Year 8 Transformations and congruence: free practice test with answers

Sample transformations and congruence questions with answers

Try each one before you open the answer.

1. Reflection

   Write the coordinates of the image of the point $(4, -2)$ after a reflection in:

   (a) the $y$-axis

   (b) the $x$-axis

   Show the answer

   Key idea: A reflection in the $y$-axis changes the sign of $x$; a reflection in the $x$-axis changes the sign of $y$.

   1. (a) $(4, -2) \to (-4, -2)$
   2. (b) $(4, -2) \to (4, 2)$

   Answer: (a) $(-4, -2)$ (b) $(4, 2)$

2. Translation

   The point $(2, -5)$ is translated $4$ units right and $3$ units up. Find its image.

   Show the answer

   Key idea: Moving right adds to $x$; moving up adds to $y$.

   1. $x$: $2 + 4 = 6$
   2. $y$: $-5 + 3 = -2$

   Answer: $(6, -2)$

3. Rotation

   The point $(3, 1)$ is rotated $90^\circ$ anticlockwise about the origin. Find its image.

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   Key idea: A rotation of $90^\circ$ anticlockwise about the origin maps $(x, y)$ to $(-y, x)$.

   1. $(3, 1) \to (-1, 3)$

   Answer: $(-1, 3)$

4. Congruent figures

   Quadrilateral $ABCD$ is congruent to quadrilateral $WXYZ$, with vertices in matching order. $AB = 7$ cm, $BC = 5$ cm, $CD = 9$ cm and $DA = 4$ cm. Find the perimeter of $WXYZ$.

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   Key idea: Congruent figures have matching sides equal, so they have equal perimeters.

   1. Perimeter of $ABCD = 7 + 5 + 9 + 4 = 25$ cm.
   2. So the perimeter of $WXYZ$ is $25$ cm.

   Answer: $25$ cm

5. Congruent triangles

   Triangle $ABC$ has $AB = 5$ cm, $BC = 8$ cm and $\angle ABC = 47°$. Triangle $PQR$ has $PQ = 5$ cm, $QR = 8$ cm and $\angle PQR = 47°$. State whether the two triangles are congruent and give the condition that applies.

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   Key idea: Two triangles are congruent if they satisfy one of the four conditions: SSS, SAS, AAS or RHS.

   1. List the matching information: $AB = PQ = 5$ cm, $BC = QR = 8$ cm, and the included angles $\angle ABC = \angle PQR = 47°$.
   2. The equal angle is between the two pairs of equal sides in each triangle.
   3. This matches the SAS (Side-Angle-Side) condition.
   4. Therefore $\triangle ABC \equiv \triangle PQR$ (SAS).

   Answer: The triangles are congruent by SAS.

6. Tessellations

   Explain why regular hexagons can tessellate on their own.

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   Key idea: Shapes tessellate when their angles fit together to make exactly $360^\circ$ at every vertex.

   1. Each interior angle of a regular hexagon is $\dfrac{(6 - 2) \times 180^\circ}{6} = 120^\circ$
   2. $360^\circ \div 120^\circ = 3$, a whole number, so three hexagons meet at each vertex with no gaps

   Answer: Each interior angle is $120^\circ$, and $3 \times 120^\circ = 360^\circ$, so three hexagons fit exactly around a point

7. Congruence and quadrilaterals

   $ABCD$ is a parallelogram. $AB = 11$ cm, $BC = 6$ cm and $\angle ABC = 115^\circ$.

   (a) Find $CD$.

   (b) Find $\angle ADC$.

   (c) Find $\angle BCD$.

   Show the answer

   Key idea: A diagonal splits a parallelogram into two congruent triangles, so opposite sides and opposite angles are equal; co-interior angles add to $180^\circ$.

   1. (a) Opposite sides are equal: $CD = AB = 11$ cm.
   2. (b) Opposite angles are equal: $\angle ADC = \angle ABC = 115^\circ$.
   3. (c) $\angle BCD = 180^\circ - 115^\circ = 65^\circ$ (co-interior angles).

   Answer: (a) $11$ cm (b) $115^\circ$ (c) $65^\circ$

8. Similar figures

   Rectangle $A$ is $4$ cm by $7$ cm. Rectangle $B$ is similar to $A$, and its shorter side is $10$ cm. Find the length of its longer side.

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   Key idea: Find the scale factor from a pair of matching sides, then multiply.

   1. Scale factor $= 10 \div 4 = 2.5$.
   2. Longer side $= 7 \times 2.5 = 17.5$ cm.

   Answer: $17.5$ cm

Free worksheets for this chapter

Worked examples and a free PDF with answers for each topic.

• Year 8 congruence and similarity worksheet
• Year 8 properties of quadrilaterals worksheet