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Class 10 · Chapter 6 · Mathematics

Triangles Cheat Sheet

Formulae

Ratio of areas of similar triangles

[△ABC][△PQR]=(ABPQ)2=(BCQR)2=(CARP)2\frac{[\triangle ABC]}{[\triangle PQR]} = \left(\frac{AB}{PQ}\right)^{2} = \left(\frac{BC}{QR}\right)^{2} = \left(\frac{CA}{RP}\right)^{2}

The ratio of the areas of two similar triangles equals the square of the ratio of any pair of corresponding sides.

Pythagoras theorem

AC2=AB2+BC2AC^{2} = AB^{2} + BC^{2}

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Theorems

Basic Proportionality Theorem (Thales theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

ADDB=AEEC\frac{AD}{DB} = \frac{AE}{EC}

In triangle ABC, DE ∥ BC with D on AB and E on AC.

Converse of the Basic Proportionality Theorem

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 in triangle ABC, then DE ∥ BC.

AAA similarity criterion

If in two triangles the corresponding angles are equal, then their corresponding sides are proportional and the triangles are similar.

∠A=∠P, ∠B=∠Q, ∠C=∠R⇒△ABC∼△PQR\angle A = \angle P,\ \angle B = \angle Q,\ \angle C = \angle R \Rightarrow \triangle ABC \sim \triangle PQR

Two equal angles (AA) are enough, because the third pair is then automatically equal.

SSS similarity criterion

If in two triangles the corresponding sides are proportional, then their corresponding angles are equal and the triangles are similar.

ABPQ=BCQR=CARP⇒△ABC∼△PQR\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP} \Rightarrow \triangle ABC \sim \triangle PQR

All three pairs of corresponding sides must be in the same ratio.

SAS similarity criterion

If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the two triangles are similar.

ABPQ=ACPR, ∠A=∠P⇒△ABC∼△PQR\frac{AB}{PQ} = \frac{AC}{PR},\ \angle A = \angle P \Rightarrow \triangle ABC \sim \triangle PQR

The equal angle must lie between the two proportional sides.

Converse of Pythagoras theorem

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

This is used to test whether a triangle is right-angled.

Definitions

Similar figures
Figures that have the same shape but not necessarily the same size.
Similar triangles
Triangles whose corresponding angles are equal and corresponding sides are proportional.
Corresponding sides
Sides of two similar figures that occupy the same relative position.
Equiangular triangles
Triangles in which corresponding angles are equal.
Congruent figures
Figures that have the same shape and the same size; congruent triangles are always similar.
Scale factor
The ratio of a side of one similar figure to the corresponding side of the other.
Hypotenuse
The side opposite the right angle in a right-angled triangle; the longest side.

Key Points

  • All congruent figures are similar, but all similar figures need not be congruent.
  • Two polygons are similar if their corresponding angles are equal and their corresponding sides are proportional.
  • The Basic Proportionality Theorem and its converse relate parallel lines to proportional division of sides.
  • Similarity can be proved using the AA (AAA), SSS or SAS criteria.
  • The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
  • In a right-angled triangle, the Pythagoras theorem gives hypotenuse² = base² + perpendicular².
  • The converse of the Pythagoras theorem is used to decide whether a triangle is right-angled.