How shapes can be proportionally scaled, when triangles are "the same shape", and the beautiful patterns of symmetry.
AA / SSS / SAS SimilarityBasic Proportionality TheoremScale Factor kLines of Symmetry8 Board Questions
Core Concepts
Similar Figures
Two figures are similar if they have exactly the same shape but can be different sizes. All corresponding angles are equal, and all corresponding sides are in the same ratio (the scale factor k).
Think of a photo: A 4×6 print and a 8×12 enlargement of the same photo are similar — every measurement doubles (k=2), every angle stays the same, but the area becomes 4× larger.
Similarity Criteria for Triangles
AA (Angle-Angle): If two angles of one triangle equal two angles of another, the triangles are similar. (The third angle must also match since angles sum to 180°.)
SSS Similarity: If all three pairs of corresponding sides are proportional (a/d = b/e = c/f = k), the triangles are similar.
SAS Similarity: If two pairs of corresponding sides are proportional AND the included angle is equal, the triangles are similar.
Properties of Similar Triangles
If △ABC ~ △DEF with scale factor k (meaning AB/DE = BC/EF = CA/FD = k):
Ratio of perimeters = k (linear — same as sides ratio)
Ratio of areas = k² (squared — area scales with the square of linear dimensions)
Basic Proportionality Theorem (Thales' Theorem)
If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. In △ABC, if DE ∥ BC (D on AB, E on AC), then AD/DB = AE/EC.
Visualise it: Imagine railway tracks (the parallel lines) crossing two roads (the sides AB and AC). The tracks cut the roads at the same ratio, no matter how far apart the roads diverge.
The converse is also true: if DE divides AB and AC proportionally (AD/DB = AE/EC), then DE ∥ BC.
Lines of Symmetry
A line of symmetry divides a figure into two mirror-image halves. A regular polygon with n sides has exactly n lines of symmetry. An equilateral triangle has 3; a square has 4; a regular hexagon has 6.
Rotational Symmetry
A figure has rotational symmetry of order n if it looks identical after rotating by 360°/n. A regular hexagon has order 6 (rotate by 60° each time). The order equals the number of sides for regular polygons.
Key Formulas
Scale Factor
k = AB/DE = BC/EF = CA/FD
All corresponding side ratios equal k for similar triangles
Area Ratio
Area(△ABC) / Area(△DEF) = k²
Area scales as the square of the linear scale factor
Basic Proportionality Theorem
AD/DB = AE/EC (when DE ∥ BC)
Also: AD/AB = AE/AC = DE/BC = 1/k
Altitude on Hypotenuse
CD² = AD · DB (geometric mean)
In right △ABC with altitude CD to hypotenuse AB
Symmetry Count
Lines of symmetry = n (regular n-gon)
Order of rotational symmetry = n for regular n-gon
Similar Triangle Scale Explorer
Enter the three sides of Triangle 1 and adjust the scale factor to see Triangle 2. The simulator checks similarity and shows area and perimeter ratios.
Interactive Similarity Explorer
Board Questions
Q1Dhaka 2024BPT Application4 Marks
In △ABC, DE ∥ BC where D is on AB and E is on AC. If AD = 3 cm, DB = 5 cm, AE = 4.5 cm, find EC.
1
By the Basic Proportionality Theorem (Thales), since DE ∥ BC: AD/DB = AE/EC.
Prove: If two triangles are equiangular (all three pairs of corresponding angles are equal), they are similar.
1
Given: △ABC and △DEF with ∠A = ∠D, ∠B = ∠E, ∠C = ∠F. To prove: AB/DE = BC/EF = CA/FD.
2
Construction: On DE, mark point P such that DP = AB. Draw PQ ∥ EF meeting DF at Q.
3
In △DPQ: since PQ ∥ EF, by corresponding angles, ∠DPQ = ∠DEF = ∠B, and ∠D = ∠D. So △DPQ ≅ △ABC by ASA (since ∠D=∠A, DP=AB, ∠DPQ=∠B → ∠PQD = ∠C).
