Triangles
Similar figures
Decide when two shapes are similar.
Print the same photo of the Taj Mahal small, then large. The two prints have the same shape but different sizes. In maths they are similar.
| Word | Same shape? | Same size? | Example |
|---|---|---|---|
| congruent | yes | yes | two copies of the same coin |
| similar | yes | not necessarily | a photo and its enlargement |
Every pair of congruent figures is similar, but similar figures need not be congruent.
Always similar: any two circles, any two squares, any two equilateral triangles. Whatever their size, they have exactly the same shape.
Not automatically similar: a small triangle and a big triangle. Same size isn't the point either. A photo of you at 10 and another at 40, printed the same size, are not similar: the shape of your face has changed.
Maps and building plans use similarity. Everything is shrunk by the same scale factor, so the shape stays true.
The test for two polygons with the same number of sides. They are similar if both of these hold:
- (i) the corresponding angles are equal, and
- (ii) the corresponding sides are in the same ratio.
You really need both. A square and a long rectangle have all angles 90°, but their sides are not in proportion, so they are not similar. A square and a rhombus have all sides in proportion, but their angles differ, so again they are not similar.
(For triangles only, one condition brings the other with it. More on that in a later lesson.)
Practise this lesson
Questions that get harder as you improve, a Tutor that helps without giving the answer away, and progress you can see. Free for every learner.
No account, no email address. Opens the real product.