Shapes and Patterns

Given: Triangular pieces cut from a rhombus (each piece is an isosceles triangle with 2 equal sides).

Activity: When you fit 2 such triangles together along different edges, you can form different triangles.

Types of triangles you can make:

1. Isosceles triangle – by placing two triangles base-to-base or side-to-side so that the resulting triangle has exactly 2 equal sides.
2. Equilateral triangle – by arranging two pieces so that all three sides of the resulting triangle are equal.

Observation about sides:
Each small triangular piece has 2 equal sides (and one different side). Such triangles are called isosceles triangles.

On folding: When you fold an isosceles triangle in half along the line of symmetry, the two base angles coincide, showing that each isosceles triangle has 2 equal angles.

Given: Triangular pieces from the rhombus (isosceles triangles).

Concept: An equilateral triangle has all three sides equal.

Answer: Yes, it is possible to make an equilateral triangle by fitting 2 of the isosceles triangular pieces together along their equal sides in the correct orientation, so that the resulting triangle has all three sides equal.

Conclusion: An equilateral triangle can be formed. All angles of an equilateral triangle are also equal (each = $60°$), as verified by paper folding.

Given: Triangular pieces from the rhombus.

Concept: A scalene triangle has no two sides equal.

Answer: Yes, it is possible to make a scalene triangle by joining two triangular pieces along a shorter or longer edge in a way that the three resulting sides are all of different lengths.

Note: Triangles that have no equal sides are called scalene triangles. In a scalene triangle, no two angles are equal either.

Conclusion: A scalene triangle can be formed from the pieces.

Given: Triangular pieces from the rhombus.

Activity: By fitting 2 triangular pieces together along different edges, you can form various quadrilaterals (4-sided shapes).

Different quadrilaterals possible:
1. Kite – two pairs of adjacent (neighbouring) sides are equal.
2. Parallelogram – opposite sides are equal and parallel.
3. Rectangle – a special parallelogram where all angles are right angles ($90°$).

About the Kite:
- Side 1 = Side 2 (one pair of adjacent sides are equal)
- Side 3 = Side 4 (another pair of adjacent sides are equal)
- These equal pairs are called adjacent sides (sides that share a common vertex/corner).

Conclusion: At least 3 different types of quadrilaterals (kite, parallelogram, rectangle) can be made.

Given: Two quadrilaterals A and B formed from the triangular pieces (both are parallelograms, with B being a rectangle).

Concept: In a parallelogram, opposite sides are equal and parallel.

Observations on measuring:

Quadrilateral A (Parallelogram):
- The pair of opposite sides are equal: top side = bottom side, and left side = right side.
- The equal pairs are opposite sides (not adjacent).
- Opposite angles are equal; adjacent angles are supplementary (add up to $180°$).

Quadrilateral B (Rectangle):
- Again, opposite sides are equal: top = bottom, left = right.
- The equal pairs are opposite sides.
- All four angles are equal and are right angles ($90°$ each).
- A rectangle is a special type of parallelogram.

Conclusion: In both A and B, opposite sides are equal. Quadrilaterals whose opposite sides are equal are called parallelograms. Parallelogram B, having all right angles, is called a rectangle.

Given: A dot/square grid.

Concept:
- A kite has two pairs of adjacent sides equal.
- A parallelogram has two pairs of opposite sides equal and parallel.

How to draw a Kite on the grid:
1. Mark a centre point.
2. Draw two pairs of line segments from adjacent vertices such that Side 1 = Side 2 and Side 3 = Side 4 (adjacent pairs equal).
3. Example: Choose vertices at grid points $(0,2)$, $(1,0)$, $(0,-1)$, $(-1,0)$ — this gives a kite shape.
4. Draw a second kite with different proportions (e.g., a taller or wider kite).

How to draw a Parallelogram on the grid:
1. Choose a starting point, draw a horizontal base of 3 units.
2. From each end, draw a slanted side of 2 units in the same direction.
3. Connect the tops — this gives a parallelogram with opposite sides equal and parallel.
4. Draw a second parallelogram with a different slant or different side lengths.

Note: Students should draw these on the actual grid provided in the textbook. The key properties to maintain are:
- Kite: two pairs of adjacent equal sides.
- Parallelogram: two pairs of opposite equal and parallel sides.

Given: 3 triangular pieces from the rhombus.

Concept: By arranging 3 triangles in different ways, we can create polygons with different numbers of sides.

(a) 3-sided shape (Triangle):
Arrange all 3 triangles so that they fit together to form a larger triangle. Place two triangles to form a parallelogram/rhombus, then attach the third on one slanted side. The result is a larger triangle (3 sides).

(b) 4-sided shape (Quadrilateral):
Arrange 3 triangles so that the outer boundary has exactly 4 sides. For example, place two triangles to form a parallelogram, then attach the third triangle on one of the shorter sides pointing outward — this can give a trapezium or parallelogram (4 sides).

(c) 5-sided shape (Pentagon):
Arrange 3 triangles so that the outer boundary has 5 sides. Place two triangles side by side and attach the third at an angle so that 5 outer edges are visible — this gives a pentagon (5 sides).

Conclusion: Using 3 triangular pieces, shapes with 3, 4, or 5 sides can be formed depending on how the pieces are arranged.

Given: 4 triangular pieces from the rhombus (the rhombus is divided into 4 congruent isosceles triangles).

Concept: By rearranging all 4 pieces, we try to form each shape.

(a) Square: Yes, it is possible. Arrange all 4 triangles so that their right-angle corners (or appropriate angles) meet at the centre, forming a square with all sides equal and all angles $90°$.

(b) Rectangle: Yes, it is possible. Arrange the 4 triangles in two pairs, each pair forming a parallelogram/rectangle, placed side by side to give a longer rectangle.

(c) Triangle: Yes, it is possible. Arrange all 4 pieces to form a larger equilateral or isosceles triangle.

(d) Pentagon (5-sided): Yes, it is possible with a suitable arrangement where 5 outer edges are visible.

(e) Hexagon (6-sided): Yes, it is possible. Arrange the 4 triangles so that the outer boundary forms a 6-sided figure.

(f) Octagon (8-sided): This is difficult/not straightforward with only 4 pieces from this particular rhombus, as the pieces may not produce 8 distinct outer edges easily. Students should try and verify.

Note: The actual results depend on the exact shape of the triangular pieces. Students are encouraged to physically try each arrangement and trace the outlines to verify.