4
Therefore PQ = BC and DQ = AC. Since PQ ∥ EF in △DEF, by BPT: DP/PE = DQ/QF → AB/PE = AC/QF.
5
Also DP/DE = DQ/DF, so AB/DE = AC/DF. Similarly (applying the same argument for side BC), AB/DE = BC/EF = CA/FD. Hence the triangles are similar. QED ✓
Q3Chittagong 2022Scale Factor — Find Sides5 Marks
△ABC ~ △DEF. AB = 4 cm, BC = 6 cm, CA = 8 cm, DE = 6 cm. Find EF and FD.
1
Scale factor: k = DE/AB = 6/4 = 3/2 = 1.5.
2
EF corresponds to BC: EF = k × BC = 1.5 × 6 = 9 cm.
3
FD corresponds to CA: FD = k × CA = 1.5 × 8 = 12 cm.
Ans
EF = 9 cm, FD = 12 cm ✓
Q4Jessore 2025Altitude on Hypotenuse6 Marks
In right △ABC with right angle at C, CD ⊥ AB. Prove △ACD ~ △ABC and find CD if AC = 6 cm and AB = 10 cm.
1
Proof: In △ACD and △ABC: ∠A is common. ∠ACD = ∠ABC (both = 90° − ∠A, since ∠ACB = 90° and ∠ADC = 90°). By AA, △ACD ~ △ABC ✓.
2
From similarity: AC/AB = CD/BC = AD/AC, so AC² = AD × AB (geometric mean relation).
3
Find BC: BC = √(AB² − AC²) = √(100 − 36) = √64 = 8 cm.
4
From ratio AC/AB = CD/BC: CD = AC × BC / AB = 6 × 8 / 10 = 48/10 = 4.8 cm.
Ans
CD = 4.8 cm ✓
Q5Comilla 2021Area Ratio5 Marks
Two similar polygons have perimeters 24 cm and 36 cm. If the area of the smaller polygon is 48 cm², find the area of the larger polygon.
1
Ratio of perimeters (= scale factor k): k = 36/24 = 3/2.
2
Ratio of areas = k²: Area(larger)/Area(smaller) = (3/2)² = 9/4.
3
Area of larger = 48 × 9/4 = 432/4 = 108 cm².
Ans
Area of larger polygon = 108 cm² ✓
Q6Barisal 2023Shadow Problem4 Marks
A 6 m pole casts a 4 m shadow. At the same time, a nearby tree casts a 14 m shadow. Find the height of the tree.
1
At the same time of day, the sun is at the same angle. The pole and its shadow form a right triangle similar to the tree and its shadow triangle.
2
Corresponding sides are proportional: height/shadow = 6/4 = tree height/14.
3
Tree height = 6 × 14 / 4 = 84 / 4 = 21 m.
Ans
Height of tree = 21 m ✓
Q7Sylhet 2024Lines of Symmetry4 Marks
How many lines of symmetry does a regular hexagon have? Describe them and explain why.
1
A regular hexagon has 6 equal sides and 6 equal interior angles (each 120°).
2
Type 1 — Vertex to vertex (3 lines): Lines passing through opposite vertices. Each line connects vertex 1 to vertex 4, vertex 2 to vertex 5, and vertex 3 to vertex 6.
3
Type 2 — Midpoint to midpoint (3 lines): Lines passing through midpoints of opposite sides. Each such line bisects one pair of opposite sides.
4
Total = 3 + 3 = 6 lines of symmetry. In general, a regular n-gon has exactly n lines of symmetry.
Ans
A regular hexagon has 6 lines of symmetry ✓
Q8Dinajpur 2022Verify BPT5 Marks
In △PQR, a line parallel to QR cuts PQ at S and PR at T. PS = 3 cm, SQ = 5 cm, PT = 4.5 cm. Verify BPT and find TR.