Reflection in Horizontal, Vertical and Diagonal Mirror Lines - a geometry worksheet for practicing shape reflections on grid paper.

To solve these reflection problems, we need to understand how shapes flip over a mirror line. Here is the step-by-step logic for each section:

Section A: Reflect the shapes

The rule for reflection is that every point on the shape must be the same distance from the mirror line as its corresponding point on the reflected shape. The reflected shape will look like a "mirror image."

1) Horizontal Mirror Line (Bottom)
* Analysis: The mirror line is horizontal. We reflect vertically.
* Key Points: The bottom edge of the triangle is 1 square above the line. The top tip is 4 squares above the line.
* Reflection: The new bottom edge will be 1 square *below* the line. The new top tip will be 4 squares *below* the line.
* Result: An inverted triangle pointing downwards.

2) Horizontal Mirror Line (Bottom)
* Analysis: Horizontal mirror line. Reflect vertically.
* Key Points: The lowest vertex is 1 square above the line. The highest vertex is 4 squares above the line.
* Reflection: The lowest vertex moves to 1 square below. The highest moves to 4 squares below.
* Result: The quadrilateral is flipped upside down.

3) Horizontal Mirror Line (Bottom)
* Analysis: Horizontal mirror line. Reflect vertically.
* Key Points: The flat bottom side is 1 square above the line. The top-left corner is 3 squares above.
* Reflection: The flat side moves to 1 square below. The top-left corner moves to 3 squares below.
* Result: The trapezoid is flipped upside down.

4) Vertical Mirror Line (Right)
* Analysis: The mirror line is vertical. We reflect horizontally (left/right).
* Key Points: The rightmost tip of the arrowhead is 1 square to the left of the line. The leftmost tail is 4 squares to the left.
* Reflection: The new rightmost tip will be 1 square to the *right* of the line. The new tail will be 4 squares to the *right*.
* Result: The arrowhead points to the right.

5) Vertical Mirror Line (Center)
* Analysis: Vertical mirror line. Reflect horizontally.
* Key Points: The shape crosses the line. The left part extends 2 squares left. The right part extends 1 square right.
* Reflection: The part that was on the left (2 squares wide) moves to the right. The part that was on the right (1 square wide) moves to the left.
* Result: The hexagon is flipped horizontally. The longer side is now on the right.

6) Diagonal Mirror Line (/)
* Analysis: The line goes from bottom-left to top-right.
* Rule: For this diagonal, you swap the horizontal and vertical distances relative to the line. Or simply, count squares perpendicular to the line.
* Key Points: The bottom-left corner is on the line (it stays put). The top-right corner is 3 squares "up and left" from the line diagonally? No, let's count grid units. The top-right corner is at grid position (relative to bottom-left of box) roughly (4,7). The line is $y=x$. Reflection swaps x and y.
* Simpler Method: Look at the top vertex. It is 3 squares above the line vertically. In the reflection, it should be 3 squares to the right of the line horizontally.
* Result: The parallelogram flips to the other side of the diagonal.

7) Diagonal Mirror Line (\)
* Analysis: The line goes from top-left to bottom-right.
* Key Points: The shape is a "U" or bracket. The top-left corner of the shape is 2 squares away from the line (diagonally).
* Reflection: Each point moves perpendicularly across the line by the same distance.
* Result: The shape flips across the diagonal. The opening of the "U" which faced up/left will now face down/right.

8) Diagonal Mirror Line (\)
* Analysis: Same diagonal direction as #7.
* Key Points: The triangle has one vertex on the line. The other two are away from it.
* Reflection: Flip the vertices across the line.
* Result: The triangle is mirrored across the diagonal.

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Section B: Draw the mirror line

Task: Find the line of symmetry between the blue shape and the orange shape.

