CBSE · Class 10 · Mathematics · Chapter 6

Triangles — Formula Sheet

Board Formulas

14 formulas

1.Similar Polygons
$Similar ⟺ corresponding angles equal and \frac{A B}{P Q} = \frac{B C}{QR} = \dots$
For polygons with the same number of sides, BOTH conditions are needed. A square and a rectangle have equal angles but are not similar; a square and a rhombus have proportional sides but are not similar.
2.Similar Triangles (Definition)
$△ A B C \sim △ P QR ⟺ ∠ A = ∠ P, ∠ B = ∠ Q, ∠ C = ∠ R and \frac{A B}{P Q} = \frac{B C}{QR} = \frac{C A}{R P}$
The symbol ~ means 'is similar to'. The order of letters fixes the correspondence A ↔ P, B ↔ Q, C ↔ R. Congruent triangles are similar with ratio 1.
3.Basic Proportionality Theorem (Thales)★
$D E ∥ B C \Rightarrow \frac{A D}{D B} = \frac{A E}{E C}$
$D$ : Point where the parallel line cuts side AB · $E$ : Point where the parallel line cuts side AC
A line drawn parallel to one side of a triangle, cutting the other two sides at distinct points, divides those sides in the same ratio.
4.BPT — Other Useful Forms
$D E ∥ B C \Rightarrow \frac{A D}{A B} = \frac{A E}{A C}, \frac{D B}{A B} = \frac{E C}{A C}$
Part-to-whole forms that follow from AD/DB = AE/EC. Use them when a whole side (AB or AC) is given.
5.Converse of BPT★
$\frac{A D}{D B} = \frac{A E}{E C} \Rightarrow D E ∥ B C$
If a line divides two sides of a triangle in the same ratio, it is parallel to the third side. Used to PROVE two lines parallel.
6.AAA Similarity Criterion
$∠ A = ∠ P, ∠ B = ∠ Q, ∠ C = ∠ R \Rightarrow △ A B C \sim △ P QR$
If corresponding angles of two triangles are equal, their corresponding sides are proportional and the triangles are similar.
7.AA Similarity Criterion★
$∠ A = ∠ P, ∠ B = ∠ Q \Rightarrow △ A B C \sim △ P QR$
Two pairs of equal angles are enough, because the third angles are then equal by the angle sum property (180°). The most-used criterion in proofs.
8.SSS Similarity Criterion★
$\frac{A B}{P Q} = \frac{B C}{QR} = \frac{C A}{R P} \Rightarrow △ A B C \sim △ P QR$
If all three pairs of corresponding sides are in the same ratio, the corresponding angles are equal and the triangles are similar.
9.SAS Similarity Criterion★
$∠ A = ∠ P, \frac{A B}{P Q} = \frac{A C}{P R} \Rightarrow △ A B C \sim △ P QR$
One pair of equal angles AND the sides including those angles in the same ratio. A non-included angle does not work.
10.Ratio of Perimeters
$\frac{Perimeter of △ A B C}{Perimeter of △ P QR} = \frac{A B}{P Q} = \frac{B C}{QR} = \frac{C A}{R P}$
Follows directly from proportional sides: if every side is multiplied by k, so is their sum. Valid for ΔABC ~ ΔPQR.
11.Corresponding Medians and Altitudes
$△ A B C \sim △ P QR \Rightarrow \frac{A D}{P M} = \frac{A B}{P Q}$
AD and PM are corresponding medians (or corresponding altitudes). Prove it by showing ΔABD ~ ΔPQM: SAS for medians, AA for altitudes.
12.Diagonals of a Trapezium
$A B ∥ D C \Rightarrow \frac{O A}{O C} = \frac{O B}{O D} = \frac{A B}{C D}$
O is the point where diagonals AC and BD meet. ΔAOB ~ ΔCOD by AA, using alternate angles.
13.Shadows at the Same Time
$\frac{h _{1}}{s _{1}} = \frac{h _{2}}{s _{2}}$
$h_{1}, h_{2}$ : Heights of the two vertical objects (m) · $s_{1}, s_{2}$ : Lengths of their shadows (m)
Vertical objects at the same place and time: the sun's rays make the same angle with the ground, so the right triangles are similar (AA).
14.Lamp-post and Shadow
$\frac{H}{d + s} = \frac{h}{s}$
$H$ : Height of the lamp above the ground (m) · $h$ : Height of the person (m) · $d$ : Distance of the person from the foot of the lamp-post (m) · $s$ : Length of the person's shadow (m)
Large triangle: lamp-post and the ground up to the shadow's tip. Small triangle: the person and the shadow. They share the angle at the shadow's tip, so they are similar (AA).