Solved Exercises on Reflections in High School Geometry

Master reflections: line of reflection, coordinate rules, geometric properties, and applications through these 5 detailed exercises.

Rules and methods, laws,...
\(r_{line}(x,y)\)
Reflection Notation
\((x,y) → (x, -y)\) (x-axis)
Common Reflection Rule
Reflection
Rigid Transformation
Preserves distance, angles
Line of Reflection
Mirror line
Perpendicular bisector
Distance Property
Equal from mirror
Point and image equidistant
Key definitions:

Reflection: A rigid transformation that flips a figure across a line (called the line of reflection), creating a mirror image. Every point and its image are equidistant from the line of reflection.

Line of Reflection: The line across which the figure is reflected. Also called the mirror line or axis of symmetry.

Pre-image: The original figure before the transformation.

Image: The figure after the transformation, denoted with prime notation (A' for transformed point A).

Reflection Methods:
  1. Identify the line of reflection: Determine the mirror line
  2. Apply coordinate rules: Use specific rules for common reflection lines
  3. Verify equidistance: Check that points are equidistant from the line
  4. Graph the result: Plot the reflected figure
  5. Confirm properties: Verify the transformation is rigid
Tip 1: Points on the line of reflection don't move!
Tip 2: Distance from point to mirror = distance from image to mirror.
Tip 3: Remember common reflection rules for axes.
Tip 4: Reflection reverses orientation (handedness).
Solution: Exercises 1 to 3
1 Reflection Across X-Axis
Exercise 1
Point P(3, -2) is reflected across the x-axis. Find the coordinates of P'.
Definition:

Reflection Across X-axis: The transformation rule is (x, y) → (x, -y). This reflects the point across the horizontal line y = 0.

Reflection Method:
  1. Identify the line of reflection: x-axis (y = 0)
  2. Apply the rule: (x, y) → (x, -y)
  3. Substitute original coordinates
  4. Calculate new coordinates
Original Point
P(3, -2)
Line of Reflection
x-axis (y = 0)
Reflected Point
P'(3, 2)
Step 1: Identify the reflection rule

For reflection across x-axis: (x, y) → (x, -y)

Step 2: Apply the rule to point P(3, -2)

Substitute x = 3 and y = -2 into the rule (x, y) → (x, -y)

Step 3: Calculate the new coordinates

P(3, -2) → P'(3, -(-2)) = P'(3, 2)

New x-coordinate: x = 3 (unchanged)

New y-coordinate: -y = -(-2) = 2

Step 4: Verify equidistance from x-axis

Distance from P to x-axis: |−2| = 2 units

Distance from P' to x-axis: |2| = 2 units

Distances are equal ✓

P'(3, 2)
Distance from x-axis preserved: 2 units
Final answer:

The coordinates of the reflected point are P'(3, 2). The point is equidistant from the x-axis as the original point.

Applied rules:

• X-axis Rule: (x, y) → (x, -y)

• Equidistance Property: Point and image equidistant from mirror line

• Rigidity: Reflections preserve all distances

2 Reflection Across Y-Axis
Exercise 2
Triangle ABC has vertices A(1, 2), B(4, 0), C(2, -3). Reflect across the y-axis. Find coordinates of A'B'C'.
Definition:

Reflection Across Y-axis: The transformation rule is (x, y) → (-x, y). This reflects the point across the vertical line x = 0.

Pre-image
A(1,2), B(4,0), C(2,-3)
Line of Reflection
y-axis (x = 0)
Image
A'(-1,2), B'(-4,0), C'(-2,-3)
Step 1: Apply y-axis reflection rule to vertex A

Rule: (x, y) → (-x, y)

A(1, 2) → A'(-1, 2)

Step 2: Apply y-axis reflection rule to vertex B

B(4, 0) → B'(-4, 0)

Step 3: Apply y-axis reflection rule to vertex C

C(2, -3) → C'(-2, -3)

Step 4: Verify equidistance from y-axis for one vertex

Distance from A to y-axis: |1| = 1 unit

Distance from A' to y-axis: |-1| = 1 unit

Distance preserved ✓

Step 5: Confirm the transformation is rigid

Since all vertices maintain the same distance from the y-axis, the transformation is rigid

Triangle A'B'C' is congruent to triangle ABC

A'(-1, 2), B'(-4, 0), C'(-2, -3)
Triangle A'B'C' ≅ Triangle ABC
Final answer:

The coordinates of the reflected triangle are A'(-1, 2), B'(-4, 0), and C'(-2, -3). The reflection preserved the shape and size of the triangle.

