Key points
When transforming a shape, either through translation, reflection or rotation, a congruent shape is produced. Enlargements create shapes that are similar.
When two shapes are the same in shape and size, they are congruentTwo shapes that are identical. shapes. When two shapes are different in size but proportionally the same shape, they are similar shapesOne shape is an enlargement of another. The angles in each shape are the same, and the side lengths are in the same proportion. All squares are mathematically similar. All circles are mathematically similar. shapes.
To understand congruence and similarity, a good understanding of the properties of polygons, including triangles and quadrilaterals, can be helpful.
Congruent shapes
Two shapes are described as congruentTwo shapes that are identical. if they are identical.
- The lengths of sides (edges) and sizes of angles must be equal between the two shapes for them to be congruent.
- reflectionA reflection is a mirror image of the shape. An image will reflect through a line, known as the line of reflection. or rotationA transformation of a shape which results in a turning effect on the shape. change the orientation of a shape but they are still congruent to the original shape.
- A great way to check if two shapes are congruent is to place one on top of the other. If they match, with no overlaps, they are congruent.
Examples
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Question
Look at the grid of shapes. How many congruent pairs can you find?
Two shapes are described as congruent if they are identical.
Shapes B and M are congruent squares.
Shapes E and J are congruent rectangles. When E is rotated 90° clockwise it has the same orientation as J.
Shapes I and L are congruent triangles. When I is rotated 90° anti-clockwise it has the same orientation as L.
Similar shapes
Two shapes are described as similar shapesOne shape is an enlargement of another. The angles in each shape are the same, and the side lengths are in the same proportion. All squares are mathematically similar. All circles are mathematically similar. if one is an enlargementA transformation of a shape which results in a shape increasing or decreasing in size. of the other. The shapes do not need to be orientationThe position of a shape in relation to a coordinate system. Orientation is the way an object is angled. the same way.
The sizes of angles must be equal between the two shapes.
The shapes must also be proportionalityA relationship that is maintained between numbers. the same. If one side on the enlarged shape doubles in length, all sides must be double the original size shape. The increase in size from one shape to another is called a scale factorThe ratio between corresponding sides in an enlargement..
The shapes may need to be rotationA transformation of a shape which results in a turning effect on the shape. to find out if they are similar.
It is possible to calculate missing lengths on similar shapes when given either the scale factor or enough information to calculate it.
Examples
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Question
Which of these two triangles are similar?
Rotating triangles into the same orientation, with the right-angle in the bottom right, makes it easier to visualise which two shapes are similar.
Triangles ABC and DEF are similar.
Triangle DEF is an enlargement scale factor 2.
Line segment EF corresponds to AC.
EF = AC × 2
EF = 3 × 2
EF = 6 cm.
Similarly, line segment DE corresponds to BC.
DE = BC × 2
DE = 4 × 2
DE = 8 cm.
Practise working with congruent and similar shapes
Quiz
Practise working with congruent and similar shapes with this quiz. You may need a pen and paper to help you with your answers.
Real-life maths
Some congruent shapes can fit together without gaps. If this is possible the shape is said to tessellate.
The construction of a beehive is a real-life example of a tessellation pattern.
The hexagon shapes fit together to form the structure. The bees use this pattern as it is an efficient use of the space which involves as little wax, the material which they use, as possible.
Similar tessellation patterns can be found in the way paths, driveways and brick walls are designed.
Game - Divided Islands
Divided Islands. gameDivided Islands
Use your maths skills to help the islanders of Ichi build bridges and bring light back to the islands in this free game from BBC Bitesize.
More on Symmetry and transformations
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