Polygons & Coordinate Transformations
Match Polygon to Feature
Draw a line to match each polygon to its key feature.
Regular or Irregular?
Sort each polygon into the correct column.
Polygon Properties
Circle the correct answer.
A regular polygon has 5 sides. Each interior angle = ?
How many lines of symmetry does a square have?
A polygon with all interior angles less than 180° is called:
Sum of interior angles of a quadrilateral:
Coordinate Transformations
Draw a line to match each transformation to its description.
Transformations on a Grid
Circle the correct new coordinates.
Point (3, 2) reflected in the y-axis
Point (1, 4) translated right 2, down 3
Point (2, 3) rotated 180° about the origin
Describe and Draw Transformations
Answer each question about transformations.
A square has vertices at (1,1), (3,1), (3,3), (1,3). Translate it right 4 and up 2. Write the new coordinates.
Describe the transformation that maps (2, 5) to (−2, 5).
Name two real-life objects that have rotational symmetry and state the order of symmetry.
Interior Angle Sum Formula
The sum of interior angles of a polygon = (n − 2) × 180°, where n is the number of sides.
Sum of interior angles of a pentagon (5 sides)
Sum of interior angles of a hexagon (6 sides)
Each interior angle of a regular octagon (8 sides)
A polygon has interior angle sum of 1080°. How many sides?
Lines of Symmetry
Draw a line to match each shape to the number of its lines of symmetry.
Rotational Symmetry Order
Sort each shape by its order of rotational symmetry.
Reflections and Translations on a Grid
Apply the transformation and choose the correct result.
Triangle with vertices (2,1),(4,1),(3,3) reflected in the x-axis. New vertex from (3,3)?
Point (−2, 4) translated right 5 and down 3. New coordinates?
Point (4, −2) reflected in the y-axis. New coordinates?
Symmetry Investigation
Answer each question about symmetry in shapes.
Draw a shape that has exactly 2 lines of symmetry but no rotational symmetry beyond order 2. Label the lines of symmetry.
A regular polygon has 10 sides. How many lines of symmetry does it have? What is its order of rotational symmetry?
Shapes and Symmetry Exploration
Look for polygons and symmetry in your surroundings.
- 1Find 5 real-life polygons at home or outside (e.g. tiles, windows, signs). Name each polygon and count its sides and angles.
- 2Fold a piece of paper to find lines of symmetry in a shape you cut out. How many lines of symmetry does it have?
- 3Design a pattern using only regular polygons. Which polygons tessellate (tile without gaps)?
- 4Plot a simple triangle on grid paper, then draw its reflection over the x-axis and its translation 3 units right. Label all vertices.
Classifying Quadrilaterals
Use properties of sides, angles, and diagonals to classify each quadrilateral.
A quadrilateral has all sides equal and all angles equal. What is it? Is it also a parallelogram? Is it also a rhombus?
A quadrilateral has opposite sides parallel but no right angles and all sides equal. What is it?
Explain why every square is a rectangle but not every rectangle is a square.
Coordinates — Plotting Points
Circle the correct coordinates of the described point.
4 right and 3 up from the origin
2 left and 5 down from the origin
On the y-axis, 7 units above the origin
Plotting and Connecting Points
Plot and connect points to form named shapes.
Plot the points A(0,0), B(4,0), C(4,3), D(0,3) on a coordinate grid. What shape do ABCD form? Calculate the perimeter and area.
Plot A(1,1), B(5,1), C(4,4), D(2,4). Join them in order. What quadrilateral does this appear to be? Explain how you can verify using the properties of that shape.
Transformations — Reflection
Describe or find the image of each reflection.
Point A is at (3, −4). What are its coordinates after reflection in: (a) the x-axis? (b) the y-axis?
Triangle with vertices (1,2),(4,2),(2,5) is reflected in the y-axis. Write the new vertices.
Tessellation Investigation
Investigate which shapes tessellate.
Explain why regular hexagons tessellate but regular pentagons do not. (Hint: consider the interior angle and what angles must add to 360° at each vertex.)
Which regular polygons can tessellate on their own? List all of them and explain why.
Lines of Symmetry
Circle the correct number of lines of symmetry.
Equilateral triangle
Regular hexagon
Rectangle (not square)
Rhombus (not square)
Midpoint and Distance on a Coordinate Plane
Find midpoints and distances between points.
Find the midpoint of the segment joining A(2, 4) and B(8, 10).
Find the midpoint of the segment joining C(−3, 1) and D(5, −7).
M is the midpoint of AB. A = (1, 3) and M = (4, 7). What are the coordinates of B?
Perimeter on a Coordinate Grid
Plot the shapes and calculate their perimeters.