Applied rules:

• Y-axis Rule: (x, y) → (-x, y)

• Uniform Reflection: Apply same rule to all points

• Congruence: Original and reflected figures are congruent

3 Reflection Across Origin
Exercise 3
Point Q(-2, 5) is reflected across the origin. Find the coordinates of Q'.
Definition:

Reflection Across Origin: The transformation rule is (x, y) → (-x, -y). This reflects the point across the point (0, 0), which is equivalent to a 180° rotation about the origin.

Original Point
Q(-2, 5)
Line of Reflection
Origin (0, 0)
Reflected Point
Q'(2, -5)
Step 1: Identify the reflection rule

For reflection across origin: (x, y) → (-x, -y)

Step 2: Apply the rule to point Q(-2, 5)

Substitute x = -2 and y = 5 into the rule (x, y) → (-x, -y)

Step 3: Calculate the new coordinates

Q(-2, 5) → Q'(-(-2), -5) = Q'(2, -5)

New x-coordinate: -x = -(-2) = 2

New y-coordinate: -y = -5

Step 4: Verify with distance preservation

Distance from origin to Q: √[(-2)² + 5²] = √[4 + 25] = √29

Distance from origin to Q': √[2² + (-5)²] = √[4 + 25] = √29

Distance preserved ✓

Step 5: Note equivalence to rotation

Reflection across origin is equivalent to 180° rotation about origin

Same result: (x, y) → (-x, -y)

Q'(2, -5)
Distance from origin preserved: √29
Final answer:

The coordinates of the reflected point are Q'(2, -5). The reflection preserved the distance from the origin, confirming this is a rigid transformation.

Applied rules:

• Origin Rule: (x, y) → (-x, -y)

• Distance Preservation: Distance from origin remains constant

• Equivalence: Same as 180° rotation about origin

Solution: Exercises 4 to 5
4 Reflection Across a Line y = x
Exercise 4
Point R(4, 1) is reflected across the line y = x. Find the coordinates of R'.
Definition:

Reflection Across Line y = x: The transformation rule is (x, y) → (y, x). This reflects the point across the diagonal line where x and y coordinates are swapped.

Original Point
R(4, 1)
Line of Reflection
y = x
Reflected Point
R'(1, 4)
Step 1: Identify the reflection rule

For reflection across line y = x: (x, y) → (y, x)

Step 2: Apply the rule to point R(4, 1)

Substitute x = 4 and y = 1 into the rule (x, y) → (y, x)

Step 3: Calculate the new coordinates

R(4, 1) → R'(1, 4)

New x-coordinate: y = 1

New y-coordinate: x = 4

Step 4: Verify equidistance from line y = x

Distance from R(4,1) to line y = x: |4-1|/√(1² + (-1)²) = 3/√2

Distance from R'(1,4) to line y = x: |1-4|/√(1² + (-1)²) = 3/√2

Distances are equal ✓

Step 5: Verify midpoint property

The midpoint of RR' lies on the line y = x

Midpoint: ((4+1)/2, (1+4)/2) = (5/2, 5/2)

Since 5/2 = 5/2, the midpoint is on y = x ✓

R'(1, 4)
Distance from line y = x preserved: 3/√2
Final answer:

The coordinates of the reflected point are R'(1, 4). The reflection preserved the distance from the line y = x, confirming this is a rigid transformation.

Applied rules:

• Line y=x Rule: (x, y) → (y, x)

• Equidistance Property: Point and image equidistant from mirror line

• Midpoint Property: Midpoint of point and image lies on mirror line

5 Reflection and Distance Preservation
Exercise 5
Points A(1, 0) and B(0, 1) are reflected across the x-axis to A' and B'. Verify that the distance between A and B equals the distance between A' and B'.
Definition:

Rigid Transformation Property: Reflections are rigid transformations, meaning they preserve distances between points. The distance between any two points in the pre-image equals the distance between their images.

