Similar triangles, criteria for similarity, areas, and Pythagoras theorem
Similar figures have the same shape but not necessarily the same size. All corresponding angles are equal and corresponding sides are proportional.
Important: All congruent figures are similar, but all similar figures are NOT congruent.
• Photographs enlarged or reduced — same shape, different size
• A map and the actual country — similar shapes at different scales
• All circles are similar (but not congruent unless same radius)
• All squares are similar (but not all rectangles)
Two triangles are similar (△ABC ~ △DEF) if:
(i) Corresponding angles are equal: ∠A = ∠D, ∠B = ∠E, ∠C = ∠F
(ii) Corresponding sides are proportional: AB/DE = BC/EF = CA/FD
Note: Both conditions automatically imply each other for triangles.
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
If DE ∥ BC in △ABC, then: AD/DB = AE/EC
If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
If AD/DB = AE/EC, then DE ∥ BC
Q: In △ABC, DE ∥ BC. If AD = 3, DB = 5, and AE = 4.5, find EC.
By BPT: AD/DB = AE/EC
3/5 = 4.5/EC → EC = (4.5 × 5)/3 = 22.5/3 = 7.5
| Criterion | Full Name | Condition |
|---|---|---|
| AA | Angle-Angle | If 2 angles of one Δ = 2 angles of another, they are similar |
| SSS | Side-Side-Side | If 3 pairs of corresponding sides are proportional |
| SAS | Side-Angle-Side | If 2 sides proportional AND included angle equal |
Q: In △ABC and △PQR, ∠A = 60°, ∠B = 70°, ∠P = 60°, ∠Q = 70°. Are they similar?
∠A = ∠P = 60° and ∠B = ∠Q = 70°
By AA criterion: △ABC ~ △PQR ✓
Note: ∠C = 180° – 60° – 70° = 50° = ∠R (third angle also equal)
Q: Check if △ABC with sides 3,4,5 is similar to △PQR with sides 6,8,10.
AB/PQ = 3/6 = 1/2 | BC/QR = 4/8 = 1/2 | CA/RP = 5/10 = 1/2
All ratios equal → By SSS criterion: △ABC ~ △PQR
The ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides.
Area(△ABC) / Area(△DEF) = (AB/DE)² = (BC/EF)² = (CA/FD)²
Q: △ABC ~ △DEF. If AB = 6 and DE = 4, find the ratio of their areas.
Area(△ABC) / Area(△DEF) = (AB/DE)² = (6/4)² = (3/2)² = 9/4
So area of △ABC is 9/4 times the area of △DEF.
In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.
If ∠B = 90° in △ABC, then: AC² = AB² + BC²
If in a triangle, the square of one side equals the sum of squares of the other two sides, then the angle opposite the first side is a right angle.
Q: A ladder 10 m long leans against a wall. If the foot of the ladder is 6 m from the wall, how high up the wall does the ladder reach?
The ladder, wall, and ground form a right triangle.
Hypotenuse = 10 m, Base = 6 m, Height = ?
h² = 10² – 6² = 100 – 36 = 64 → h = 8 m
Answer: The ladder reaches 8 m up the wall.
DE ∥ BC → AD/DB = AE/EC
Area₁/Area₂ = (side₁/side₂)²
AC² = AB² + BC²
(∠B = 90°)
AA / SAS / SSS