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The Line y = x

Part of the Transformations & Constructions section of Coddy's Geometry journey. Lesson 8 of 61.

The line y = x runs diagonally through (0, 0), (1, 1), (2, 2) and so on. Reflecting in it swaps the coordinates: (x, y) → (y, x).

$Triangle KLM reflected in the diagonal line y = x$

Why does a swap work? Take K(1, 4) and K'(4, 1). The segment between them goes 3 right and 3 down, so it crosses the mirror at right angles. Its midpoint is (5/2, 5/2), which is on y = x. Both conditions of a reflection hold: straight across, and the same distance on each side.

So to draw a reflection in y = x, swap each corner's coordinates and place the image corner there.

Challenge

Draw the image of triangle ABC after the reflection in the line y = x. Drag A', B' and C' into place, then press Check.

Try it yourself

$Move the shape onto its target$

Move the shape onto its target

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Transformations & Constructions

1Translations

Sliding a Shape Draw the Image Image Coordinates The Translation Rule Recap: Translations

4Combining Moves

One Move, Then Another Order Matters Two Mirrors Name the Move Recap: Combining Moves

7Drawing on Squared Paper

Lengths by Pythagoras Perpendicular Steps Squares by Area Right Triangles by Area Midpoints on the Grid Recap: Squared Paper

10Perpendiculars and Parallels

Perpendicular From a Point Perpendicular at a Point Parallel Through a Point Tools You Have Earned Recap: Parallels

2Reflections

Reflecting in a Line The Two AxesThe Line y = xFind the Mirror Recap: Reflections

5Dilations

Scaling From a Center Shrinking a Shape Dilating About a Point Negative Scale Factors Recap: Dilations

8Compass and Straightedge

Circles Keep Distances Why the Sides Match Angles You Built A 60° Angle Recap: Equilateral Triangle

11Constructed Shapes

A Square on a Segment The Regular Hexagon Copy an Angle A Tangent From a Point Recap: Constructed Shapes