WEEK 4

 

6.1 Geometrical properties

6.1.2 Understand and use similarity and congruence (1)

In this week's tasks we compare shapes and try to determine whether they are similar. It turns out in most of our examples they are not!

 

Monday: This task is quite subtle. It turns out that the red and blue rectangles are not similar but the red and blue triangles are.

Many students will think the rectangles are similar. This is not surprising as they look very much 'alike'!

How can one show the rectangles are not similar? One way is to look at the differences in length between the red edges and the corresponding blue edges. It turns out that the difference is the same (10 cm) for the horizontal edges as for the vertical edges. But as the horizontal edges are longer, their difference would have to be more than for the vertical edges for the shapes to be similar.

Another approach is to imagine using the same wooden strips to keep putting a new frame around the existing frame - as the frames get larger, they look more and more ‘square’ since, for both the red and the blue rectangles, the difference the horizontal and vertical edges stays the same but the relative difference gets smaller. We consider this situation in the next task.

A third approach would be to seek a centre of enlargement. As corresponding edges are parallel, there should be one if the shapes are similar - but it turns out that the lines going through corresponding points don't meet at a single point.

For the triangular frame, the red and blue triangles are similar. Corresponding angles are equal for the two shapes, which is a sufficient condition for triangles to be similar - though this doesn't hold for other polygons. [Notice that this time the difference in length between corresponding horizontal edges is not the same as for corresponding vertical edges (or, indeed, corresponding slanting edges).]

Tuesday: Here we keep increasing the horizontal and vertical edges by the same, fixed amount. Do students discern that as the rectangles get larger, their shapes gets closer to that of a square? And if they do, do they appreciate that this means the rectangles are not similar?

 

Wednesday: Here we compare shapes whose sides are not parallel. This makes it difficult to discern whether corresponding angles are equal - though we might be able to spot the corresponding pair of right angles; as we know in the case of triangles, equal corresponding angles would be sufficient to determine that the shapes are similar. 

It is also difficult to see whether corresponding sides are in proportion. However, some students might spot that within each triangle, one of the sides containing the right angle is twice the length of the other. Knowing two sides and the included angle (SAS) of a triangle defines a triangle uniquely; knowing the ratio of two sides and knowing the included angle, defines the triangle's shape uniquely, though not its size. So the two triangles are similar.

Thursday: Here two pairs of corresponding sides are parallel, but not the third pair - so the triangles are not similar, though they may, at first sight, look fairly alike.

Friday: Here corresponding pairs of sides are parallel so corresponding angles will be equal. However, this is a necessary but not sufficient condition for similarity when shapes have more than three sides.

So  how might we tell whether the two shapes are similar? As corresponding sides are parallel it is fairly easy to compare lenghts. Notice, for example, that the blue top-right side is twice the length of the red top-right side. Is this true for other corresponding sides?

Or we could look for a centre of enlargement, as corresponding sides are parallel. Do lines through corresponding points meet at a single point?