A triangle has vertices A(0,0), B(4,0), and C(0,3). Calculate the perimeter. (Hint: find the length of the hypotenuse using Pythagoras.)
A quadrilateral has vertices (0,0), (6,0), (6,4), (0,4). What is its perimeter and area? What type of quadrilateral is it?
Quadrant Recognition and Plotting
Answer each question about the four quadrants of the coordinate plane.
In which quadrant does each point lie? (a) (3, −2) (b) (−4, 5) (c) (−1, −8) (d) (6, 7)
Plot 4 points, one in each quadrant, that form the vertices of a rectangle. Label all coordinates.
Diagonal Properties of Quadrilaterals
Investigate the diagonals of different quadrilaterals.
For each quadrilateral below, state whether its diagonals are (a) equal length, (b) bisect each other, (c) perpendicular. Square, Rectangle, Rhombus, Parallelogram, Trapezium.
Slope (Gradient) of a Line
Circle the correct gradient. Gradient = (rise ÷ run) = (y₂ − y₁) ÷ (x₂ − x₁).
Points (0,0) and (4,8): gradient = ?
Points (1,3) and (5,7): gradient = ?
Points (2,6) and (6,2): gradient = ?
Gradient of Parallel and Perpendicular Lines
Investigate gradients of parallel and perpendicular lines.
Two lines have gradients 3 and 1/3. Multiply these gradients: 3 × (1/3) = ? What does this suggest about the lines?
A line passes through (0,1) and (4,9). What is its gradient? Write the equation of this line in the form y = mx + c.
Coordinate Patterns
A point starts at (0, 0) and moves right 1 and up 3 each step. Fill in the missing coordinates.
Classifying Triangles by Coordinates
Use coordinate geometry to classify triangles.
A triangle has vertices A(0,0), B(6,0), C(3,3). Is it scalene, isosceles, or equilateral? Justify your answer by calculating the side lengths.
Vertices D(0,0), E(4,0), F(4,4). What type of triangle is DEF? What are its angle sizes?
Real-World Coordinates — Map Reading
Apply coordinate ideas to map reading and navigation.
On a coordinate grid (1 unit = 1 km), a park is at (3, 7) and a school is at (9, 3). Find the midpoint between them. A bus stop is to be placed at this midpoint — give its coordinates.
If walking 1 km horizontally and 1 km vertically, use Pythagoras to find the straight-line distance between points (0,0) and (3,4). How does this compare to the walking distance via right-angled streets?
Design a Shape with Given Properties
Use your coordinate and geometry knowledge to design a shape.
Design a quadrilateral on a coordinate grid that is a parallelogram but NOT a rectangle. Label all four vertices, find all side lengths, and confirm opposite sides are parallel by checking gradients.
Using Coordinates to Verify a Parallelogram
Use gradients to verify a quadrilateral is a parallelogram.
Quadrilateral with vertices A(0,0), B(4,1), C(6,5), D(2,4). Calculate the gradient of AB, BC, CD, and DA. Which pairs are parallel? What quadrilateral type is ABCD?
Pythagorean Distance Between Points
Use Pythagoras to find the distance between two points.
Find the distance between A(0,0) and B(3,4).
Find the distance between P(1,1) and Q(5,4).
Two towns are at (2,3) and (8,11) on a map (1 unit = 10 km). What is the straight-line distance between them?
Quadrilateral Properties
Match each quadrilateral to its distinguishing property.
Congruence and Transformations
Connect transformations to congruence.
Explain why translations, rotations, and reflections all preserve congruence.
Shape A has vertices (1,2),(3,2),(3,5),(1,5). Shape B has vertices (5,−1),(7,−1),(7,2),(5,2). Are they congruent? Describe the transformation.
Vector Notation for Translations
Use vector notation to describe translations.
Describe each translation as a vector: (a) right 3, up 5 (b) left 4, down 2 (c) no horizontal movement, up 6
A triangle is translated by [−3, 4]. Vertex A was at (2, −1). Where is A'?
Shape P → Q by [5,−3]. Shape Q → R by [−2,6]. What single vector maps P to R?
Design a Tiling Pattern
Create a tessellation using coordinate transformations.
A regular polygon tessellates if its interior angle divides evenly into 360°. Calculate for 3, 4, 5, 6, 7, and 8 sided regular polygons. Which ones tessellate?
Vector Addition
Circle the correct result.
Translation [3,2] then [−1,4] = combined vector?
Point (4,3) after translation [−6,−3] = ?
Point (−2,5) after translation [4,−7] = ?
Coordinate Geometry — Final Challenge
Solve this multi-step coordinate geometry problem.
