Reflection

Part ofMathsSymmetry and transformations

Key points

An image of a trapezium reflected in a vertical mirror line.
Shape A'B'C'D' is a reflection of shape ABCD.

A is one of the four types of . A shape can be reflected across a line of reflection to create an image. The line of reflection is also called the mirror line.

  • Each on the original shape is the same distance from the mirror line to its corresponding vertex on the image.

  • The new shape is to the original shape.

  • When reflecting shapes in non-vertical mirror lines, rotating the paper to make the mirror line vertical can help to visualise a problem.

  • When reflecting shapes, a good understanding of symmetry and naming and plotting coordinates can be helpful.

An image of a trapezium reflected in a vertical mirror line.
Shape A'B'C'D' is a reflection of shape ABCD.
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Reflecting in vertical and horizontal lines

  • Reflections can be shown on a grid or on sets of axes.

  • To work out the position of the image after a reflection:

    1. Pick a vertex on the shape (object).

    2. Work out the perpendicular distance to the mirror line by counting the amount of squares on the grid.

    3. Count the same perpendicular distance from the mirror line to the opposite side of the mirror line. This will be the position of the reflected vertex.

    4. Repeat the process for additional vertices.

  • If a vertex is on the mirror line, it will be under the reflection.

Examples

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  1. Example one. An image of a square grid. The grid has a length of sixteen squares and a width of ten squares. A trapezium has been drawn to the left of the square grid. The trapezium, starting from the top right vertex, A, has sides of length, four squares down, four squares to the left, a diagonal side of length four squares up and two squares to the right, and a final side of length two squares to the right. Each vertex is labelled A, B, C, and D, in a clockwise direction. There is a vertical dashed line, labelled mirror line, two squares to the right of side A B. Shape A B C D is coloured blue and the mirror line is coloured pink.

    Reflect shape ABCD in the vertical mirror line.

Question

Quadrilateral ABCD is reflected in the vertical line \(x\) = 5. What are the co-ordinates for the reflection of vertex D?

The image shows a set of axes. The horizontal axis is labelled x. The values go up in ones from zero to eight. The vertical axis is labelled y. The values go up in ones from zero to eight. Quadrilateral A B C D has been plotted and has vertices with coordinates, A equals four comma seven, B equals five comma six, C equals four comma two, and D equals three comma four. A vertical dashed line passing through the five on the x axis has been drawn. Written above the line: mirror line, x equals five. Quadrilateral A B C D is coloured blue and the mirror line is coloured pink.

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Reflecting in diagonal lines

  • To reflect a shape in a diagonal line, a small adjustment to the method needs to be made. Since the mirror line is a diagonal, the perpendicular distance is also diagonal.

  • To reflect a shape on a diagonal line:

    1. Pick a vertex on the object shape.

    2. Work out the perpendicular distance to the mirror line. This is done by drawing a line from the vertex which passes through the diagonal of each square on the grid until the mirror line is reached.

    3. Count the same perpendicular distance from the mirror line to the opposite side of the mirror line. This will be the position of the reflected vertex.

    4. Repeat the process for additional vertices.

Examples

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  1. Example one. An image of a square grid. The grid has a length of sixteen squares and a width of ten squares. A right angled triangle has been drawn to the left of the square grid. The triangle, starting from the top left vertex, A, has sides of length, five squares to the right, a diagonal side of length three squares down and five squares to the left, and a final side of length three squares up. Each vertex is labelled A, B, and C, in a clockwise direction. There is a diagonal dashed line, labelled mirror line. The line passes through points one square to the right of vertex B and three squares below vertex C.

    Reflect triangle ABC in the diagonal mirror line.

Question

Shape PQRS is reflected in the diagonal line, \(y\) = \(x\). In the reflected image, which coordinates would be invariant?

The image shows a set of axes. The horizontal axis is labelled x. The values go up in ones from zero to eight. The vertical axis is labelled y. The values go up in ones from zero to eight. Quadrilateral P Q R S has been plotted and has vertices with coordinates, P equals one comma six, Q equals three comma seven, R equals six comma six, and S equals two comma two. A diagonal dashed line passing through the coordinates zero comma zero and eight comma eight has been drawn. Written right of the line: mirror line, y equals x. Quadrilateral P Q R S is coloured blue, and the mirror line is coloured pink.

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Practise reflecting shapes

Quiz

Practise reflecting shapes with this quiz. You may need a pen and paper to help you work out your answers.

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Real-life maths

An image of a snooker table with a white ball, blue ball, and red ball. Two dashed lines are drawn to show the path of the white ball if it were being bounced off the edge of the table. The angle the white ball makes with the table before the collision and the angle the white ball is reflected at are drawn. These angles are the same size. Written left: angle of incidence. Written left: angle of reflection.
When the ball hits the cushion of a snooker table, the angle of incidence is the same as the angle of reflection.

In the game of snooker, a player may choose to bounce the cue ball off the edge of the table (cushion) to strike the ball they need to hit.

The of the ball after it hits the cushion is a reflection. The angle of incidence is equal to the angle of reflection.

An image of a snooker table with a white ball, blue ball, and red ball. Two dashed lines are drawn to show the path of the white ball if it were being bounced off the edge of the table. The angle the white ball makes with the table before the collision and the angle the white ball is reflected at are drawn. These angles are the same size. Written left: angle of incidence. Written left: angle of reflection.
When the ball hits the cushion of a snooker table, the angle of incidence is the same as the angle of reflection.
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