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Reflection and Congruence

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 10 Pathway: N/A

First Principles

Objective: Perform reflections across vertical, horizontal, and axis mirror lines on a coordinate plane, and verify geometric congruence between original and image figures.

Interactive Reflection & Congruence Visualizer

Drag the mirror line position or toggle axis to watch shape reflection & vertex coordinates update live!

A A' x=k

(a) Concrete Scenario: Imagine standing in front of a flat mirror or observing a reflection of a building on a still lake in Naivasha. Every point on the physical object corresponds to a twin point in the reflected image, sitting at an identical perpendicular distance on the opposite side of the mirror line.

(b) Key Properties of Reflection:

  • Perpendicular Bisector: The mirror line is the perpendicular bisector of the line segment connecting any original point \(P\) to its image \(P'\).
  • Isometry (Distance Preserved): Reflection preserves side lengths, perimeter, area, and internal angles.
  • Geometric Congruence: Because all side lengths and angles remain identical, the image figure is strictly congruent to the original figure (\(\triangle ABC \cong \triangle A'B'C'\)).
  • Reversed Orientation: Reflection flips orientation (handedness)—clockwise vertices become counter-clockwise.

(c) Algebraic Mapping Rules: \[\text{Reflect across } y\text{-axis } (x=0): \quad (x, y) \implies (-x, y)\] \[\text{Reflect across } x\text{-axis } (y=0): \quad (x, y) \implies (x, -y)\] \[\text{Reflect across vertical line } x=k: \quad (x, y) \implies (2k - x, y)\] \[\text{Reflect across horizontal line } y=k: \quad (x, y) \implies (x, 2k - y)\]

Pro-Tip: Congruence means identical size and shape! Reflection changes orientation/location, but NEVER changes perimeter or area.

Key Formulas

\((x, y) \xrightarrow{\text{reflect } x=k} (2k - x, y)\) — Formula for reflection across vertical mirror line \(x = k\).
\((x, y) \xrightarrow{\text{reflect } y=k} (x, 2k - y)\) — Formula for reflection across horizontal mirror line \(y = k\).
\(\text{Distance}(P, \text{line}) = \text{Distance}(P', \text{line})\) — Equal perpendicular distance to the mirror line.
\(\triangle ABC \cong \triangle A'B'C' \implies A'B' = AB, \; B'C' = BC, \; C'A' = CA\) — Corresponding sides of congruent figures are equal.
\(\text{Perimeter}(\text{Image}) = \text{Perimeter}(\text{Original}), \quad \text{Area}(\text{Image}) = \text{Area}(\text{Original})\)

Worked Examples

Example 1 (Easy - Reflection Distance): A town matatu stop at point \(A(2, -4)\) is reflected across the \(y\)-axis to a new location \(A'\). What is the straight-line distance between \(A\) and \(A'\)?
  1. Mirror line is the \(y\)-axis (line \(x = 0\)).
  2. Apply reflection mapping: \((x, y) \implies (-x, y)\). Point \(A(2, -4) \implies A'(-2, -4)\).
  3. The points share the same \(y\)-coordinate (\(-4\)).
  4. Distance = \(|x_2 - x_1| = |2 - (-2)| = 2 + 2 = 4\) units.
  5. Answer: \(4\) units.
Example 2 (Medium - Perimeter of Reflected Triangle): Right-angled triangle \(ABC\) has vertices \(A(0,0)\), \(B(3,0)\), and \(C(0,4)\). It is reflected across the vertical line \(x = 5\) to form triangle \(A'B'C'\). Find the perimeter of \(A'B'C'\).
  1. Since reflection is an isometry, the image \(A'B'C'\) is congruent to \(ABC\).
  2. Find side lengths of original \(\triangle ABC\):
    Leg \(AB = |3 - 0| = 3\).
    Leg \(AC = |4 - 0| = 4\).
    Hypotenuse \(BC = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5\).
  3. Perimeter of \(ABC = 3 + 4 + 5 = 12\).
  4. By congruence, \(\text{Perimeter}(A'B'C') = \text{Perimeter}(ABC) = 12\).
  5. Answer: \(12\).
Example 3 (Hard - Distance Between Parallel Reflected Shapes): A rectangular plot (shamba) measures \(12\text{ m}\) by \(8\text{ m}\). It is reflected across a line parallel to its \(12\text{ m}\) side, located \(5\text{ m}\) away from the nearest edge of the plot. What is the shortest distance between the original plot and its reflected image?
  1. Let the nearest edge of the rectangle sit at distance \(d = 5\text{ m}\) from the mirror line.
  2. Upon reflection, the image edge will sit at an equal distance of \(5\text{ m}\) on the opposite side of the mirror line.
  3. Total separation distance between nearest edges = \(5\text{ m} + 5\text{ m} = 10\text{ m}\).
  4. Answer: \(10\text{ m}\).

Common Mistakes

Mistake Expecting reflection across a mirror line to change the size, perimeter, or area of a geometric shape.
Correction Reflection is a rigid transformation (isometry). It changes position and orientation, but preserves side lengths, angles, and area. Original and image are strictly congruent!
Mistake Shifting both \(x\) and \(y\) coordinates when reflecting across a vertical line \(x = k\).
Correction Across a vertical mirror line, ONLY the \(x\)-coordinate changes (\(x' = 2k - x\)); the \(y\)-coordinate remains unchanged!

Real World

Architectural Symmetry: Gateways and church facades across Nairobi use vertical line symmetry where the left wing is a reflected congruent copy of the right wing.
Bilateral Biological Symmetry: Animal and human skeletal structures show bilateral reflection symmetry across the sagittal plane.
Woodworking & Furniture Joinery: Carpenters cut congruent mirror-image components for cabinet doors and window frames.

Practice

On a town map drawn on a coordinate grid, a matatu stop is located at point (2, -4). The stop is reflected across the y-axis to a new location. What is the distance between the original stop and its reflected location? (Type only the number, e.g., 4)
Review the concepts above.
Triangle ABC has vertices A(0,0), B(3,0), and C(0,4). It is reflected across the vertical line x = 5 to obtain triangle A'B'C'. What is the perimeter of triangle A'B'C'? (Type only the number, e.g., 12)
Review the concepts above.
A landmark point A is located 8 metres north of a straight east-west road line. The road acts as a mirror line. Point A is reflected across the road line to form point A'. What is the distance between A and A' in metres? (Type only the number, e.g., 16)
Review the concepts above.
A rectangular shamba measures 12 m by 8 m. It is reflected across a mirror line parallel to its 12-m side that is 5 m away from the rectangle. What is the shortest distance between the original rectangle and its reflected image in metres? (Type only the number, e.g., 10)
Review the concepts above.
Omondi, a carpenter, cuts two congruent triangular wooden boards for a shelf. The first triangle has interior angles measuring 40 degrees and 60 degrees. What is the measure of the third angle of the second triangle in degrees? (Type only the number, e.g., 80)
Review the concepts above.
A point P(3, 7) is reflected across the horizontal mirror line y = 4 to form point P'. What is the y-coordinate of point P'? (Type only the number, e.g., 1)
Review the concepts above.