Chapter in a nutshell: Two triangles are similar if they have the same shape — equal corresponding angles and proportional corresponding sides. The key results are the similarity criteria (AA, SSS, SAS), the Basic Proportionality (Thales) Theorem, and the fact that the ratio of areas of similar triangles = the square of the ratio of corresponding sides.
1. Key Ideas & Definitions
- Similar triangles ($\sim$): corresponding angles equal and corresponding sides in the same ratio.
- Congruent is a special case of similar with ratio 1 (same size and shape).
- Used in maps, models and scale drawings (scale factor = ratio of corresponding lengths).
2. Similarity Criteria (theorems)
- AA (AAA): two angles of one triangle equal two angles of the other → similar.
- SSS: all three pairs of corresponding sides proportional → similar.
- SAS: one angle equal and the including sides proportional → similar.
3. Key Theorems
- Basic Proportionality Theorem (BPT / Thales): a line drawn parallel to one side of a triangle divides the other two sides in the same ratio: if $DE\parallel BC$, then $\dfrac{AD}{DB}=\dfrac{AE}{EC}$.
- Converse of BPT: if a line divides two sides in the same ratio, it is parallel to the third side.
- Area theorem: the ratio of the areas of two similar triangles equals the square of the ratio of any pair of corresponding sides (or medians, or altitudes).
4. Formulas (quick reference — all of them)
- For $\triangle ABC\sim\triangle PQR$: $\dfrac{AB}{PQ}=\dfrac{BC}{QR}=\dfrac{CA}{RP}=k$ (scale factor).
- Ratio of areas: $\dfrac{\text{ar}(ABC)}{\text{ar}(PQR)}=\left(\dfrac{AB}{PQ}\right)^2=k^2$.
- Ratio of perimeters / medians / altitudes = ratio of corresponding sides = $k$.
- (Prerequisite) Mid-point theorem: the segment joining the mid-points of two sides is parallel to and half the third side.
- (Prerequisite) Pythagoras theorem: in a right triangle, $\text{hyp}^2=\text{base}^2+\text{height}^2$.
- (Maps/models) real length = map length × scale; area scales by (scale)².
5. Worked Example (one, for the method)
Q. Two similar triangles have corresponding sides 3 cm and 5 cm. Find the ratio of their areas. Solution: Ratio of areas = $(3/5)^2 = 9/25$.6. Common Mistakes to Avoid
- Using a side ratio for the area ratio — areas go as the square of the side ratio.
- Matching the wrong corresponding sides/angles (write the similarity with vertices in order).
- Confusing similar (same shape) with congruent (same shape and size).
- Forgetting BPT needs the line parallel to a side.
7. Likely Exam Questions (with crisp answers)
- When are two triangles similar? → Equal corresponding angles and proportional corresponding sides.
- State the AA criterion. → Two angles of one equal two angles of the other.
- State the Basic Proportionality Theorem. → A line parallel to one side divides the other two sides in the same ratio.
- State the area theorem for similar triangles. → Ratio of areas = square of the ratio of corresponding sides.
- Ratio of areas if sides are in ratio 2:3? → 4:9.
- State the converse of BPT. → If a line divides two sides proportionally, it is parallel to the third side.
- Difference between similar and congruent? → Similar = same shape; congruent = same shape and size.
- Ratio of perimeters of similar triangles equals? → The ratio of corresponding sides.
- State the mid-point theorem. → The line joining mid-points of two sides is parallel to and half the third side.
- If areas are 16:25, what is the side ratio? → 4:5.
Extended & Prerequisite Formulas (full)
- Criteria AA, SSS, SAS; corresponding sides ratio = k; areas ratio = k²; perimeters/medians/altitudes ratio = k.
- BPT: DE∥BC ⇒ AD/DB=AE/EC; converse holds.
- Prereq: Pythagoras hyp²=base²+height²; mid-point theorem (segment = ½ third side, parallel); area of triangle ½·b·h.