Video Transcript
Two points 𝐴 and 𝐵 have coordinates two, four and four, four, respectively. Line segment 𝐴𝐵 is reflected in the 𝑦-axis to line segment 𝐴 prime 𝐵 prime. Determine the coordinates of 𝐴 prime and 𝐵 prime. Is the length of line segment 𝐴𝐵 greater than, less than, or equal to the length of line segment 𝐴 prime 𝐵 prime?
We’ve been given the coordinates of two points 𝐴 and 𝐵 and told that the line segment connecting these points is reflected in the 𝑦-axis. We first need to determine the coordinates of the images of points 𝐴 and 𝐵 following this reflection.
One way to approach this problem would be to visualize the effect this transformation has using a sketch of the two points on a coordinate grid. When we reflect a point in the 𝑦-axis, it appears on the opposite side of the 𝑦-axis at the same horizontal distance from the mirror line as the original point but in the opposite direction. The vertical height, and hence the 𝑦-coordinate of the point, is unchanged. But the 𝑥-coordinate changes from either positive to negative or negative to positive. We can therefore identify that the coordinates of 𝐴 prime and 𝐵 prime will be negative two, four and negative four, four, respectively.
We could also recall the general result that a reflection in the 𝑦-axis maps a point 𝑃 with coordinates 𝑥, 𝑦 to the point 𝑃 prime with coordinates negative 𝑥, 𝑦. The 𝑦-coordinate is unchanged, but the 𝑥-coordinate changes sign. Applying this mapping to points 𝐴 and 𝐵 confirms the coordinates of 𝐴 prime and 𝐵 prime as negative two, four and negative four, four.
The second part of the question asks us to determine the relationship between the lengths of line segment 𝐴𝐵 and line segment 𝐴 prime 𝐵 prime. As these are both horizontal line segments, we can easily determine from our sketch that the length of each line segment is the difference between the 𝑥-coordinates of its endpoints, which in each case is two units. Alternatively, we can recall one of the key properties of reflection in a line, which is that the lengths of line segments are preserved. So the length of line segment 𝐴𝐵 is equal to the length of its image.
Therefore, our answers are that the coordinates of 𝐴 prime are negative two, four. The coordinates of 𝐵 prime are negative four, four. And the length of line segment 𝐴𝐵 is equal to the length of line segment 𝐴 prime 𝐵 prime.
