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Reflecting 2D shapes

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visibility 188update 5 months agobookmarkshare

🎯 In this topic you will

Sketch the reflection of a two-dimensional shape on a grid.

🧠 Key Words

mirror line
reflection

Show Definitions

mirror line: A line that acts as a boundary so that a shape and its image are the same distance from it on opposite sides.
reflection: A transformation that flips a shape over a mirror line to produce a reversed image.

Reflecting Shapes Accurately

I n this section you will improve the accuracy of how you reflect shapes over a mirror line by using a grid of squares.

Creating Patterns and Designs 🎨

W hen you know how to reflect a shape, you can create interesting patterns and designs.

❓ EXERCISES

$1.$ Sketch this shape and the mirror line on plain paper. Draw the reflection of the shape over the mirror line.

👀 Show answer

To draw the reflection, copy the shape on the opposite side of the mirror line so that each point of the image is the same perpendicular distance from the mirror line as the original shape. The reflected shape should be the same size and shape, but flipped vertically across the mirror line.

📘 Worked example

Reflect this shape over the mirror line on the grid.

This is a vertical mirror line.

The vertex A is two squares from the mirror line. Its reflection will be two squares from the mirror line, on the other side of the mirror.

The vertex B is also two squares from the mirror line. Its reflection will be two squares from the mirror line, on the other side of the mirror.

The vertex C is three squares from the mirror line. Its reflection will be three squares from the mirror line, on the other side of the mirror.

Join the vertices to make the reflection of the whole shape.

Answer:

To reflect a shape in a vertical mirror line, count the number of squares from each vertex to the mirror line. Place each reflected vertex the same distance from the mirror line on the opposite side, then join the points in the same order.

❓ EXERCISES

2. Draw each shape and mirror line on squared paper.

Reflect each shape by counting the squares between the vertices and the mirror line.

👀 Show answer

To reflect each shape, count the horizontal or vertical distance in squares from each vertex to the mirror line, then place the corresponding reflected vertex the same distance on the opposite side of the mirror line. Join the reflected vertices in the same order.

3. When a shape is reflected over a mirror line that is along one of its edges, the original shape and the reflected shape make a new shape together.

Example:

The parallelogram and the reflected parallelogram combined have made a hexagon.

a. For each shape predict what the new combined shape will be after the reflection.

b. Sketch and reflect the shapes to check your predictions.

👀 Show answer

Reflecting a shape along one of its edges produces a combined shape that is symmetrical along that edge. For example, a square reflected along one side forms a rectangle, a triangle reflected along its base forms a kite, and other polygons combine to form larger symmetric polygons depending on their original shape.

4. Raj has reflected this pentagon over the vertical mirror line. He has drawn red lines between each original vertex and its reflected vertex. Describe the red lines.

👀 Show answer

The red lines are horizontal, perpendicular to the vertical mirror line, and each one is bisected by the mirror line. This means the mirror line cuts each red line exactly in half at a right angle.

5. Draw your own shape and a horizontal mirror line on squared paper. Reflect the shape over the mirror line. Draw lines between each original vertex and its reflected vertex. Describe the lines.

👀 Show answer

The lines joining each vertex to its reflection are vertical, perpendicular to the horizontal mirror line, and each line is cut exactly in half by the mirror line.

6. The vertices of this rectangle have coordinates $(0,2)$, $(0,5)$, $(1,2)$ and $(1,5)$. Conjecture what will happen then investigate to find the coordinates of the vertices of the shape when it is reflected over the mirror line.

👀 Show answer

When reflected over the vertical mirror line at $x = 3$, the $x$-coordinates change symmetrically. The new vertices are $(6,2)$, $(6,5)$, $(5,2)$ and $(5,5)$.

🧠 Think like a Mathematician

a. With your small group, think carefully about the following question:

Does a reflected shape have a greater, smaller or the same area as the original shape?

b. Investigate this question by reflecting rectangles.

c. Make a poster showing your conclusions. Include a generalisation about what you have found out.

d. Swap your poster with another group and assess the understanding shown in their poster.

Has the group answered the question set in the investigation?
Has the group used diagrams to illustrate or demonstrate their answer?
Has the group written a convincing explanation?
What advice can you give to help the group improve their poster?

👀 Show answer

A reflected shape always has the same area as the original shape. Reflection changes the position and orientation of a shape but does not change its size. This is true for all shapes, including rectangles, triangles, and irregular shapes.

📘 What we've learned

We learned how to reflect a $2$D shape over a mirror line on a grid.
We learned that each vertex of a shape is reflected the same perpendicular distance from the mirror line on the opposite side.
We practiced reflecting shapes using both vertical and horizontal mirror lines.
We learned that a reflection changes the position and orientation of a shape but keeps its size and area the same.
We learned that the line joining a point and its reflection is perpendicular to the mirror line and is bisected by it.

Reflecting 2D shapes

🎯 In this topic you will

🧠 Key Words

Reflecting Shapes Accurately

Creating Patterns and Designs 🎨

❓ EXERCISES

❓ EXERCISES

🧠 Think like a Mathematician

📘 What we've learned

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