Question 6

We can form different quadrilaterals by joining two identical isosceles triangles along one of their sides.

Step 1 — Understanding the Triangles

Let us consider one of the given cardboard triangles. It is an isosceles triangle. It has two sides of length 8 cm and one side of length 6 cm. The 6 cm side is the base, and the 8 cm sides are the equal legs.

Step 2 — First Way: Joining along the 6 cm side

We can join the two triangles by making their 6 cm sides (bases) common. Imagine placing the two triangles so their 6 cm bases perfectly overlap. The resulting shape is a quadrilateral. The four outer sides of this quadrilateral will be the 8 cm legs from each triangle. So, all four sides of this quadrilateral are 8 cm long. A quadrilateral with all four sides equal in length is called a rhombus.

Step 3 — Second Way: Joining along an 8 cm side (Parallelogram)

We can also join the two triangles by making one of their 8 cm sides common. There are two distinct ways to orient the second triangle when joining along an 8 cm side. Let us consider the orientation where the 6 cm sides of the two triangles end up opposite each other. The resulting quadrilateral will have sides of length 8 cm, 6 cm, 8 cm, and 6 cm in sequence. In this quadrilateral, the opposite sides are equal in length ($8 \text{ cm}$ and $8 \text{ cm}$, and $6 \text{ cm}$ and $6 \text{ cm}$). A quadrilateral with opposite sides equal in length is called a parallelogram.

Step 4 — Third Way: Joining along an 8 cm side (Kite)

Let us consider the other orientation when joining along an 8 cm side. This time, the 6 cm sides of the two triangles end up adjacent to each other. The resulting quadrilateral will have sides of length 8 cm, 6 cm, 6 cm, and 8 cm in sequence. In this quadrilateral, there are two pairs of equal-length adjacent sides ($8 \text{ cm}$ and $8 \text{ cm}$, and $6 \text{ cm}$ and $6 \text{ cm}$). A quadrilateral with two pairs of equal-length adjacent sides is called a kite.

Answer

(i) The different ways they can be joined to get a quadrilateral are:

1. By joining them along their 6 cm side.
2. By joining them along one of their 8 cm sides, with the 6 cm sides opposite each other.
3. By joining them along one of their 8 cm sides, with the 6 cm sides adjacent to each other.

(ii) The quadrilaterals formed and their justifications are:

1. Rhombus: When joined along the 6 cm side, all four outer sides are the 8 cm legs of the triangles. So, all four sides of the quadrilateral are 8 cm. A quadrilateral with all sides equal is a rhombus.
2. Parallelogram: When joined along an 8 cm side (with 6 cm sides opposite), the resulting quadrilateral has opposite sides equal (one pair of 8 cm sides and one pair of 6 cm sides). A quadrilateral with opposite sides equal is a parallelogram.
3. Kite: When joined along an 8 cm side (with 6 cm sides adjacent), the resulting quadrilateral has two pairs of equal-length adjacent sides (one pair of 8 cm sides and one pair of 6 cm sides). A quadrilateral with two pairs of equal-length adjacent sides is a kite.