Understanding Reflection Across y = x
A reflection across y = x is a coordinate swap. The original x value becomes the new y value. The original y value becomes the new x value. This rule works because the line y = x is a perfect diagonal mirror. Every point on that line has equal coordinates. Points away from the line move to the opposite side.
Why This Calculator Helps
Manual reflection is easy for one point. It becomes slower with decimals, negatives, and many rows. This calculator handles single points and batch entries. It also reports distance from the mirror line, displacement, midpoint, and matrix form. These extra values help students check each step. They also help teachers prepare answer keys.
Coordinate Details
The reflected point keeps the same distance from the mirror line. The midpoint between the original and reflected points lies on y = x. The segment joining both points is perpendicular to the mirror line. These properties are useful checks. If any check fails, the coordinates were likely copied incorrectly.
Batch Use Cases
Batch mode is useful for polygons, triangle vertices, graph transformations, and classroom tables. Enter one point per line. You may write x,y or label,x,y. Labels make exported files easier to read. The calculator rounds values using your selected precision. It keeps formulas visible, so the result stays transparent.
Practical Notes
Reflection across y = x does not change shape size. It changes orientation. A square remains a square. A triangle remains congruent. Slopes are transformed because coordinate roles are exchanged. For example, a horizontal segment may become vertical. A point already on y = x does not move. This happens when x equals y. Use the distance field to confirm that case.
Checking Graph Work
Use the original and reflected coordinates together. Plot both points on the same grid. Draw the line y = x. The line should cut the joining segment exactly halfway. It should also meet that segment at a right angle. For polygons, reflect every vertex in the same order. Then connect the new vertices. The reflected figure should match the original figure in size. Only its position and orientation should change. These checks make errors easier to see before submission and grading review.