Triangles: Important Concepts for CBSE Class 10
The chapter 'Triangles' in Class 10 Mathematics primarily deals with the concept of **similarity of triangles** and related theorems, along with the **Pythagoras Theorem**. Mastering these concepts is crucial for solving geometry problems.
1. Similar Figures
- Congruent Figures: Same shape and same size.
- Similar Figures: Same shape but not necessarily the same size.
Two polygons of the same number of sides are similar if:- All corresponding angles are equal.
- All corresponding sides are in the same ratio (proportional).
- All congruent figures are similar, but the converse is not true.
2. Similarity of Triangles
Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.
Criteria for Similarity of Triangles:
- AAA (Angle-Angle-Angle) Similarity: If in two triangles, corresponding angles are equal, then their corresponding sides are proportional, and hence the triangles are similar. (AA similarity is a corollary: if two angles are equal, the third must be equal).
- SSS (Side-Side-Side) Similarity: If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal, and hence the triangles are similar.
- SAS (Side-Angle-Side) Similarity: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
3. Basic Proportionality Theorem (BPT) / Thales Theorem
- Statement: 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.
In $\\triangle ABC$, if $DE \\parallel BC$, then $\\frac{AD}{DB} = \\frac{AE}{EC}$. - Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
4. Areas of Similar Triangles
- Theorem: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
If $\\triangle ABC \\sim \\triangle PQR$, then $\\frac{Area(\\triangle ABC)}{Area(\\triangle PQR)} = \\left(\\frac{AB}{PQ}\\right)^2 = \\left(\\frac{BC}{QR}\\right)^2 = \\left(\\frac{CA}{RP}\\right)^2$.
5. Pythagoras Theorem
- Statement: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In $\\triangle ABC$ right-angled at $B$, $AC^2 = AB^2 + BC^2$. - Converse of Pythagoras Theorem: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
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