Question 5

We can join two equilateral triangles along one of their sides to form a quadrilateral.

Step 1 — Properties of equilateral triangles

Let us consider two cardboard cutouts. Each cutout is an equilateral triangle. Its side length is 8 cm. In an equilateral triangle, all three sides are equal. Also, all three interior angles are equal. Each interior angle measures $60^\circ$.

Step 2 — Joining the triangles

We join the two equilateral triangles. We place one side of the first triangle against one side of the second triangle. This common side will be 8 cm long. This forms a new closed shape. This shape has four outer sides. It also has two diagonals.

Step 3 — Identifying the quadrilateral

The resulting shape is a quadrilateral. Let us examine its properties. All four outer sides are 8 cm long. This is because they are sides of the original equilateral triangles. A quadrilateral with all four sides equal is a rhombus. Let us also check the angles. Each angle of an equilateral triangle is $60^\circ$. Two angles of the quadrilateral are $60^\circ$ each. These are the angles at the "tips" of the joined triangles. The other two angles are formed by joining two $60^\circ$ angles. So, these angles are $60^\circ + 60^\circ = 120^\circ$. The quadrilateral has angles of $60^\circ$, $120^\circ$, $60^\circ$, $120^\circ$. Opposite angles are equal ($60^\circ = 60^\circ$ and $120^\circ = 120^\circ$). This confirms it is a parallelogram. Since all sides are equal, it is a rhombus. It is not a square because its angles are not all $90^\circ$.

Answer

(i) Yes, we can join them to get a quadrilateral. (ii) The type of quadrilateral formed is a rhombus. (iii) Justification: All four outer sides of the quadrilateral are 8 cm long. This is because they are the sides of the original equilateral triangles. A quadrilateral with all four sides equal is defined as a rhombus. Also, the opposite angles of the quadrilateral are equal ($60^\circ$ and $120^\circ$), which is a property of a rhombus.