Question Video: Determining the Image of a Line Segment after a Reflection in a Straight Line Mathematics

Video Transcript

Which of the following represents the image of line segment 𝐴𝐵 after a reflection in line 𝑀?

We’ve been given five diagrams that each show line segment 𝐴𝐵 and line 𝑀. We need to determine which of the five diagrams shows the correct image of line segment 𝐴𝐵 after a reflection in line 𝑀.

One of the key properties of reflections in a line is that they preserve the perpendicular distance of all points from the mirror line. So, when we’re looking for the image of line segment 𝐴𝐵, it should be on the opposite side of the mirror line but at the same perpendicular distance away. In fact, the mirror line is the perpendicular bisector of the line segment connecting each point and its image.

Let’s look at diagram (A). Taking the diagonal of the smaller squares on the grid to be our unit of measure, we can see that points 𝐴 and 𝐴 prime are each at a perpendicular distance of two and a half units from the mirror line. So this looks good so far. But when we consider points 𝐵 and 𝐵 prime, they’re at perpendicular distances of 1.5 and 2.5 units from the mirror line. So the perpendicular distances are not the same. We can therefore rule out option (A).

In option (B), the images of points 𝐴 and 𝐵 are in the wrong place. The images appear to have been labeled the wrong way round. Line segments 𝐴𝐴 prime and 𝐵𝐵 prime are not perpendicular to the mirror line. So this figure can’t represent a reflection of line segment 𝐴𝐵 in line 𝑀.

In option (C), points 𝐴 and 𝐴 prime are the same perpendicular distance from the mirror line, but points 𝐵 and 𝐵 prime aren’t. So this isn’t the correct image either.

In option (D), the perpendicular distances from the mirror line are the same for points 𝐴 and 𝐴 prime, which are each 2.5 units, and for points 𝐵 and 𝐵 prime, which are each 1.5 units. So option (D) does show the correct image of line segment 𝐴𝐵 after a reflection in line 𝑀.

For due diligence, we should check the perpendicular distances in option (E). We may be able to judge by eye that they are not the same. But if we add the distances to the diagram, this confirms it.

The correct figure which shows the image of line segment 𝐴𝐵 after a reflection in line 𝑀 is option (D).