Animated Diagram — Triangles

Scale factor k = 1.5

💡 The two triangles are similar — same shape, different size. Drag the slider to change the scale factor k. Corresponding angles stay equal; corresponding sides multiply by k. When k = 1 the triangles are congruent.

Position of DE = 0.5

💡 DE is drawn parallel to BC. As you slide DE up and down, notice that AD/DB always equals AE/EC — that is the Basic Proportionality Theorem.

Three ways to prove two triangles are similar

📐

AA (or AAA)

∠A=∠P, ∠B=∠Q

Two pairs of equal angles → similar (third angle follows).

📏

SSS

AB/PQ = BC/QR = CA/RP

All three side-pairs in the same ratio → similar.

🔗

SAS

∠A=∠P & AB/PQ=AC/PR

Equal angle with proportional including sides → similar.

ar(ABC) / ar(PQR) = (AB/PQ)²
Ratio of areas = square of ratio of corresponding sides

Area / Side Ratio Converter

side 1

side 2

Enter the ratio of corresponding sides to get the ratio of areas (and vice-versa).

Reverse: Areas → Side Ratio

area 1

area 2

Enter two areas of similar triangles to get the ratio of their sides.

c² = a² + b²
In a right triangle, hypotenuse² = sum of squares of the legs

side a = 3

side b = 4

💡 Click any real-life situation to see how triangles solve it.

🌳Heights & Shadows

Pattern: A short object and a tall object cast shadows at the same time → similar triangles.
height / shadow = constant Example: 6 m pole → 4 m shadow. Tower → 28 m shadow.
6/4 = h/28 → h = 42 m

🪜Ladder against a Wall

Pattern: Ladder, wall and ground form a right triangle.
ladder² = height² + base² Example: 10 m ladder reaches 8 m high.
base = √(10²−8²) = 6 m

🌉Width of a River

Pattern: Construct a triangle on the bank similar to one spanning the river.
unknown width = known side × ratio Example: Similar triangles with ratio 1:3 and near side 15 m → width = 45 m.

🗼Distance Between Pole Tops

Pattern: Height difference & horizontal gap form a right triangle.
d = √(gap² + (h₂−h₁)²) Example: Poles 6 m & 11 m, 12 m apart.
d = √(12² + 5²) = 13 m

🗺️Maps & Scale Drawings

Pattern: A map is similar to the real region.
actual = map distance × scale factor Example: 1 cm on map = 50 km. 6.5 cm → 325 km.

📐Diagonal of a Rectangle

Pattern: The diagonal splits a rectangle into two right triangles.
diagonal² = length² + breadth² Example: 12 cm × 5 cm rectangle.
diagonal = √(144+25) = 13 cm