Question 7

When two identical triangles are joined along a common side, that side becomes a diagonal of the resulting quadrilateral.

Step 1 — Identify the common sides

The given scalene triangle has three different side lengths. These lengths are 6 cm, 9 cm, and 12 cm. We can join the two identical triangles along any of these three common sides. For each common side, there are two main ways to join the triangles, leading to different quadrilaterals.

Step 2 — Ways to form a Kite

A kite is a quadrilateral where two pairs of equal-length sides are adjacent to each other. We can form a kite by reflecting one triangle across the common side.

• Joining along the 6 cm side: Let us take the first triangle, $\triangle ABC$, with sides $AB=6 \text{ cm}$, $BC=9 \text{ cm}$, and $AC=12 \text{ cm}$. We join it with an identical triangle, $\triangle ABD'$, by making $AB$ their common side. The resulting quadrilateral is $ACBD'$. Its sides are $AC = \mathbf{12 \text{ cm}}$, $CB = \mathbf{9 \text{ cm}}$, $BD' = \mathbf{9 \text{ cm}}$, and $D'A = \mathbf{12 \text{ cm}}$. Since $AC = D'A$ and $CB = BD'$, this quadrilateral has two pairs of adjacent equal sides.

  $\boxed{\text{Kite}}$ The common side $AB = \mathbf{6 \text{ cm}}$ is one of its diagonals.

  <DIAGRAM: A kite named ACBD'. The diagonal AB is 6 cm. The sides AC and AD' are 12 cm. The sides BC and BD' are 9 cm. Triangle ACB and triangle ADB' are shown, sharing side AB.>

• Joining along the 9 cm side: We join $\triangle ABC$ with an identical triangle, $\triangle BCD'$, by making $BC$ their common side. The resulting quadrilateral is $ABD'C$. Its sides are $AB = \mathbf{6 \text{ cm}}$, $BD' = \mathbf{12 \text{ cm}}$, $D'C = \mathbf{12 \text{ cm}}$, and $CA = \mathbf{6 \text{ cm}}$. Since $AB = CA$ and $BD' = D'C$, this quadrilateral has two pairs of adjacent equal sides.

  $\boxed{\text{Kite}}$ The common side $BC = \mathbf{9 \text{ cm}}$ is one of its diagonals.

• Joining along the 12 cm side: We join $\triangle ABC$ with an identical triangle, $\triangle AD'C$, by making $AC$ their common side. The resulting quadrilateral is $ABD'C$. Its sides are $AB = \mathbf{6 \text{ cm}}$, $BD' = \mathbf{9 \text{ cm}}$, $D'C = \mathbf{9 \text{ cm}}$, and $CA = \mathbf{6 \text{ cm}}$. Since $AB = CA$ and $BD' = D'C$, this quadrilateral has two pairs of adjacent equal sides.

  $\boxed{\text{Kite}}$ The common side $AC = \mathbf{12 \text{ cm}}$ is one of its diagonals.

Step 3 — Ways to form a Parallelogram

A parallelogram is a quadrilateral where both pairs of opposite sides are equal in length. We can form a parallelogram by rotating one triangle 180 degrees around the midpoint of the common side.

• Joining along the 6 cm side: Let us take the first triangle, $\triangle ABC$, with sides $AB=6 \text{ cm}$, $BC=9 \text{ cm}$, and $AC=12 \text{ cm}$. We join it with an identical triangle, $\triangle BAD$, by making $AB$ their common side. The resulting quadrilateral is $ACBD$. Its sides are $AC = \mathbf{12 \text{ cm}}$, $CB = \mathbf{9 \text{ cm}}$, $BD = \mathbf{12 \text{ cm}}$, and $DA = \mathbf{9 \text{ cm}}$. Since opposite sides are equal ($AC=BD$ and $CB=DA$), this quadrilateral is a parallelogram.

  $\boxed{\text{Parallelogram}}$ The common side $AB = \mathbf{6 \text{ cm}}$ is one of its diagonals.

  <DIAGRAM: A parallelogram named ACBD. The diagonal AB is 6 cm. The sides AC and BD are 12 cm. The sides BC and AD are 9 cm. Triangle ACB and triangle ADB are shown, sharing side AB.>

• Joining along the 9 cm side: We join $\triangle ABC$ with an identical triangle, $\triangle DCB$, by making $BC$ their common side. The resulting quadrilateral is $ABDC$. Its sides are $AB = \mathbf{6 \text{ cm}}$, $AC = \mathbf{12 \text{ cm}}$, $CD = \mathbf{6 \text{ cm}}$, and $DB = \mathbf{12 \text{ cm}}$. Since opposite sides are equal ($AB=CD$ and $AC=DB$), this quadrilateral is a parallelogram.

  $\boxed{\text{Parallelogram}}$ The common side $BC = \mathbf{9 \text{ cm}}$ is one of its diagonals.

• Joining along the 12 cm side: We join $\triangle ABC$ with an identical triangle, $\triangle CDA$, by making $AC$ their common side. The resulting quadrilateral is $ABCD$. Its sides are $AB = \mathbf{6 \text{ cm}}$, $BC = \mathbf{9 \text{ cm}}$, $CD = \mathbf{6 \text{ cm}}$, and $DA = \mathbf{9 \text{ cm}}$. Since opposite sides are equal ($AB=CD$ and $BC=DA$), this quadrilateral is a parallelogram.

  $\boxed{\text{Parallelogram}}$ The common side $AC = \mathbf{12 \text{ cm}}$ is one of its diagonals.

Answer

(i) There are 6 different ways to join the two triangles to get a quadrilateral:

1. Joining along the 6 cm side to form a kite.
2. Joining along the 6 cm side to form a parallelogram.
3. Joining along the 9 cm side to form a kite.
4. Joining along the 9 cm side to form a parallelogram.
5. Joining along the 12 cm side to form a kite.
6. Joining along the 12 cm side to form a parallelogram.

(ii) We can identify 6 different quadrilaterals obtained by joining the triangles:

1. A Kite with two adjacent sides of 12 cm and two adjacent sides of 9 cm. (Formed by joining along the 6 cm side).
2. A Parallelogram with adjacent sides of 12 cm and 9 cm. (Formed by joining along the 6 cm side).
3. A Kite with two adjacent sides of 6 cm and two adjacent sides of 12 cm. (Formed by joining along the 9 cm side).
4. A Parallelogram with adjacent sides of 6 cm and 12 cm. (Formed by joining along the 9 cm side).
5. A Kite with two adjacent sides of 6 cm and two adjacent sides of 9 cm. (Formed by joining along the 12 cm side).
6. A Parallelogram with adjacent sides of 6 cm and 9 cm. (Formed by joining along the 12 cm side).