Treat a motion as a function on the plane

Try

Move a paper triangle three units right and two units down without turning it. Every point moves by the same displacement, not merely its labeled corners. In coordinates this translation is T(x,y)=(x+3,y−2). If A=(−1,4), then T(A)=(2,2). If B=(2,4), then T(B)=(5,2): the horizontal side still has length 3.

A geometric transformation is a function that assigns an image to every point in its domain. The motions here act on the entire plane and are invertible. A rigid motion preserves distances and angles; translation, rotation and reflection are our basic examples. We use the Euclidean properties of these constructions explicitly, rather than defining congruence by a criterion that we later pretend to prove.

Predict

Predict the image of A if you first translate by (3,−2) and then reflect across the x-axis. Compare with reflecting first and translating second. Decide before calculating whether the two sequences must agree simply because each individual step preserves lengths. Preservation of a quantity does not mean operations commute.

Also predict what a horizontal stretch H(x,y)=(2x,y) does to a horizontal side, a vertical side and a sloping side. An operation can preserve selected lengths or angles while failing to be rigid.

Experiment

Use the table as a manual coordinate machine. Substitute the same point (−1,4) into each rule, then repeat with (2,4). Draw both results on squared paper using the same unit length on each axis.

Operation	Coordinate rule	Image of (−1,4)
Translate by (3,−2)	(x+3,y−2)	(2,2)
Reflect across x-axis	(x,−y)	(−1,−4)
Rotate 90° counterclockwise about origin	(−y,x)	(−4,−1)
Rotate 180° about origin	(−x,−y)	(1,−4)

The two-step sequences predicted above produce (2,−2) and (2,−6), respectively. Record intermediate points, not only the final output: that trace exposes an ordering error immediately. A rotation about a different center O requires subtracting O, rotating the displacement, and adding O back. Rotating the original coordinates directly uses the wrong center.

Understand

For two points P and Q, write their coordinate difference as (Δx,Δy). Translation leaves this difference unchanged. Reflection across the x-axis changes it to (Δx,−Δy); a quarter-turn changes it to (−Δy,Δx). In every case its squared length remains Δx²+Δy². Thus the coordinate rules verify distance preservation for arbitrary points, not just our sample triangle.

Angles are also preserved by the geometric constructions: translations move both rays with unchanged direction; rotations turn both through the same angle; reflections reverse handedness while keeping the separation of reflected rays. In a coordinate model, the same rules preserve the dot product ΔxΔu+ΔyΔv as well as lengths, which gives a further verification without using any triangle-congruence criterion. That optional algebraic check is not a prerequisite for following the ray constructions.

A reflection across a line fixes its points and sends each other point to the opposite side so that the line is the perpendicular bisector of the point-image segment. A rotation fixes its center and turns every ray from it through the stated angle, keeping radius fixed. These descriptions specify all points. Their inverses are reflection again, rotation through the negative angle, and translation by the negative vector. The inverse of a composition reverses the order of steps.

The stretch H has no such distance guarantee: (0,0) to (1,0) changes from length 1 to length 2. Even angle preservation fails generally. Rays in directions (1,1) and (1,−1) are perpendicular, while their stretched directions (2,1) and (2,−1) are not. One counterexample is enough to disprove rigidity.

Apply

Write an inverse procedure for “reflect across the y-axis, then translate by (4,1).” Undo the translation first, then reflect. Verify the result on a point and then algebraically for (x,y). Explain why checking one point alone could conceal an error, especially if that point lies on a reflection axis or at a rotation center.

When mapping a segment, use the full point rule A+t(B−A), not an animation of its endpoints. These affine motion rules map it onto the segment between the images, and their inverses ensure no points are omitted.

Continue

Use these motions next to find symmetries of entire figures. A symmetry must return the whole set to itself; moving one vertex onto another is only the beginning of that test. Keep operation order and the difference between point matching and rigidity in your written work.