y \in [1, 3]$ . The closest corner to $B_1(0, 1.5)$ is $W_1(3, 1)$ :

$\text{distance}(B_1, W_1) = \sqrt{(3 - 0)^2 + (1 - 1.5)^2} = \sqrt{9 + 0.25} = \sqrt{9.25} \approx 3.04 \text{ ft}$

(You will study this distance formula formally on the next two pages.)

Step 3 — Compare door reach to wardrobe distance.

Door reach = 2.5 ft. Closest wardrobe corner = ≈ 3.04 ft away.

Since $2.5 < 3.04$ , the door does not reach the wardrobe at any swing angle. The arc of swept points stays entirely inside the disk of radius 2.5 ft around $B_1$ , and the wardrobe lies outside this disk.

Step 4 — Suggestion if the door were widened.

If the door is made wider — say $w$ ft — it will start hitting the wardrobe as soon as $w \ge 3.04$ ft (≈ 93 cm). The cleanest fix is to move the hinge upward (closer to $B_2$ ) so that the door's swept disk shifts away from the wardrobe.

Answer: No, the 2.5-ft door does not hit the wardrobe at any swing angle. If the door were widened past about 3 ft, the hinge would need to be raised to keep the swept arc clear.

Spatial ReasoningLevel 4 · Evaluation

An architect designs a wider bathroom door — width 4 ft — hinged at the same point $B_1(0, 1.5)$ . The wardrobe stays at the same position. The architect insists, 'The door is fine. It only hits the wardrobe at one specific angle, so we just won't open it that far.' Why is this argument unsound?

Bridging Science and Society — Architectural CAD

Modern architectural CAD software (such as AutoCAD, Revit, or India's BharatCAD) automatically performs the door-swing collision check we just did by hand. Every door object in the software carries a swept-region polygon, and the software flags any collision with another object in the room as soon as a piece of furniture is dragged within range.

Practice Yourself — More Coordinate Geometry of the House

Try these on graph paper before peeking at hints:

The corners of the bed are $S_1(1, 5)$ , $S_2(7, 5)$ , $S_3(7, 9)$ , $S_4(1, 9)$ . Find its length and width.
A new bookshelf is to be placed with corners at $(9, 1), (11, 1), (11, 4), (9, 4)$ . Will it overlap with the wardrobe (whose corners are $W_1(3, 1), W_2(7, 1), W_3(7, 3), W_4(3, 3)$ )?
Reiaan's school chair is placed at $(9, 5), (10, 5), (10, 6), (9, 6)$ . What is the chair's footprint area?
The diagonal of the bed connects $S_1(1, 5)$ to $S_3(7, 9)$ . How long is this diagonal? (Hint: this needs the distance formula — you'll meet it on the next page.)
A new entry mat lies along the bottom wall from $(5, 0)$ to $(8, 0)$ . What part of the room door $D_1 R_1$ does the mat cover?

Worked answers: 1. length = 4 ft, width = 6 ft. 2. Both occupy the strip $y \in [1, 3]$ but the bookshelf is at $x \in [9, 11]$ while the wardrobe is at $x \in [3, 7]$ — no overlap. 3. 1 sq ft. 4. $\sqrt{(7-1)^2 + (9-5)^2} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21$ ft. 5. The room door covers $x \in [7.5, 11.5]$ and the mat covers $[5, 8]$ , so the overlap is $x \in [7.5, 8]$ — half a foot.

Quick Check

Q1.Three corners of a rectangle are $(2, 1)$ , $(7, 1)$ , and $(7, 5)$ . Where is the fourth corner?

AI Generation Prompt

✦Think About It

Real architects and designers do their work on coordinate paper. Every door, every window, every piece of furniture is a set of numbers before it is anything else.

If the bathroom door swings into the bedroom, how do you check whether it will hit the wardrobe?

✦सूत्रात्मक सिद्धांत

From the Indian Mathematical Tradition — On Careful Surveying

प्रत्येकं स्थानं संख्यया वेदितव्यम्।

एकस्य अपि क्षेपः सर्वम् असत्यं करोति॥

(pratyekaṃ sthānaṃ saṃkhyayā veditavyam / ekasya api kṣepaḥ sarvam asatyaṃ karoti)

'हर जगह को संख्या से जानना चाहिए। एक भी ग़लती सब कुछ बिगाड़ देती है।'

'Every position must be known by its number. Even one error throws everything off.'

Reiaan's apartment — the whole grid

Key labelled points (units: 1 grid step = 1 foot):

$O = (0, 0)$ — corner where bedroom meets bathroom (origin)
$A = (12, 0)$ , $B = (12, 10)$ , $C = (0, 10)$ — three other bedroom corners
$W_1, W_2, W_3, W_4$ — corners of the wardrobe
$S_1, S_2, S_3, S_4$ — corners of the bed
$B_1 = (0, 1.5)$ and $B_2 = (0, 4)$ — endpoints of the bathroom door (a vertical segment on the y-axis)
$D_1, R_1$ — endpoints of the room (entrance) door, on the bottom wall
$F$ — top-left corner of the bathroom; $H$ — left wall corner; $P, R$ — far corners of the bathroom in the negative-x region

Using this single coordinate plane, we can now answer NCERT Exercise Set 1.2 Questions 1 and 2 with no measuring tape — just arithmetic.

