Congruent and Similar Shapes in Patterns and Designs

1. Congruent shapes

Two shapes are congruent if they have exactly the same shape and the same size. Imagine cutting one out and laying it on the other: if it fits exactly, they are congruent. They can be in different positions, turned round or even flipped over, as long as every matching side and every matching angle are equal.

Congruence comes from three transformations, which you have met. A translation (slide), a reflection (flip) and a rotation (turn) all keep every length and angle the same. So the image of a shape under any of these is congruent to the object.

▶ Triangle ABC and its reflection in the y-axis. Check: AB = 3, AC = 4 and BC = 5 on both triangles.

▶ A turn of 90° about the origin: the triangle has been turned but its sides are still 3, 4, 5.

2. Similar shapes and the scale factor

Two shapes are similar if one is an enlargement (or reduction) of the other. They have the same shape but not always the same size. For similar shapes: (1) all matching angles are equal; (2) all matching sides are in the same ratio. That ratio is the scale factor.

▶ Enlargement of triangle ABC by scale factor 2 from the origin. Notice that every side is doubled: 2 becomes 4, and the angles stay the same.

In the animation, AB = 2 units and A′B′ = 4 units, so the scale factor is $4÷2=2$. The sides AC and A′C′ show the same: $4÷2=2$. Because the angles are equal and the sides are in the same ratio, the triangles are similar.

1. 1List the corresponding sides in the same order (for example, shortest to shortest).
2. 2Divide each side of the second shape by the matching side of the first.
3. 3If all the answers are the same, the shapes are similar and that answer is the scale factor.
4. 4If they are different, the shapes are not similar (unless they are not the same type of shape at all).

Areas: if the lengths are multiplied by the scale factor $k$, the area is multiplied by $k^2$. In the animation $k=2$, so the area is multiplied by $4$. For a scale factor of 3, the area grows 9 times.

3. Verifying on the coordinate plane

On the coordinate plane you can verify congruence and similarity by working out lengths from the coordinates. A horizontal side from (1, 1) to (4, 1) has length $4-1=3$. A vertical side from (1, 1) to (1, 5) has length $5-1=4$. A slanted side is the longest side of a right-angled triangle, so use the 3-4-5 pattern or Pythagoras: here, $\sqrt{3^2+4^2}=5$.

Take triangle ABC with A(1, 1), B(4, 1) and C(1, 5). It has sides AB = 3, AC = 4, BC = 5 and a right angle at A. Compare it with these triangles:

▶ Enlargement by scale factor 3. Notice that the lines through the centre and each point pass through the image points; A(1,1) → A′(3,3).

How to verify congruence: calculate all the sides and compare them (SSS), check that the matching angles are equal, and make sure the shapes can be matched by a slide, a flip or a turn. How to verify similarity: compare matching angles and divide matching sides to see that the ratios are all the same.

4. Congruence and similarity in real life

Carpets and fabric: A carpet maker in Kumasi weaves the same diamond many times across a mat, and every diamond is congruent to the others. A smaller copy of the diamond woven in the border is similar to the main diamond, with a scale factor such as $\frac{1}{2}$. Kente cloth patterns are built from repeated shapes in different sizes.

Plans and photos: an architect's plan of a school block is similar to the real building: all lengths are in the same ratio, such as $1:200$. A passport photo and its enlargement keep the same shape, and a photocopier set to 150% has scale factor 1.5. Because the area changes by $k^2$, a photo enlarged with scale factor 1.5 uses $1.5^2=2.25$ times the paper.

▶ A square tile repeated by a translation: all copies are congruent. The pattern is built by sliding.

Worked examples

Try each one yourself first, then open the solution and compare every step.