Reflecting Shapes on the Cartesian Plane

By the end of this lesson, learners should be able to

1. Reflect geometric shapes across diagonal mirror lines y = x and y = -x on a Cartesian plane.
2. Construct reflected images across horizontal, vertical, and general linear mirror lines y = mx + c.
3. Determine image coordinates for multi-vertex polygons, letters, and circles under reflection.

Key vocabulary

Cartesian plane

A two-dimensional surface formed by perpendicular x and y axes intersecting at an origin.

We plot object points and their mirror image vertices on a grid Cartesian plane.

Image point

The new point produced after transforming an original object point across a mirror line.

Reflecting point L(-4, 1) across y = -x yields the image point L'(-1, 4).

Mirror line

A straight line acting as an axis of reflection for transforming geometric shapes.

The mirror line y = x + 2 bisects perpendicular line segments joining object and image.

Perpendicular distance

The shortest distance measured at right angles from a point to a line.

Each vertex and its reflected image lie at equal perpendicular distance from the mirror line.

Congruence

The state of two geometric figures having identical size and shape dimensions.

A reflected shape and its original object exhibit opposite congruence across the mirror line.

Orientation

The relative physical arrangement or order of vertices in a geometric shape.

A reflection changes clockwise vertex orientation into an anti-clockwise image orientation.

Lesson preview

Slide 1 · Reflecting Across Diagonal Lines y = -x and y = xOpening · 12 min

When reflecting figures on a Cartesian plane across diagonal lines such as $y=-x$ or $y=x$, perpendicular projection lines run diagonally at ${45}^{\circ }$ angles to the axes. For a reflection in $y=-x$, every point $(x,y)$ transforms into $(-y,-x)$. Measuring equal perpendicular distances along lines of gradient $1$ allows us to locate image vertices precisely. Connecting these images creates a congruent figure with reversed orientation.

• The line $y=-x$ passes through the origin with a negative gradient, acting as a diagonal mirror line.
• Reflecting a point $(x,y)$ across $y=-x$ yields the image point $(-y,-x)$ on the Cartesian plane.
• Perpendicular projection lines connecting an object point to its reflected image have a gradient of $1$.
• The perpendicular distance from any vertex to $y=-x$ equals the perpendicular distance to its image.

Worked example. Identify the coordinate transformation rule for reflection in $y=-x$, which maps $(x,y)$ to $(-y,-x)$. Apply the rule to vertex $L(-4,1)$: $x=-4,y=1⟹L^{\mathrm{\prime }}(-1,-(-4))=L^{\mathrm{\prime }}(-1,4)$. Apply the rule to vertex $M(-1,1)$: $x=-1,y=1⟹M^{\mathrm{\prime }}(-1,-(-1))=M^{\mathrm{\prime }}(-1,1)$. Since $M$ lies on the mirror line $y=-x$, $M^{\mathrm{\prime }}=M$ (an invariant point). Apply the rule to vertex $N(-2,4)$: $x=-2,y=4⟹N^{\mathrm{\prime }}(-4,-(-2))=N^{\mathrm{\prime }}(-4,2)$. Plot $L^{\mathrm{\prime }}(-1,4)$, $M^{\mathrm{\prime }}(-1,1)$, and $N^{\mathrm{\prime }}(-4,2)$ on the grid and join them with straight lines. Answer: The vertices of the reflected image triangle $L^{\mathrm{\prime }}{M}^{\mathrm{\prime }}{N}^{\mathrm{\prime }}$ are $L^{\mathrm{\prime }}(-1,4)$, $M^{\mathrm{\prime }}(-1,1)$, and $N^{\mathrm{\prime }}(-4,2)$..

Check question: A quadrilateral has a vertex at $E(3,-5)$. What are the coordinates of its image $E^{\mathrm{\prime }}$ after a reflection in the mirror line $y=x$?

1. $E^{\mathrm{\prime }}(-5,3)$
2. $E^{\mathrm{\prime }}(5,-3)$
3. $E^{\mathrm{\prime }}(-3,5)$
4. $E^{\mathrm{\prime }}(3,5)$

Show answer

$E^{\mathrm{\prime }}(-5,3)$. Reflection in the diagonal line $y=x$ swaps the x and y coordinates of any point $(x,y)\to (y,x)$. Therefore, $E(3,-5)$ maps directly to $E^{\mathrm{\prime }}(-5,3)$.

Slide 2 · Reflecting Shapes Across Horizontal, Vertical, and Linear LinesModel · 14 min

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• All 3 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 10 lessons in this sub-strand

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All 11 lessons in Reflection and Congruence

1. 1Identifying Lines of Symmetry in Plane FiguresIdentify lines of symmetry in plane figures
2. 2Identifying Lines of Symmetry in Plane FiguresIdentify lines of symmetry in plane figures
3. 3Properties of Reflection in Plane GeometryDetermine the properties of reflection in different situations
4. 4Properties of Reflection in Plane GeometryDetermine the properties of reflection in different situations
5. 5Reflecting Shapes on Plane Surfaces and Cartesian PlanesDraw an image given an object and a mirror line on plane surface and Cartesian plane
6. 6Reflecting Shapes on the Cartesian PlaneDraw an image given an object and a mirror line on plane surface and Cartesian plane
7. 7Determining the Equation of a Mirror LineDetermine the equation of the mirror line given an object and its image
8. 8Determining the Equation of a Mirror LineDetermine the equation of the mirror line given an object and its image
9. 9Direct and Opposite Congruence in Plane FiguresCarry out congruence tests for triangles
10. 10Carrying Out Congruence Tests for TrianglesCarry out congruence tests for triangles
11. 11Real-Life Applications of Reflection, Line Symmetry, and CongruencePromote use of reflection and congruence in real-life situations