Shapes and Patterns — NCERT Solutions

Given: A standard tangram set (7 pieces cut from a square): 2 large right-angled triangles, 1 medium right-angled triangle, 2 small right-angled triangles, 1 square, 1 parallelogram.

Similarities:

• All pieces are flat (2-dimensional) shapes.
• All pieces are made up of right angles or $45°$ angles.
• All pieces can be combined to form a large square.

Differences:

• They differ in shape: some are triangles, one is a square, one is a parallelogram.
• They differ in size: there are large, medium, and small triangles.
• The parallelogram is the only piece that is not a triangle or square.

Conclusion: The pieces are similar in that they all have straight sides and specific angles ($45°$ and $90°$), but differ in shape and size.

Given: Tangram pieces — triangles, square, parallelogram.

Observation:

• All triangles in the tangram are right-angled isosceles triangles: they have one $90°$ angle and two $45°$ angles.
• The square has all four angles equal to $90°$.
• The parallelogram has two angles of $45°$ and two angles of $135°$.

Conclusion: The angles in all tangram pieces are either $45°$, $90°$, or $135°$. The triangles all have the same set of angles ($45°$-$45°$-$90°$).

Given: Tangram pieces.

Observation:

• The sides of all tangram pieces are related to each other — they are either equal in length or one is $\sqrt{2}$ times the other (due to the $45°$-$45°$-$90°$ triangle property).
• The small triangles have the shortest sides.
• The medium triangle's hypotenuse equals the side of the large triangle's leg.
• The square's side equals the leg of the medium triangle.
• The parallelogram's sides equal the legs of the small triangles.

Conclusion: The sides of the tangram pieces are all related — they are multiples or $\sqrt{2}$ multiples of the smallest unit length, showing a consistent proportional relationship throughout the set.

Given: Three views of a cube (front, top, side — each showing a different face with a pattern or marking).

Concept: A net of a cube is a 2D cross-shaped (or T-shaped) layout of 6 squares that folds into a cube. Each square in the net represents one face of the cube.

Method:

1. Identify which face is the front, which is the top, and which is the side from the three given views.
2. In the net, the faces are arranged in a cross pattern. The centre square is typically the bottom; the square above it is the front; the square to the left is the left side; the square to the right is the right side; the square above the front is the top; the square below the bottom is the back.
3. Place the markings/patterns from each view onto the corresponding square in the net.

Note: Since the actual images of the cube views are not visible here, students should physically look at the three views provided in the textbook, identify the pattern on each face, and draw/write the correct pattern on each face of the net in the correct position and orientation.

Conclusion: The net should have all 6 faces correctly labelled/drawn so that when folded, the cube matches all three given views.

Given: Large cube frames made of small cubes, with some small cubes removed from the interior or corners.

Concept:

• First find the total number of small cubes in the complete solid cube.
• Then count the small cubes remaining in the frame.
• Cubes removed = Total cubes − Remaining cubes.

Note: Since the actual images (a), (b), (c) are not visible in the OCR, the method is as follows:

General Method:

1. Determine the size of the large cube (e.g., $3 \times 3 \times 3 = 27$ small cubes total).
2. Count the small cubes visible in the frame structure.
3. Subtract: Removed = Total − Remaining.

Example for a $3 \times 3 \times 3$ cube frame (only edges remaining, all interior and face-centre cubes removed):

• Total = $27$
• Edge cubes only = $8$ corners $+ 12$ edges $\times 1 = 20$
• Removed = $27 - 20 = 7$

Students should apply this method to each figure (a), (b), (c) as shown in their textbook and count accordingly.

Given: A large cube made of $27$ small cubes ($3 \times 3 \times 3$). The outside of the large cube is painted red.

Concept: Depending on the position of each small cube in the large cube, different numbers of its faces are exposed (painted).

