IGCSE Maths / Transformation and similarity

Finding the mirror line from object and image revision notes

Edexcel International GCSE Maths (4MA1), chapter 14: Transformation and similarity, specification point 5.2e. Free lesson.

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A reflection line lies halfway between each point on a shape and its matching point on the image, at right angles to the joins. Find the halfway points and use them to identify the mirror line, then give its equation.

How the mirror relates to the object and image

Match each corner of the object to the same corner on its image. The mirror cuts the join between each matching pair in half, at right angles. This means the halfway points all lie on the mirror line. Join every corner to its matching image corner. Find the halfway point of each join. These points show where the mirror is. The direction of the joins helps identify the mirror: horizontal joins go with an upright mirror, vertical joins with a horizontal mirror, and slanting joins with a diagonal mirror.

Finding an upright mirror

In this example, the matching corners have the same y-coordinate, so their joins are horizontal. The mirror must be upright. Each halfway point has x-coordinate 1, so the mirror is x = 1.

Finding a diagonal mirror

Here the joins between matching corners are slanting, so the mirror is diagonal. The coordinates of each halfway point are equal, which means the points lie on y = x. A halfway point is (−2, −2), since it is midway between A (−5, 1) and A′ (1, −5). The other matching pairs also have halfway points with equal coordinates.

Telling the two diagonal lines apart

The two diagonal lines can look similar, so test a halfway point. On y = x, the coordinates match, such as (3, 3). On y = −x, they are opposites, such as (3, −3). Use a halfway point from the object and image to decide which diagonal it lies on. A point with equal coordinates fits y = x. A point with opposite coordinates fits y = −x.

Write the answer as an equation

A line drawn on a grid is not a complete description. State the equation of the mirror. You can check your answer by making sure the mirror is the same perpendicular distance from a corner and its image. For a corner at (1, 2) and its image at (7, 2), the join is horizontal, so the mirror is upright. The halfway x-coordinate is (1 + 7)/2 = 4, so the equation is x = 4.