Take two cardboard cutouts of a scalene triangle with sides , , and . What are the different ways they can be joined to get a quadrilateral? Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.

Understand the Question We have two identical scalene triangles with side lengths , , and . When joining two congruent triangles along a common matching side, the shared side becomes a diagonal of the resulting quadrilateral. For each of the side lengths, there are distinct ways to join them: 1. Reflection across the common side: results in adjacent pairs of equal sides, forming a Kite. 2. Rotation along the common side: results in opposite pairs of equal sides, forming a Parallelogram. In total, there are possible quadrilaterals. Part (i) What are the different ways they can be joined to get a quadrilateral? Step 1 · Count the Combinations The two triangles have distinct side lengths: , , and . To form a quadrilateral with a full edge-to-edge fit, the triangles must be joined along sides of equal length. For each of the sides, there are orientations (reflection and rotation): 1. Along the side ( ways: Kite and Parallelogram) 2. Along the side ( ways: Kite and Parallelogram) 3. Along the side ( ways: Kite and Parallelogram) Answer > (i) There are different ways to join the two triangles to form a quadrilateral. Part (ii) Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral. Step 1 · Identify Kites (Reflection) A kite has two distinct pairs of adjacent equal-length sides. Reflecting one triangle across the common side yields: Joining along the side: Consider with , , and joined with along common side . The resulting quadrilateral has sides , , , and . Since and , adjacent sides are equal, forming a Kite. Joining along the side: Joining with along common side forms quadrilateral with sides and . Since adjacent sides are equal, it is a Kite. Joining along the side: Joining with along common side forms quadrilateral with sides and . Since adjacent sides are equal, it is a Kite. Step 2 · Identify Parallelograms (Rotation) A parallelogram has both pairs of opposite sides equal in length. Rotating one triangle by along the common side yields: Joining along the side: Joining with identical along common side forms quadrilateral . Its sides are and . Since opposite sides are equal, it is a Parallelogram. Joining along the side: Joining with along common side forms quadrilateral . Its sides are and . Since opposite sides are equal, it is a Parallelogram. Joining along the side: Joining with along common side forms quadrilateral . Its sides are and . Since opposite sides are equal, it is a Parallelogram. Answer > (ii) The identified quadrilaterals are: 1. Kite with adjacent sides and (common side ) 2. Parallelogram with adjacent sides and (common side ) 3. Kite with adjacent sides and (common side ) 4. Parallelogram with adjacent sides and (common side ) 5. Kite with adjacent sides and (common side ) 6. Parallelogram with adjacent sides and (common side ) Common Mistakes Overlooking Orientations: Assuming there are only quadrilaterals by only considering one way to join each pair of equal sides, forgetting that reflection gives a kite while rotation gives a parallelogram. Confusing Kite and Parallelogram Properties: A kite requires adjacent sides to be equal (, ), whereas a parallelogram requires opposite sides to be equal (, ). Mismatched Side Alignment: Trying to join unequal sides (e.g. to ), which does not form a single simple quadrilateral.

Quadrilaterals | A

Question 7

Take two cardboard cutouts of a scalene triangle with sides $6\text{ cm}$ , $9\text{ cm}$ , and $12\text{ cm}$ .

What are the different ways they can be joined to get a quadrilateral?
Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.

Solution

Understand the Question

We have two identical scalene triangles with side lengths $6\text{ cm}$ , $9\text{ cm}$ , and $12\text{ cm}$ .
When joining two congruent triangles along a common matching side, the shared side becomes a diagonal of the resulting quadrilateral.
For each of the $3$ $3$ side lengths, there are $2$ distinct ways to join them:
1. Reflection across the common side: results in adjacent pairs of equal sides, forming a Kite.
2. $180^\circ$ Rotation along the common side: results in opposite pairs of equal sides, forming a Parallelogram.
In total, there are $3 \times 2 = 6$ possible quadrilaterals.

(i) What are the different ways they can be joined to get a quadrilateral?

Step 1 · Count the Combinations

The two triangles have $3$ distinct side lengths: $6\text{ cm}$ , $9\text{ cm}$ , and $12\text{ cm}$ .