AI Generation Prompt

📸 Fig. 1.5 — Reiaan's whole apartment. Bedroom in $x \geq 0$, bathroom in $x \leq 0$. Origin at the bedroom-bathroom corner. Each grid step is 1 foot.

Question 1: Where does the study table go?

NCERT Exercise Set 1.2, Q1. Place Reiaan's rectangular study table with three of its feet at the points $(8, 9)$ , $(11, 9)$ and $(11, 7)$ . Answer the three sub-questions below.

Q1 (i)Where will the fourth foot be?

NCERT

Three feet of a rectangular table are at $(8, 9)$ , $(11, 9)$ , $(11, 7)$ . Where is the fourth foot?

Solution

Step 1 — Identify the rectangle.

A rectangle has four corners; opposite sides are equal and parallel to each other. Let me name the three given corners $T_1(8, 9)$ , $T_2(11, 9)$ , $T_3(11, 7)$ .

Step 2 — Spot the right-angle.

$T_1 \to T_2$ : same y-coordinate (9), so this side is horizontal of length $|11 - 8| = 3$ .
$T_2 \to T_3$ : same x-coordinate (11), so this side is vertical of length $|9 - 7| = 2$ .

So the right angle of the rectangle is at $T_2(11, 9)$ . The remaining corner $T_4$ must be diagonally opposite to it.

Step 3 — Place the fourth corner.

The fourth corner sits below $T_1$ and to the left of $T_3$ . Its x-coordinate matches $T_1$ 's x ( $x = 8$ ); its y-coordinate matches $T_3$ 's y ( $y = 7$ ). So:

$T_4 = (8, 7)$

Answer: The fourth foot is at $(8, 7)$ .

Q1 (ii)Is this a good spot for the table?

NCERT

Is the table at $(8, 9), (11, 9), (11, 7), (8, 7)$ in a good spot inside Reiaan's bedroom?

Solution

Step 1 — Check the bounds.

The bedroom occupies $0 \le x \le 12$ and $0 \le y \le 10$ . All four feet are within these bounds, so the table fits.

Step 2 — Check for collisions with other furniture.

Bed corners (from Fig. 1.5): roughly $S_1(1, 5)$ to $S_3(7, 9)$ . The bed's region is $x \in [1, 7]$ , $y \in [5, 9]$ . Our table's region is $x \in [8, 11]$ , $y \in [7, 9]$ . The two regions do not overlap (the table starts at $x = 8$ , beyond the bed's right edge at $x = 7$ ). ✓
Wardrobe corners (from Fig. 1.5): $W_1(3, 1)$ to $W_3(7, 3)$ . The wardrobe is far below the table and far to its left. ✓

Step 3 — Practical reasoning.

Answer: Yes, the table fits inside the bedroom and does not collide with the bed or wardrobe — it is a reasonable spot.

Q1 (iii)Width? Length? Height?

NCERT

What is the width of the table? The length? Can you make out the height of the table from the floor plan?

Solution

Step 1 — Width and length from coordinates.

Using the corners $(8, 7), (11, 7), (11, 9), (8, 9)$ :

Horizontal sides (along x): length $= |11 - 8| = 3$ feet.
Vertical sides (along y): length $= |9 - 7| = 2$ feet.

By convention, the longer side is the length and the shorter side is the width. So:

Length = 3 feet (≈ 91 cm)
Width = 2 feet (≈ 61 cm)

Step 2 — Height?

Answer: Length = 3 ft, Width = 2 ft. The height cannot be determined from a 2-D floor plan.

Question 2: Will the bathroom door hit the wardrobe?

This is a real architectural question — and now you have the tools to settle it without ever stepping into the room.

Loading simulator…

Q2Door-swing analysis

NCERT

Solution

Step 1 — How far can the door reach?

Step 2 — How close is the wardrobe to $B_1$ ?

The wardrobe occupies $x \in [3, 7]$ and $y \in [1, 3]$ . The closest corner to $B_1(0, 1.5)$ is $W_1(3, 1)$ :

$\text{distance}(B_1, W_1) = \sqrt{(3 - 0)^2 + (1 - 1.5)^2} = \sqrt{9 + 0.25} = \sqrt{9.25} \approx 3.04 \text{ ft}$

(You will study this distance formula formally on the next two pages.)

Step 3 — Compare door reach to wardrobe distance.

Door reach = 2.5 ft. Closest wardrobe corner = ≈ 3.04 ft away.

Step 4 — Suggestion if the door were widened.

Answer: No, the 2.5-ft door does not hit the wardrobe at any swing angle. If the door were widened past about 3 ft, the hinge would need to be raised to keep the swept arc clear.

Spatial ReasoningLevel 4 · Evaluation

Bridging Science and Society — Architectural CAD

Quick Check

Q1.Three corners of a rectangle are $(2, 1)$ , $(7, 1)$ , and $(7, 5)$ . Where is the fourth corner?