Playing with Constructions

Given: The central line AB = 8 cm, and the wave is made of half circles placed along AB.

Concept: A half circle (semicircle) is drawn with its diameter lying on the central line. If the total length AB = 8 cm is divided equally into two half circles, each half circle has a diameter of 4 cm.

Working:
- Each half circle sits on a portion of AB as its diameter.
- Diameter of each half circle = $\frac{8}{2} = 4$ cm
- Radius = $\frac{\text{Diameter}}{2} = \frac{4}{2} = 2$ cm

Answer:
- The radius to be taken in the compass = 2 cm
- The length of AX (the diameter of the first half circle) = 4 cm

Activity-based question — Sample with central line = 12 cm

Given: Let the central line AB = 12 cm.

Steps of Construction:
1. Draw a line segment AB = 12 cm.
2. Divide AB into equal parts. For two waves, mark point X such that AX = 6 cm (midpoint).
3. For the first wave: Place the compass tip at the midpoint of AX (i.e., 3 cm from A), set radius = 3 cm, and draw a semicircle above AB.
4. For the second wave: Place the compass tip at the midpoint of XB (i.e., 3 cm from X), set radius = 3 cm, and draw a semicircle below AB.
5. The wavy wave is complete.

Observation: The radius used = 3 cm, and each half circle has diameter = 6 cm. Students may choose any even length for the central line and divide accordingly to get identical waves.

Given: Waves that are arcs smaller than a semicircle (less than half circles), as seen in the neck of 'A Person'.

Concept: An arc smaller than a semicircle is drawn by placing the compass centre below (or above) the central line, so that the circle's arc crosses the line at two points, producing a shallower curve.

Steps of Construction:
1. Draw the central line AB of a chosen length, say 8 cm.
2. Mark points P and Q on AB such that PQ = 4 cm (the span of one wave). Let P be at 2 cm from A and Q at 6 cm from A.
3. To draw an arc smaller than a semicircle over PQ: Choose a radius larger than half of PQ (i.e., radius r > 2 cm, say $r = 3$ cm).
4. Find the centre: It lies on the perpendicular bisector of PQ, below the line AB. Using compass, mark the centre $O_1$ at the intersection of the perpendicular bisector of PQ and a point below AB at the appropriate distance.
5. Place compass tip at $O_1$, set radius = 3 cm, and draw the arc from P to Q above the line.
6. For the second identical wave: Repeat the same process for the next segment of equal length on AB, keeping the same radius and the same offset for the centre.

Key to identical waves: Use the same radius and place the centre at the same perpendicular distance from AB for both waves. This ensures both arcs are congruent.

Note: Multiple trials may be needed to find the right centre position so the arc looks like the one in the figure.

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°.