Class 10 NCERT Maths Chapter 6: Triangles
1. Introduction
In this chapter, we focus on Similar Figures—shapes that have the same form but not necessarily the same size. That means they have identical angles and proportional sides, even if they’re scaled differently. (based on NCERT explanation of “similar figures”) :contentReference[oaicite:0]{index=0}
2. Similarity of Triangles
Triangles are similar if:
- Their corresponding angles are equal.
- Their corresponding sides are in the same ratio.
3. Criteria for Similarity
- AA (Angle–Angle) – Two angles of one triangle equal two of another.
- SSS (Side–Side–Side) – Corresponding sides in proportion.
- SAS (Side–Angle–Side) – One angle equal, with proportional sides around it.
4. Basic Proportionality Theorem (BPT)
If a line drawn parallel to one side of a triangle intersects the other two sides, it divides those sides proportionally. \[ \text{If } DE \parallel BC \text{ in } \triangle ABC,\ \frac{AD}{DB} = \frac{AE}{EC} \]
5. Areas of Similar Triangles
The ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = \left(\frac{AB}{DE}\right)^2 \]
6. Pythagoras’ Theorem & Converse
In a right-angled triangle: \[ \text{(Hypotenuse)}^2 = \text{(Base)}^2 + \text{(Height)}^2 \] Conversely, if this equality holds in a triangle, it must be right-angled.
7. Examples
Example 1 (BPT):
In \(\triangle ABC\), line \(DE \parallel BC\) and \(D\) is on \(AB\), \(E\) on \(AC\).
\[
\frac{AD}{DB} = \frac{AE}{EC}
\]
Example 2 (Pythagoras):
In right \(\triangle ABC\) with \(\angle B = 90^\circ\), if \(AB = 5\) cm and \(BC = 12\) cm, then:
\[
AC = \sqrt{5^2 + 12^2} = 13\ \text{cm}
\]