Question Video: Identifying the Mirror Line Used given a Point and Its Image after a Reflection | Nagwa Question Video: Identifying the Mirror Line Used given a Point and Its Image after a Reflection | Nagwa

Question Video: Identifying the Mirror Line Used given a Point and Its Image after a Reflection Mathematics • First Year of Preparatory School

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Victoria reflected point 𝑃 and point 𝑄 across one of the dotted lines in the grid. Complete the following: She reflected point 𝑃 across line _ and point 𝑄 across line _.

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Video Transcript

Victoria reflected point 𝑃 and point 𝑄 across one of the dotted lines in the grid. Complete the following. She reflected point 𝑃 across line blank and point 𝑄 across line blank.

We’ve been told that Victoria reflected two points 𝑃 and 𝑄 across one of the dotted lines in the grid. And we need to determine which line she reflected each point across. We can see points 𝑃 and 𝑄 on the grid as well as their images 𝑃 prime and 𝑄 prime after their reflections. The two possible mirror lines are labeled line 𝐿 and line 𝑀.

To answer this question, we need to recall that a reflection preserves the perpendicular distance of a point from the mirror line. Each of the mirror lines on the grid are vertical. So the perpendicular distance of each point from either mirror line will be horizontal. If we connect points 𝑃 and 𝑃 prime, then the mirror line must intersect this line segment at its midpoint so that points 𝑃 and 𝑃 prime are the same perpendicular distance from the mirror line. In fact, the mirror line will be the perpendicular bisector of the line segment 𝑃𝑃 prime.

If we consider line 𝑀, we can see that point 𝑃 is four squares away horizontally, whereas 𝑃 prime is two squares away horizontally. Therefore, line 𝑀 cannot be the mirror line used for point 𝑃 as the perpendicular distances are different. If, however, we consider the perpendicular distance between line 𝐿 and the points 𝑃 and 𝑃 prime, we can see that in each case this perpendicular distance is three squares. So this tells us that Victoria reflected point 𝑃 across line 𝐿.

Next, we consider point 𝑄. And we can draw in the line segment connecting 𝑄 and 𝑄 prime. This time, we can see that line 𝑀 passes through the midpoint of this line segment and that points 𝑄 and 𝑄 prime are each two squares horizontally away from line 𝑀. So, for point 𝑄, the mirror line was line 𝑀.

By considering the perpendicular distance of each point and its image from the possible mirror lines, we’ve found that Victoria reflected point 𝑃 across line 𝐿 and point 𝑄 across line 𝑀.

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