Reflecting Shapes on Plane Surfaces and Cartesian Planes

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

1. Construct reflected images of plane figures by drawing perpendicular lines to a mirror line and measuring equal distances.
2. Determine the image of line segments intersecting mirror lines at given angles.
3. Reflect geometric figures on a Cartesian plane across vertical lines $x=k$ and horizontal lines $y=k$.

Key vocabulary

Mirror line

A fixed line across which an object is reflected to form its image.

Each point on the image sits at an equal perpendicular distance from line m.

Perpendicular line

A line that meets or crosses another line at a right angle.

We construct a line at right angles from each vertex through line m.

Invariant point

A point that does not change position after a transformation.

Points lying on line m are invariant points during reflection.

Image segment

The straight line segment formed by reflecting an object segment across a line.

The image segment forms the same angle with line m as the original segment.

Cartesian plane

A two-dimensional grid defined by perpendicular x-axis and y-axis.

We plot triangle vertices on the Cartesian plane to perform coordinate reflections.

Lesson preview

Slide 1 · Drawing Reflected Images on a Plane SurfaceOpening · 12 min

Reflecting an object on a plane surface requires locating the image of each vertex across line m. From each corner, construct a straight line at 90° to line m and project it across to the opposite side. Measure the exact perpendicular distance from the corner to line m, then mark the image point at that identical distance on the opposite side. Joining these image points creates a congruent figure. Use Figure 1 to observe how perpendicular lines help locate image points at identical distances.

• Every point and its reflected image lie on a line at right angles to line m.
• The perpendicular distance from an object point to line m equals the distance to its image.
• Reflecting an object vertex by vertex preserves all line lengths, angle measures, and shape dimensions.
• The image formed by reflection on a plane surface is congruent to the original object.

Worked example. Construct right-angled lines from corners A, B, C, and D across line m to the right side. Along each perpendicular line, measure the same distance from mirror line m as the object point: 4 cm for A', 2 cm for B', 5 cm for C', and 3 cm for D'. Mark image vertices A', B', C', D' then join them in order with straight line segments to form image quadrilateral A'B'C'D'. Since A is 4 cm to the left of line m and A' is 4 cm to the right along the perpendicular, distance AA' = 4 cm + 4 cm = 8 cm. Answer: Image vertices are located at equal perpendicular distances on the opposite side of line m; total distance AA' is 8 cm..

Check question: A point P is located 3.5 cm away from a mirror line m. Point P' is its reflected image in line m. What is the angle formed between the line segment PP' and mirror line m, and what is the length of line segment PP'?

1. Angle is ${90}^{\circ }$ and length PP' is 7 cm
2. Angle is ${90}^{\circ }$ and length PP' is 3.5 cm
3. Angle is ${180}^{\circ }$ and length PP' is 7 cm
4. Angle is ${45}^{\circ }$ and length PP' is 3.5 cm

Show answer

Angle is ${90}^{\circ }$ and length PP' is 7 cm. The line segment connecting an object point to its reflected image always meets the mirror line at a right angle (${90}^{\circ }$). Since P is 3.5 cm from the mirror line and P' is located an equal distance on the opposite side, the total length of PP' is $3.5\text{ cm}+3.5\text{ cm}=7$ cm.

Slide 2 · Reflecting Angled Line Segments and FiguresModel · 14 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• 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