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

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

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