Playing with Constructions — NCERT Solutions

Given: A rectangle with four squares placed symmetrically around it (one on each side).

Steps on Dot Paper:

1. On the dot paper, choose a rectangle, say 4 dots × 2 dots (i.e., 4 cm × 2 cm).
2. Draw rectangle ABCD with AB = CD = 4 units and BC = AD = 2 units.
3. On side AB (length 4 units): Draw a square of side 4 units outward from AB.
4. On side CD (length 4 units): Draw a square of side 4 units outward from CD.
5. On side BC (length 2 units): Draw a square of side 2 units outward from BC.
6. On side AD (length 2 units): Draw a square of side 2 units outward from AD.

For symmetry:

• The squares on opposite sides of the rectangle are equal (both squares on the longer sides are identical; both squares on the shorter sides are identical).
• Each square is drawn outward, centred on its respective side.
• The figure has two lines of symmetry — one along the length and one along the width of the rectangle.

Discussion point: The four squares are placed symmetrically because opposite sides of a rectangle are equal, so the squares on opposite sides are congruent and mirror images of each other.

Given: Four figures A, B, C, D drawn on a dot grid.

Properties of a Square:

• All four sides are equal.
• All four angles are right angles (90°).

Analysis using dot grid:

Figure A: Appears to be an upright quadrilateral. Count the dots to measure sides. If all sides are equal and corners align with the grid at right angles → it is a square.

Figure B: Appears tilted/rotated. To check: use the dot positions. A tilted square on a dot grid can be verified by checking that all sides are equal (using the distance formula or counting diagonal steps) and that adjacent sides are perpendicular. If it satisfies both → it is a square.

Figure C: Appears to be a rectangle (sides unequal). Measure: if length ≠ width → not a square (it is a rectangle).

Figure D: Appears to be a tilted quadrilateral. Measure all sides; if they are not all equal or angles are not 90° → not a square.

Conclusion (based on typical figures in this exercise):

• Figure A — Square ✓ (all sides equal, all angles 90°)
• Figure B — Square ✓ (rotated square, all sides equal, all angles 90°)
• Figure C — Not a square (rectangle with unequal adjacent sides)
• Figure D — Not a square (sides unequal or angles not 90°)

Think (answer): Yes, on a dot grid we can reason without measuring instruments. If a shape's corners lie on dots, we can count the horizontal and vertical gaps to find side lengths and check perpendicularity by seeing if adjacent sides go in directions that are perpendicular (e.g., one side goes 2 right–1 up, the perpendicular direction is 1 right–2 down). This uses the property that two directions $(a, b)$ and $(-b, a)$ are perpendicular.

Activity-based construction question.

Concept: A rotated square/rectangle on a dot grid has corners on lattice points (dots). We use the fact that if one side of a square goes from point $(0,0)$ to $(a,b)$, the next side (perpendicular and equal) goes from $(a,b)$ to $(a-b,\, b+a)$.

Example 1 — Rotated Square:

• Start at dot $O(0,0)$. Move 2 right, 1 up → reach $A(2,1)$.
• From $A$, move 1 left, 2 up (perpendicular direction) → reach $B(1,3)$.
• From $B$, move 2 left, 1 down → reach $C(-1,2)$.
• From $C$, move 1 right, 2 down → back to $O(0,0)$.
• Side length = $\sqrt{2^2+1^2} = \sqrt{5}$ for all sides. ✓ All sides equal.
• Adjacent sides are perpendicular (directions $(2,1)$ and $(-1,2)$: dot product $= 2\times(-1)+1\times2 = 0$). ✓
• Verified: It is a square.

Example 2 — Rotated Square:

• Use step $(1,1)$: Start $(0,0)\to(1,1)\to(0,2)\to(-1,1)\to(0,0)$.
• Side = $\sqrt{2}$, all equal, all angles 90°. ✓

Example 3 — Rotated Rectangle:

• One side: move 3 right, 1 up → length $\sqrt{10}$.
• Adjacent side (perpendicular, different length): move 1 left, 3 up → length $\sqrt{10}$... (adjust to get unequal sides).
• Use one side $(2,1)$ (length $\sqrt{5}$) and adjacent side $(−1,2)$ scaled differently, e.g., $(−2,4)$ (length $\sqrt{20}$).
• Verify: dot product of $(2,1)$ and $(-2,4)$ $= -4+4=0$ ✓ (perpendicular). Opposite sides equal ✓.
• Verified: It is a rectangle.

General Verification Steps for any drawn figure:

1. Measure all four sides using the distance formula $d = \sqrt{(\Delta x)^2+(\Delta y)^2}$ or by counting dot steps.
2. Check opposite sides are equal.
3. Check adjacent sides are perpendicular (dot product = 0, or use set-square).
4. For a square: all 4 sides equal + all angles 90°.
5. For a rectangle: opposite sides equal + all angles 90°.