Question diagram

To form a quadrilateral with a full edge-to-edge fit, the triangles must be joined along sides of equal length. For each of the $3$ sides, there are $2$ orientations (reflection and rotation):

Along the $6\text{ cm}$ side ( $2$ ways: Kite and Parallelogram)
Along the $9\text{ cm}$ side ( $2$ ways: Kite and Parallelogram)
Along the $12\text{ cm}$ side ( $2$ ways: Kite and Parallelogram)

$\text{Total ways} = 3 \times 2 = 6$

Answer

(i) There are $6$ different ways to join the two triangles to form a quadrilateral.

(ii) Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.

Step 1 · Identify Kites (Reflection)

A kite has two distinct pairs of adjacent equal-length sides. Reflecting one triangle across the common side yields:

Joining along the $6\text{ cm}$ side: Consider $\triangle ABC$ with $AB = 6\text{ cm}$ , $BC = 9\text{ cm}$ , and $AC = 12\text{ cm}$ joined with $\triangle ABD'$ along common side $AB$ . The resulting quadrilateral $ACBD'$ has sides $AC = 12\text{ cm}$ , $CB = 9\text{ cm}$ , $BD' = 9\text{ cm}$ , and $D'A = 12\text{ cm}$ . Since $AC = D'A$ and $CB = BD'$ , adjacent sides are equal, forming a Kite.* Joining along the $9\text{ cm}$ side: Joining $\triangle ABC$ with $\triangle BCD'$ along common side $BC$ forms quadrilateral $ABD'C$ with sides $AB = CA = 6\text{ cm}$ and $BD' = D'C = 12\text{ cm}$ . Since adjacent sides are equal, it is a Kite.
Joining along the $12\text{ cm}$ side: Joining $\triangle ABC$ with $\triangle AD'C$ along common side $AC$ forms quadrilateral $ABD'C$ with sides $AB = CA = 6\text{ cm}$ and $BD' = D'C = 9\text{ cm}$ . Since adjacent sides are equal, it is a Kite.

Step 2 · Identify Parallelograms (Rotation)

A parallelogram has both pairs of opposite sides equal in length. Rotating one triangle by $180^\circ$ along the common side yields:

Joining along the $6\text{ cm}$ side: Joining $\triangle ABC$ with identical $\triangle BAD$ along common side $AB$ forms quadrilateral $ACBD$ . Its sides are $AC = BD = 12\text{ cm}$ and $CB = DA = 9\text{ cm}$ . Since opposite sides are equal, it is a Parallelogram.* Joining along the $9\text{ cm}$ side: Joining $\triangle ABC$ with $\triangle DCB$ along common side $BC$ forms quadrilateral $ABDC$ . Its sides are $AB = CD = 6\text{ cm}$ and $AC = DB = 12\text{ cm}$ . Since opposite sides are equal, it is a Parallelogram.
Joining along the $12\text{ cm}$ side: Joining $\triangle ABC$ with $\triangle CDA$ along common side $AC$ forms quadrilateral $ABCD$ . Its sides are $AB = CD = 6\text{ cm}$ and $BC = DA = 9\text{ cm}$ . Since opposite sides are equal, it is a Parallelogram.

Answer

(ii) The $6$ identified quadrilaterals are:

Kite with adjacent sides $12\text{ cm}$ and $9\text{ cm}$ (common side $6\text{ cm}$ )
Parallelogram with adjacent sides $12\text{ cm}$ and $9\text{ cm}$ (common side $6\text{ cm}$ )
Kite with adjacent sides $6\text{ cm}$ and $12\text{ cm}$ (common side $9\text{ cm}$ )
Parallelogram with adjacent sides $6\text{ cm}$ and $12\text{ cm}$ (common side $9\text{ cm}$ )
Kite with adjacent sides $6\text{ cm}$ and $9\text{ cm}$ (common side $12\text{ cm}$ )
Parallelogram with adjacent sides $6\text{ cm}$ and $9\text{ cm}$ (common side $12\text{ cm}$ )

Common Mistakes

Overlooking Orientations: Assuming there are only $3$ quadrilaterals by only considering one way to join each pair of equal sides, forgetting that reflection gives a kite while rotation gives a parallelogram.
Confusing Kite and Parallelogram Properties: A kite requires adjacent sides to be equal ( $AC = AD'$ , $BC = BD'$ ), whereas a parallelogram requires opposite sides to be equal ( $AC = BD$ , $BC = AD$ ).
Mismatched Side Alignment: Trying to join unequal sides (e.g. $6\text{ cm}$ to $9\text{ cm}$ ), which does not form a single simple quadrilateral.

Question 7

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