Change the scale factor (ratio) k to grow or shrink a triangle. Side lengths grow exactly by the scale factor, but area grows by the square of the scale factor. And when the scale factor becomes exactly 1:1, the two triangles overlap completely — that relationship is called congruence.
Enlarge or shrink a photo and the shape stays the same, right? A relationship like that — same shape, different size — is called similarity. And when the scale factor becomes exactly 1:1, so even the size matches, that's called congruence — in other words, congruence is a special case of similarity.
What's interesting about similarity is how the area changes. Double the side lengths and the area doesn't double — it becomes 4 times (2²) as large; triple them and the area becomes 9 times (3²). Length scales with the ratio, but area scales with the ratio squared.
Change the scale factor k with the slider and compare, in numbers, exactly how side length and area each change by a different multiple.
Even for the same shape (similar), when the sizes differ, the ratio of side lengths (perimeter ratio) equals the scale factor, but the ratio of areas (area ratio) becomes the scale factor squared. Only when the scale factor is exactly 1 do the shapes match in both shape and size — congruence.
Satisfy just one of these three conditions and two triangles are similar. △ABC is the reference triangle, and △DEF is a triangle built with scale factor k. Change the values and press "overlap" to check that the two triangles are, in the end, the same shape.
At the same moment, under the same sun, a stick and a tree both make the same angle with their shadows. So the triangle formed by the stick and the triangle formed by the tree are AA similar, and setting up a proportion lets you find the tree's height without ever measuring it directly.
The two words can be confusing, but the core idea is simple — congruence is a special case of similarity (exactly when the scale factor is 1).
| Congruence | Similarity | |
|---|---|---|
| Meaning | Shape and size are exactly the same | Only shape matches; size can differ |
| Corresponding angles | Equal | Equal |
| Corresponding side lengths | Equal (ratio = 1:1) | Constant ratio (ratio = 1:k) |
| Ratio of perimeters | 1:1 | 1:k |
| Ratio of areas | 1:1 | 1:k² |
| Triangle conditions | SSS · SAS · ASA · RHS · RHA | AA · SAS (similarity) · SSS (similarity) |
| Symbol | △ABC ≡ △DEF | △ABC ∽ △DEF |