Chapter Summary: Triangles

A one-page recap of every key idea, theorem and formula in this chapter. Read this the night before the exam.

1. Similar Figures

  • Two figures are similar if they have the same shape (not necessarily the same size).
  • Two triangles are similar if: (a) their corresponding angles are equal, and (b) their corresponding sides are in the same ratio (proportional).
  • All congruent figures are similar, but all similar figures need not be congruent.

2. Basic Proportionality Theorem (BPT / Thales)

  • If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides them in the same ratio: if DE∥BCDE \parallel BC then ADDB=AEEC\dfrac{AD}{DB} = \dfrac{AE}{EC}.
  • Converse: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side.

3. Criteria for Similarity

  • AA (or AAA): two pairs of corresponding angles equal ⇒ similar.
  • SSS: all three pairs of corresponding sides proportional ⇒ similar.
  • SAS: one pair of angles equal and the sides including them proportional ⇒ similar.

4. Areas of Similar Triangles

  • The ratio of areas of two similar triangles equals the square of the ratio of any pair of corresponding sides (also of corresponding medians, altitudes, angle bisectors, perimeters): ar(△ABC)ar(△PQR)=(ABPQ)2=(BCQR)2=(CARP)2.\frac{\text{ar}(\triangle ABC)}{\text{ar}(\triangle PQR)} = \left(\frac{AB}{PQ}\right)^2 = \left(\frac{BC}{QR}\right)^2 = \left(\frac{CA}{RP}\right)^2.

5. Pythagoras Theorem

  • In a right triangle, hypotenuse2=base2+height2\text{hypotenuse}^2 = \text{base}^2 + \text{height}^2.
  • Converse: if in a triangle the square of one side equals the sum of squares of the other two, the angle opposite the first side is a right angle.
  • Useful Pythagorean triples: (3,4,5)(3,4,5), (5,12,13)(5,12,13), (8,15,17)(8,15,17), (7,24,25)(7,24,25), (9,40,41)(9,40,41) and their multiples.

Common mistakes to avoid

  • Don't forget to name the criterion (AA/SSS/SAS) when proving similarity.
  • Area ratio is the square of the side ratio — students often forget to square.
  • Match corresponding vertices in the right order when writing △ABC∼△PQR\triangle ABC \sim \triangle PQR.

Final tip: Most board questions need just BPT, one similarity criterion, and Pythagoras. Master those three and you cover the chapter.