Quadrilateral ABCD: A(0,0), B(6,0), C(8,4), D(2,4). (a) Find the midpoints of diagonals AC and BD. (b) Do the diagonals bisect each other? (c) What type of quadrilateral is ABCD? (d) Find the area of ABCD.
Distance Patterns — Growing Right Triangles
A right triangle has one leg = n and the other leg = n+1. Fill in the hypotenuse lengths (to 1 decimal).
Symmetry in Coordinates
Explore symmetry using coordinate geometry.
A shape has vertices A(2,3), B(5,3), C(5,7), D(2,7). Find the line of symmetry of this shape. Write its equation. What type of shape is ABCD?
A triangle has vertices (0,0), (4,0), (2,4). Find all lines of symmetry. How many does it have?
Enlargement and Scale Factor
Investigate enlargement as a transformation.
A triangle has vertices A(1,1), B(3,1), C(2,3). It is enlarged by scale factor 2 from the origin. Write the new vertices A', B', C'.
Explain why enlargement does NOT produce a congruent shape but DOES produce a similar shape.
Scale Factor and Similar Shapes
Investigate similar shapes using scale factors.
Rectangle A is 3 cm × 5 cm. Rectangle B is 6 cm × 10 cm. What is the scale factor from A to B? Are they similar? Are they congruent?
Triangle P has sides 4, 6, 8 cm. Triangle Q has sides 6, 9, 12 cm. What is the scale factor from P to Q? Find the ratio of their areas.
Map Scale Problems
Apply scale factor to map reading problems.
A map has scale 1:50 000. A distance of 4 cm on the map represents what actual distance in km?
Two cities are 120 km apart. On a map with scale 1:2 000 000, how far apart are they in cm?
Similar Shapes — Circle Correct
Circle the correct answer.
Two similar rectangles: the smaller is 3×2 cm. Scale factor = 4. Larger is:
Small triangle area = 9 cm². Scale factor = 3. Large triangle area = ?
Two similar pentagons: side ratio 2:5. Perimeter of smaller = 20 cm. Perimeter of larger = ?
Connecting Transformations to Similarity
Explain the connection between enlargement and similarity.
Explain why an enlargement from the origin always produces a similar shape (same angles, proportional sides). How is this different from a translation, rotation, or reflection?
Golden Ratio and Polygons
Investigate the golden ratio in pentagons.
A regular pentagon has side length 1 cm. The diagonal has length approximately 1.618 cm. What is the ratio diagonal:side? What is this ratio called?
Research: where does the golden ratio appear in nature? Name at least 2 examples.
Coordinates on a Growing Grid
A square grows each step. Its bottom-left corner stays at (0,0) and its side increases by 1. Give the top-right vertex coordinates.
Geometry Investigation — Regular vs Irregular
Compare regular and irregular polygons.
A regular octagon has side length 5 cm. What are its interior and exterior angles? What is its angle sum? A kite also has 4 sides — what are possible interior angles for a kite?
Describe two ways you could determine whether an irregular quadrilateral is a parallelogram using only a ruler and coordinate methods.
3D Coordinates Preview
Extend coordinate ideas to three dimensions.
A rectangular box has one corner at (0,0,0) and the opposite corner at (4,3,2). How long is the main diagonal? (Hint: use 3D Pythagoras: d = √(x²+y²+z²).)
In 3D coordinates, the point (2,5,3) means 2 right, 5 forward, 3 up. What point is equidistant from (0,0,0) and (4,0,0) and lies on the z-axis?
Scale Factor Problems
Circle the correct answer.
Map scale 1:100 000. A road is 3 cm on the map. Real length = ?
Scale factor 5. Original area = 4 cm². New area = ?
Two similar shapes: scale factor 3:1. Perimeter of larger = 24 cm. Smaller perimeter = ?
Sort by Interior Angle
Sort these regular polygons from smallest to largest interior angle.
Interior Angle Sequence
Fill in the missing interior angle for each regular polygon.
Regular Polygons — Sides vs Angle Sum
Read the picture graph showing angle sums for different polygons, then answer the questions.
| Triangle (3) | |
| Quad (4) | |
| Pentagon (5) | |
| Hexagon (6) |
If each icon = 90°, what is the angle sum of a triangle?
How does the angle sum increase as the number of sides increases?
Types of Quadrilaterals Survey
Count the quadrilateral types found in a school room, then answer questions.
| Item | Tally | Total |
|---|---|---|
Square | ||
Rectangle | ||
Rhombus | ||
Other |
Match Transformation to Effect
Draw a line to match each transformation to what it does to coordinates.
Mixed Polygon Questions
Circle the correct answer.
A regular polygon has interior angle 144°. How many sides?
Interior angle sum of a 7-gon:
Exterior angle of a regular pentagon: