296. Mirror Across the X-Axis · Reflecting a point across the x-axis using construction tools
P′ is the reflection of P across the x-axis — same x-coordinate, opposite y-coordinate.
Stuck? Ask Guru
For subscribers · Selina ICSE: Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)
What this lesson covers
Try to break it
Drag P across the x-axis and watch P′ cross with it. When P is on the x-axis itself, P′ meets P — that′s an invariant point. Observe how the distances d1 and d2 stay equal, and the right angle at the axis stays true no matter where P goes.
How you build it
Reflect a point across the x-axis.
- Draw the segment AB.
- Construct the perpendicular bisector of AB — the point where the bisector meets AB is O.
- Mark point P.
- With centre O, draw an arc of radius OP.
- Mark point P′ where the arc cuts the bisector below the x-axis.
The proof, step by step
Prove that reflecting in the x-axis leaves the x-coordinate unchanged and flips the sign of the y-coordinate.
- The x-axis is the perpendicular bisector of the segment PP′.
- The horizontal distance from the y-axis remains unchanged.
- The vertical distance from the x-axis is equal for P and P′.
- The y-coordinate changes sign while the x-coordinate stays the same.
Worked example
A point P is reflected in the x-axis to get P′(-3, 7). What are the coordinates of the original point P?
Reflection in the x-axis changes only the sign of the y-coordinate. Since P′ is (-3, 7), the original point P must have coordinates (-3, -7).
- (3, -7)
- (-3, -7) — correct
- (3, 7)
- (-3, 7)