Step-by-Step Deduction:
1. Identify Corresponding Points: Pick a clear corner on the blue shape and find its matching corner on the orange shape.
* Let's look at the "inner corner" where the blue and orange shapes almost touch near the center.
* Blue inner corner: Let's set the bottom-left of the grid as (0,0). The inner corner of the blue shape is at approximately $(6, 5)$.
* Orange inner corner: The matching corner on the orange shape is at approximately $(7, 4)$.
2. Find the Midpoint: The mirror line must pass exactly halfway between these points.
* Midpoint between $(6,5)$ and $(7,4)$ is $(6.5, 4.5)$.
3. Check Another Point:
* Top tip of Blue shape: approx $(7, 7)$.
* Bottom tip of Orange shape: approx $(6, 3)$? No, let's look closer.
* Let's look at the far left tip of the blue shape: $(3, 5)$.
* The corresponding far right tip of the orange shape: $(9, 3)$? No, that doesn't look symmetric.
* Let's re-evaluate the geometry visually.
* The blue shape is shifted left and up. The orange shape is shifted right and down.
* If we draw a line connecting the top-most point of the blue shape and the bottom-most point of the orange shape, the mirror line cuts perpendicularly through the middle.
* Actually, looking at the overlap, the grey square suggests they are reflections of each other across a specific axis.
* Let's trace the diagonal from top-left to bottom-right passing through the center of the grid area occupied by the shapes.
* Let's test a Diagonal Line (\) going from top-left to bottom-right.
* Take the top-left vertex of the blue shape: Grid coordinate $(3, 6)$ (assuming 10x10 grid, counting from bottom-left).
* Take the corresponding bottom-right vertex of the orange shape: Grid coordinate $(8, 1)$? No.
* Let's try a simpler visual approach.
* Point A (Blue, far left): 3 units from left edge, 5 units from bottom.
* Point A' (Orange, far right): 8 units from left edge, 3 units from bottom.
* Midpoint x: $(3+8)/2 = 5.5$. Midpoint y: $(5+3)/2 = 4$.
* Point B (Blue, top): 7 units from left, 7 units from bottom.
* Point B' (Orange, bottom): 6 units from left, 3 units from bottom? No, the orange shape bottom tip is at $(6, 2)$?
* Let's look at the central intersection. The blue shape and orange shape share a boundary along a diagonal? No, they are separate.
* Let's look at the vector between centers.
* Center of Blue approx: $(5.5, 5.5)$.
* Center of Orange approx: $(6.5, 3.5)$.
* This implies the mirror line is likely a diagonal with a negative slope (top-left to bottom-right).
* Let's verify the line passing through $(5, 5)$ and $(6, 4)$. Equation: $y = -x + 10$.
* Test Point: Blue Left Tip $(3, 5)$. Reflection across $y=-x+10$:
* New x = $10 - 5 = 5$.
* New y = $10 - 3 = 7$.
* Does the orange shape have a point at $(5, 7)$? No, the orange shape is lower down.
* Let's try the line $y = -x + 9$ (passing through $(4,5)$ and $(5,4)$).
* Test Point: Blue Left Tip $(3, 5)$.
* Reflection: $x' = 9 - 5 = 4$. $y' = 9 - 3 = 6$. Point $(4,6)$.
* Is there an orange point at $(4,6)$? No.
* Let's look really closely at Section B again.
* Blue Shape vertices: $(3,5), (5,5), (5,6), (7,6), (7,7), (3,7)$... wait, it's a polygon.
* Let's pick the "inner" corner of the blue shape (the bottom-right most point of the main body before the tab). It looks like $(6,5)$.
* The corresponding "inner" corner of the orange shape (top-left most point) looks like $(7,4)$.
* Midpoint: $(6.5, 4.5)$.
* Slope of connection: $(4-5)/(7-6) = -1/1 = -1$.
* Therefore, the Mirror Line is perpendicular to this connection. Perpendicular slope to $-1$ is $1$.
* So the mirror line has a slope of $1$ (bottom-left to top-right).
* Equation: Passes through $(6.5, 4.5)$ with slope $1$.
* $y - 4.5 = 1(x - 6.5) \Rightarrow y = x - 2$.
* Let's test this line: $y = x - 2$.
* Test Blue Point $(3, 5)$ [Left tip].
* Reflect across $y = x - 2$.
* Formula for reflection across $y=x+b$: $x' = y-b, y' = x+b$. Here $b=-2$.
* $x' = 5 - (-2) = 7$.
* $y' = 3 + (-2) = 1$.
* Result Point: $(7, 1)$.
* Does the orange shape have a vertex at $(7, 1)$? Looking at the grid, the bottom tip of the orange shape is indeed at column 7, row 1 (if we count grid lines carefully).
* Test Blue Point $(7, 7)$ [Top tip].
* $x' = 7 - (-2) = 9$.
* $y' = 7 + (-2) = 5$.
* Result Point: $(9, 5)$.
* Does the orange shape have a vertex at $(9, 5)$? The right-most tip of the orange shape is at column 9, row 5. Yes.

Conclusion for Section B: The mirror line is a diagonal line going from bottom-left to top-right. Specifically, it passes through intersections such as $(2,0), (3,1), (4,2), (5,3), (6,4), (7,5)$ etc. On the visible grid, you would draw a line starting from the bottom edge (between x=2 and x=3) going up to the right edge.

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Final Answer:

Section A Solutions:
1. Reflected Shape: An inverted triangle. The base is 1 square below the red line, and the tip points down, ending 4 squares below the red line.
2. Reflected Shape: An inverted quadrilateral. The lowest point is now 1 square below the line, and the highest point is 4 squares below the line.
3. Reflected Shape: An inverted trapezoid. The parallel side is 1 square below the line.
4. Reflected Shape: The arrowhead points to the right. The tip is 1 square to the right of the vertical line.
5. Reflected Shape: The hexagon is flipped horizontally. The wider half is now on the right side of the vertical line.
6. Reflected Shape: The parallelogram is reflected across the diagonal. The top-left vertex moves to the bottom-right relative to the line.
7. Reflected Shape: The U-shape is reflected across the diagonal. The opening faces towards the bottom-right.
8. Reflected Shape: The triangle is reflected across the diagonal.

Section B Solution:
* Mirror Line: Draw a diagonal line sloping upwards from left to right (positive slope).
* Position: The line passes through the grid intersections (3, 1), (4, 2), (5, 3), (6, 4), and (7, 5). (Assuming the bottom-left corner of the grid is 0,0). Visually, start at the bottom edge of the grid, 2 squares in from the left, and draw a line diagonally up to the right edge, 5 squares up from the bottom.