Congruence & Similarity

Two pages from the photocopier line up edge to edge perfectly; the outline of a country in a textbook picture and on a wall map differ hugely in size yet match in shape perfectly. The first relationship is congruence, the second similarity — the two most important kinds of "sameness" in geometry.

Congruent: able to coincide exactly

Two figures are congruent if one can be slid, turned (and flipped if needed) to land exactly on the other: same shape and same size. For triangles, three pairs of equal sides are enough — lock the three sides and the shape cannot budge, the very rigidity of triangles at work. Congruent figures are copies from the same stamp: corresponding angles equal, corresponding sides equal.

Similar: same shape, different size

Two figures are similar when their shapes match but their sizes may differ. The rules are clean:

All corresponding angles are equal;
All corresponding sides change by the same multiplier, called the scale factor.

Scale the sides by $k$ and the perimeter scales by $k$ too — but the area scales by $k^2$ , because length and width both change at once.

A triangle scaled by 3

3, 4, 5 scaled up ×3

A right triangle with sides 3, 4, 5 (since $3^2+4^2=5^2$ ) is scaled by a factor of 3. Its sides become 9, 12, 15.

Perimeter: originally $3+4+5=12$ , now $9+12+15=36$ — exactly $12 \times 3$ .

Area: originally $\frac{3 \times 4}{2}=6$ , now $\frac{9 \times 12}{2}=54$ — exactly $6 \times 9$ ; the area grew by $3^2=9$ times!

InteractiveScaling Lab

How to use

Leg a3Leg b4

3² + 4² = 9 + 16 = 25 = c²

Hypotenuse c c = √25 ≈ 5.00整数勾股数 · Pythagorean triple!

Drag the two legs: whatever you do, it stays a right triangle — the shape holds; but the sides, perimeter and area all follow your hand — the size changes. That is similarity in its most visible form.

Maps, models and blueprints

Similarity is everywhere: a 1:100000 map shrinks the real world 100000 times, so 1 cm on paper means 1 km on the ground; scale models, part drawings and photo resizing all rely on the same rule — corresponding angles equal, corresponding sides proportional. Hold on to it, add the craft of ratios, and you can convert freely between worlds of different sizes.

Check yourself

Quick quiz

0 / 3 correct0 / 3 correct

1. Two figures are congruent. What does that guarantee?
2. A triangle with sides 3, 4, 5 is scaled by 3. What is the longest side now?
3. A figure is enlarged by a factor of 3. Its area becomes