Original Points
A(1,0), B(0,1)
Line of Reflection
x-axis (y = 0)
Reflected Points
A'(1,0), B'(0,-1)
Step 1: Find coordinates of A' and B'

For reflection across x-axis: (x, y) → (x, -y)

A(1, 0) → A'(1, -0) = A'(1, 0)

B(0, 1) → B'(0, -1) = B'(0, -1)

Step 2: Calculate distance AB using distance formula

AB = √[(0-1)² + (1-0)²] = √[(-1)² + 1²] = √[1 + 1] = √2

Step 3: Calculate distance A'B' using distance formula

A'B' = √[(0-1)² + (-1-0)²] = √[(-1)² + (-1)²] = √[1 + 1] = √2

Step 4: Compare distances

Distance AB = √2 units

Distance A'B' = √2 units

AB = A'B' ✓

Step 5: Verify individual distance preservation

Distance from x-axis to A: |0| = 0

Distance from x-axis to A': |0| = 0

Distance from x-axis to B: |1| = 1

Distance from x-axis to B': |-1| = 1

Individual distances to mirror preserved ✓

Step 6: Confirm rigid transformation property

Since all distances are preserved, the reflection is a rigid transformation

This confirms that reflections preserve all geometric properties

AB = A'B' = √2 units
Reflection preserves distances
Final answer:

The distance between A and B is √2 units, and the distance between A' and B' is also √2 units. This verifies that reflection preserves distances, confirming it is a rigid transformation.

Applied rules:

• Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

• Rigidity Property: Reflections preserve all distances

• Verification: Compare distances before and after transformation

Key Concepts: Reflections

Reflection Fundamentals

📊
Common Reflection Rules

For common lines of reflection:

Across x-axis: (x,y) → (x, -y)
Across y-axis: (x,y) → (-x, y)
Across y=x: (x,y) → (y, x)
Across origin: (x,y) → (-x, -y)
Line of Reflection Properties

Key properties of the mirror line:

Perpendicular bisector of segment connecting point and image

Points on line remain fixed

Properties Preserved

Reflections preserve all geometric properties:

• Distance from mirror line

• Distance between points

• Angle measures

• Parallelism

• Congruence

Reflection Process
1
Identify Mirror Line
2
Apply Rule
3
Calculate New Coordinates
4
Verify Equidistance
5
Graph Result
Reflection Lines
y = 0 (x-axis)
x = 0 (y-axis)
y = x
y = -x
x = h (vertical)
y = k (horizontal)
Rigid Transformation
Preserves Distance
Preserves Angles
Preserves Shape
Preserves Size
Creates Congruence
Reflection = Flip!

Questions & Answers

Question: How do I remember the different reflection rules? They seem so similar!

Answer: The reflection rules follow a pattern based on which coordinate changes:

  • Across x-axis (y = 0): y-coordinate changes sign → (x, y) → (x, -y)
  • Across y-axis (x = 0): x-coordinate changes sign → (x, y) → (-x, y)
  • Across y = x: coordinates swap places → (x, y) → (y, x)
  • Across origin: both coordinates change sign → (x, y) → (-x, -y)

Remember: The coordinate that's changing corresponds to the axis or line being reflected across. For y = x, think of it as "swapping" since the line has equal x and y values.

Question: What happens when I reflect a point that's already on the line of reflection?

Answer: When you reflect a point that's on the line of reflection, the point doesn't move! This is because the point is equidistant from itself (distance = 0) and lies on the mirror line.

For example, if you reflect point P(3, 0) across the x-axis (y = 0), the result is still P(3, 0).

This is a fundamental property of reflections: all points on the line of reflection are fixed points of the transformation. This is why the line of reflection acts as a "fold" - points on the fold don't move when you fold along that line.

Question: I heard that reflections change the "orientation" of a figure. What does that mean?

Answer: Orientation refers to the "handedness" or the order in which points appear around a figure. Reflections reverse orientation, which is why they're called "improper" isometries.

For example, if you have a triangle with vertices labeled A, B, C going clockwise, after reflection, the corresponding vertices A', B', C' will go counterclockwise (or vice versa).

This is different from translations and rotations, which preserve orientation. You can think of it like reading text in a mirror - the letters appear reversed. This is why reflections are unique among rigid transformations in that they change the figure's "handedness."

Question: How do I verify that a reflection is correct? What should I check?

Answer: To verify a reflection is correct, check these three things:

  1. Equidistance Property: The original point and its image must be the same distance from the line of reflection
  2. Perpendicular Property: The line segment connecting the point and its image must be perpendicular to the line of reflection
  3. Midpoint Property: The midpoint of the segment connecting the point and its image must lie on the line of reflection

For example, if reflecting point P across line l to get P', measure the distance from P to l and from P' to l - they should be equal. Then check that PP' is perpendicular to l and that the midpoint of PP' is on l.