Take two cardboard cutouts of an isosceles triangle with sidelengths , , and . What are the different ways they can be joined to get a quadrilateral? What quadrilaterals are these? Justify your answers.

Quadrilaterals | A

Question 6

Take two cardboard cutouts of an isosceles triangle with sidelengths $8\text{ cm}$ , $8\text{ cm}$ , and $6\text{ cm}$ .

What are the different ways they can be joined to get a quadrilateral?
What quadrilaterals are these? Justify your answers.

Solution

Understand the Question

Two identical isosceles triangles with sides $8\text{ cm}$ , $8\text{ cm}$ , and $6\text{ cm}$ can be joined along any pair of matching equal sides to form a quadrilateral.
Since the triangles have one side of $6\text{ cm}$ $6 cm$ and two sides of $8\text{ cm}$ $8 cm$ :
- Joining along the $6\text{ cm}$ side produces $1$ quadrilateral.
- Joining along an $8\text{ cm}$ side produces $2$ different quadrilaterals depending on the relative orientation of the $6\text{ cm}$ sides.
We classify and justify each resulting shape based on the relationships between its side lengths.

Step 1 · Join Along the $6\text{ cm}$ Side

Diagram 1

Place the two triangles such that their $6\text{ cm}$ bases coincide. Diagram 2

The common side is the $6\text{ cm}$ base.
The four outer sides of the resulting quadrilateral are the four $8\text{ cm}$ legs.

Since all four sides are equal ( $8\text{ cm}$ each), the quadrilateral formed is a rhombus.

Step 2 · Join Along an $8\text{ cm}$ Side with $6\text{ cm}$ Sides Opposite

Place the two triangles such that one $8\text{ cm}$ side of each triangle coincides, with the two $6\text{ cm}$ sides lying opposite to each other. Diagram 3

The sequence of outer side lengths is $8\text{ cm}$ , $6\text{ cm}$ , $8\text{ cm}$ , and $6\text{ cm}$ .
Opposite sides are equal in length ( $8\text{ cm} = 8\text{ cm}$ and $6\text{ cm} = 6\text{ cm}$ ).

Since both pairs of opposite sides are equal, the quadrilateral formed is a parallelogram.

Step 3 · Join Along an $8\text{ cm}$ Side with $6\text{ cm}$ Sides Adjacent

Place the two triangles such that one $8\text{ cm}$ side of each triangle coincides, with the two $6\text{ cm}$ sides lying adjacent to each other.

The sequence of outer side lengths is $8\text{ cm}$ , $8\text{ cm}$ , $6\text{ cm}$ , and $6\text{ cm}$ .
The shape has two distinct pairs of adjacent equal sides ( $8\text{ cm}, 8\text{ cm}$ and $6\text{ cm}, 6\text{ cm}$ ).

Since there are two pairs of equal adjacent sides, the quadrilateral formed is a kite.

Answer

There are 3 ways to join the triangles to form quadrilaterals:

Rhombus (joined along the $6\text{ cm}$ base; all four sides are $8\text{ cm}$ )
Parallelogram (joined along an $8\text{ cm}$ side with $6\text{ cm}$ sides opposite; opposite sides are equal)
Kite (joined along an $8\text{ cm}$ side with $6\text{ cm}$ sides adjacent; two pairs of adjacent sides are equal)

Common Mistakes

Missing the Third Orientation: Overlooking that joining along an $8\text{ cm}$ side can be done in two distinct ways (one giving opposite $6\text{ cm}$ sides, the other giving adjacent $6\text{ cm}$ sides).
Joining Unequal Sides: Attempting to join an $8\text{ cm}$ side of one triangle to the $6\text{ cm}$ base of the other, which does not produce a valid flat quadrilateral with full matching edges.

Question 6

More questions in A

Which Quad?