Analysis of positions:

(a) Three faces painted — Corner cubes:
Corner cubes are at the 8 corners of the large cube. Each corner cube has exactly 3 faces on the outside.
$$\boxed{8 \text{ small cubes have 3 faces painted red}}$$

(b) Two faces painted — Edge cubes (non-corner):
Each edge of the large cube has $3 - 2 = 1$ small cube in the middle (not at corners). There are 12 edges in a cube.
$$12 \times 1 = \boxed{12 \text{ small cubes have 2 faces painted red}}$$

(c) One face painted — Face cubes (centre of each face):
Each face of the large cube has a $3 \times 3$ grid, with $1$ centre cube that is not on any edge. There are 6 faces.
$$6 \times 1 = \boxed{6 \text{ small cubes have 1 face painted red}}$$

(d) No faces painted — Interior cube:
The cube completely inside, not touching any outer face.
$$\boxed{1 \text{ small cube has no faces painted red}}$$

Verification: $8 + 12 + 6 + 1 = 27$ ✓ (matches total number of small cubes)

Given: 7 shapes in a line — 2 squares, 2 triangles, 1 circle, 1 hexagon, 1 rectangle.

Clues analysis (step by step):

Step 1 — Clue (e): Circle is to the left of a square.
$$\text{Circle} \rightarrow \ldots \rightarrow \text{Square}$$

Step 2 — Clue (a): A square is between the circle and the rectangle.
$$\text{Circle} - \text{Square} - \text{Rectangle}$$

Step 3 — Clue (b): The rectangle is between the square and the triangle.
$$\text{Circle} - \text{Square} - \text{Rectangle} - \text{Triangle}$$

Step 4 — Clue (c): The two triangles are next to the square. We already have one triangle to the right of the rectangle. The second triangle must also be next to a square. Since we have 2 squares, the second square must have a triangle next to it.

From clues so far: $\text{Circle} - \text{Square}_1 - \text{Rectangle} - \text{Triangle}_1$

The second triangle must be next to the second square. Place: $\text{Triangle}_2 - \text{Square}_2$ or $\text{Square}_2 - \text{Triangle}_2$.

Step 5 — Clue (d): The hexagon is to the right of a triangle.

Combining all clues, the arrangement that satisfies everything is:

$$\boxed{\text{Circle} - \text{Square} - \text{Rectangle} - \text{Square} - \text{Triangle} - \text{Triangle} - \text{Hexagon}}$$

Verification:

• (a) Square (position 2) is between Circle (1) and Rectangle (3). ✓
• (b) Rectangle (3) is between Square (2) and Square (4) — and Square (4) leads to Triangle. ✓
• (c) Both triangles (positions 5 and 6) are next to Square (position 4). ✓
• (d) Hexagon (7) is to the right of Triangle (6). ✓
• (e) Circle (1) is to the left of Square (2). ✓

Final arrangement (left to right):
$$\text{Circle} - \text{Square} - \text{Rectangle} - \text{Square} - \text{Triangle} - \text{Triangle} - \text{Hexagon}$$

Given: 3D solid models — Icosahedron and Dodecahedron (made from nets at the end of the book).

Concept: These are regular polyhedra (Platonic solids).

Icosahedron:

• Shape of faces: Equilateral triangles
• Number of faces: 20
• Do all faces look the same? Yes — all 20 faces are congruent equilateral triangles.
• Faces meeting at each vertex: 5 triangles meet at every vertex.
• Same number at each vertex? Yes — 5 faces meet at each vertex.
• Number of edges: 30

Counting edges: Each triangle has 3 edges; $20 \times 3 = 60$ edge-sides, but each edge is shared by 2 faces, so edges $= 60 \div 2 = 30$.

Dodecahedron:

• Shape of faces: Regular pentagons
• Number of faces: 12
• Do all faces look the same? Yes — all 12 faces are congruent regular pentagons.
• Faces meeting at each vertex: 3 pentagons meet at every vertex.
• Same number at each vertex? Yes — 3 faces meet at each vertex.
• Number of edges: 30

Counting edges: Each pentagon has 5 edges; $12 \times 5 = 60$ edge-sides, but each edge is shared by 2 faces, so edges $= 60 \div 2 = 30$.

Other solid shapes with all faces the same (Platonic Solids):

• Tetrahedron (4 equilateral triangular faces)
• Cube (6 square faces)
• Octahedron (8 equilateral triangular faces)

In all Platonic solids, the same number of faces meet at each vertex. ✓

How to count edges without missing or repeating: Go face by face, mark each edge as you count it, ensuring each edge is counted only once (since it is shared between exactly 2 faces).