Line Segment
Transformation Properties
Discover how rotations, reflections, and translations preserve the length of line segments and keep lines straight.
What Is a Transformation?
A transformation is a rule that moves every point of a figure to a new location on the coordinate plane. Think of it as a precise instruction: "take this shape and move, flip, or turn it."
What Is a Line Segment?
A line segment is the part of a line between two endpoints. Unlike a full line, it has a definite length that we can measure.
Why Does Preserving Length Matter?
When we move shapes around, we need to know: did the shape change, or just its position? If lengths stay the same, the figure is congruent to the original: same size, same shape.
The Core Rule
This means: if you start with a straight segment \(\overline{AB}\) of length \(d\), after any rigid motion the image \(\overline{A'B'}\) is still straight and still has length \(d\).
Translation Preserves Length
A translation slides every point by the same vector \(\langle a, b \rangle\). Since every point moves identically, the distance between any two points stays the same.
Reflection Preserves Length
A reflection flips every point across a line of reflection. Each point lands on the opposite side, equally far from the line. Distances between points remain unchanged.
Rotation Preserves Length
A rotation turns every point around a center of rotation by a given angle. Each point stays the same distance from the center, so all distances between points are preserved.
Rigid vs. Non-Rigid Motions
Not every transformation preserves length. Dilations (scaling) change the size of a figure, so they are not rigid motions. Only the three rigid motions keep segments the same length.
| Property | Rigid Motions | Dilation |
|---|---|---|
| Types | Translation, Reflection, Rotation | Enlargement or Reduction |
| Preserves Length? | ✓ Yes | ✕ No |
| Preserves Shape? | ✓ Yes | ✓ Yes |
| Lines Stay Straight? | ✓ Yes | ✓ Yes |
| Result is Congruent? | ✓ Yes | ✕ No (similar) |
Lines Map to Lines
Rigid motions also guarantee that straight things stay straight. A line never becomes a curve, and a segment never bends. This is true regardless of the transformation.
How to Apply This Rule
Example 1: Translation
Segment \(\overline{AB}\) has endpoints \(A(1, 2)\) and \(B(4, 6)\). It is translated by \(\langle 3, -1 \rangle\). What is the length of the image \(\overline{A'B'}\)?
Example 2: Reflection
Segment \(\overline{PQ}\) with \(P(2, 5)\) and \(Q(6, 5)\) is reflected over the x-axis. What is the length of \(\overline{P'Q'}\)?
Example 3: Rotation
Segment \(\overline{MN}\) has \(M(0, 0)\) and \(N(3, 4)\). It is rotated \(180^\circ\) about the origin. Find the length of \(\overline{M'N'}\).
Common Mistakes to Avoid
Summary of Properties
| Transformation | Lines → Lines? | Length Preserved? | Rigid Motion? |
|---|---|---|---|
| Translation | ✓ Yes | ✓ Yes | ✓ Yes |
| Reflection | ✓ Yes | ✓ Yes | ✓ Yes |
| Rotation | ✓ Yes | ✓ Yes | ✓ Yes |
| Dilation | ✓ Yes | ✕ No | ✕ No |
Question 1 of 5
A line segment of length 7 cm is translated 5 units to the right and 3 units down. What is the length of the image segment?
Question 2 of 5
Which of the following transformations does NOT preserve the length of a line segment?
Question 3 of 5
Segment \(\overline{AB}\) is reflected over a line. The image is \(\overline{A'B'}\). Which statement is always true?
Question 4 of 5
Point \(A(1, 3)\) and point \(B(5, 6)\) form segment \(\overline{AB}\). After a \(90^\circ\) counterclockwise rotation about the origin, what is \(A'B'\)?
Question 5 of 5
A straight line is rotated \(45^\circ\) about a point. The image is: