Reflecting A Triangle Over A Line Segment

🔑 Key Concepts

• A reflection flips a figure across a line of reflection.
• The reflected point must be the same distance from the line of reflection as the original point.
• When reflecting across a horizontal line, the \(x\)-coordinate stays the same.
• The \(y\)-coordinate changes based on how far the point is above or below the line of reflection.
• A reflection is a rigid transformation because the image keeps the same size and shape.

✏️ Worked Examples

Visual Reference

Method 1: Count From the Line

The line of reflection is line segment \(XY\), which lies on the horizontal line \(y=4\).

Point \(A(-2,8)\) is \(4\) units above \(y=4\).

\(8-4=4\)

So \(A'\) must be \(4\) units below \(y=4\).

\(4-4=0\)

The \(x\)-coordinate stays the same.

\(A'=(-2,0)\)

Reflect Points \(B\) and \(C\)

Point \(B(-6,5)\) is \(1\) unit above \(y=4\).

\(5-4=1\)

Move \(1\) unit below \(y=4\):

\(4-1=3\)

\(B'=(-6,3)\)

---

Point \(C(-2,5)\) is also \(1\) unit above \(y=4\).

\(C'=(-2,3)\)

Method 2: Use the Horizontal Reflection Rule

For a reflection across the horizontal line \(y=k\), use:

\((x,y)\rightarrow(x,\,2k-y)\)

Here, the line of reflection is \(y=4\), so \(k=4\).

\(A(-2,8)\rightarrow(-2,2(4)-8)\)

\(A'\rightarrow(-2,0)\)

\(B(-6,5)\rightarrow(-6,2(4)-5)\)

\(B'\rightarrow(-6,3)\)

\(C(-2,5)\rightarrow(-2,2(4)-5)\)

\(C'\rightarrow(-2,3)\)

---

New coordinates:
\(A'=(-2,0)\), \(B'=(-6,3)\), \(C'=(-2,3)\)

🧠 Math Vocabulary

• Reflection: a transformation that flips a figure across a line.
• Line of Reflection: the mirror line that the figure is reflected across.
• Horizontal Line: a line that runs left and right, such as \(y=4\).
• Rigid Transformation: a transformation that keeps the same size and shape.
• Line of Symmetry: a line that divides a figure into two matching mirror-image parts.
• Image: the new figure after a transformation.
• Preimage: the original figure before a transformation.
• Corresponding Points: matching points on the preimage and image, such as \(A\) and \(A'\).
• Transformation: a movement or change of a figure on the coordinate plane.

💡 Main Idea

When reflecting across a horizontal line, each image point must be the same distance on the opposite side of the line of reflection. The \(x\)-coordinate stays the same, while the \(y\)-coordinate changes based on the location of the horizontal line.

📚 What You Should Already Know

Students should know how to plot ordered pairs, identify horizontal and vertical lines, read coordinates from a graph, and count distance above and below a line on the coordinate plane.

🚀 What Comes Next

After reflecting across horizontal lines, students can practice reflections across vertical lines, the \(x\)-axis, the \(y\)-axis, and compare reflections with translations, rotations, and dilations.

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