Triangles Class 10 ||Maths|| Chapter 6 NCERT Notes

1. Similar Figures

Two figures are said to be similar if they have the same shape, though their sizes may be different.
Examples:
- All circles are similar.
- All squares are similar.
- Two polygons are similar if their corresponding angles are equal, and their corresponding sides are in proportion.

2. Similarity of Triangles

Triangles are similar if:

Their corresponding angles are equal.
Their corresponding sides are proportional.

This leads us to the AA (Angle-Angle) similarity criterion:

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

3. Basic Proportionality Theorem (Thales Theorem)

Statement: If a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those two sides in the same ratio.

Let’s say:

In triangle ABC, DE is parallel to BC and intersects AB at D and AC at E.
According to the Basic Proportionality Theorem:
$\frac{A D}{D B} = \frac{A E}{E C}$

This theorem helps in proving the similarity of triangles and solving many geometrical problems.

4. Criteria for Similarity of Triangles

There are three main criteria for determining if two triangles are similar:

AA (Angle-Angle) Criterion:
If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
SSS (Side-Side-Side) Criterion:
If the corresponding sides of two triangles are in proportion, then the triangles are similar.
$\frac{A B}{D E} = \frac{B C}{E F} = \frac{C A}{F D}$
SAS (Side-Angle-Side) 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 triangles are similar.
$\frac{A B}{D E} = \frac{A C}{D F} and ∠ A = ∠ D$

5. Areas of Similar Triangles

If two triangles are similar, the ratio of the areas of the two triangles is equal to the square of the ratio of their corresponding sides.

If △ABC ∼ △DEF, then:

$\frac{Area of △ A B C}{Area of △ D E F} = {(\frac{A B}{D E})}^{2}$

6. Pythagoras Theorem

The Pythagoras theorem is a fundamental relation in Euclidean geometry among the three sides of a right-angled triangle.

Statement: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

For triangle ABC (right-angled at B):

$A C^{2} = A B^{2} + B C^{2}$

This theorem is also used to solve problems involving right triangles in various geometrical scenarios.

7. Converse of Pythagoras Theorem

Statement: 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.
If AC² = AB² + BC², then ∠B = 90°.

This is used to prove if a triangle is a right-angled triangle.

8. Application of Similarity in Right Triangles

When a perpendicular is drawn from the right angle of a right-angled triangle to the hypotenuse, the two triangles formed are similar to the whole triangle and to each other.

Consider triangle ABC where ∠B = 90°, and BD is drawn perpendicular to AC.

△ABD ∼ △ABC
△BDC ∼ △ABC

9. Theorem on Right-Angled Triangles (Pythagorean Triplet)

If a, b, and c are three positive integers such that:

$c^{2} = a^{2} + b^{2}$

Then a, b, c form a Pythagorean triplet. For example, (3, 4, 5), (6, 8, 10), and (5, 12, 13) are common Pythagorean triplets.

10. Important Formulae

Pythagoras Theorem:
$H y p o t e n u s e^{2} = B a s e^{2} + P e r p e n d i c u l a r^{2}$
Ratio of Areas of Similar Triangles:
$\frac{Area of first triangle}{Area of second triangle} = {(\frac{corresponding sides}{corresponding sides})}^{2}$
Basic Proportionality Theorem (Thales Theorem):
If DE is parallel to BC, then:
$\frac{A D}{D B} = \frac{A E}{E C}$

Important Theorems and Their Applications

Basic Proportionality Theorem helps in solving problems where triangles have parallel lines.
Pythagoras Theorem is applied in problems involving right-angled triangles.
Similarity Criteria (AA, SSS, SAS) are used to prove the similarity of triangles in various geometrical figures.

Solved Example Problems

Let’s look at one example problem to illustrate the application of these theorems.

Example 1:

In triangle ABC, DE is parallel to BC. If AD = 3 cm, DB = 4 cm, and AE = 2.4 cm, find EC.

Solution: By the Basic Proportionality Theorem (BPT):

$\frac{A D}{D B} = \frac{A E}{E C}$

Substitute the known values:

$\frac{3}{4} = \frac{2.4}{E C}$

Cross-multiply:

$E C = \frac{4 \times 2.4}{3} = \frac{9.6}{3} = 3.2 cm$

Thus, EC = 3.2 cm.

Chapter Highlights:

Similarity of triangles is a major concept, with three important criteria: AA, SSS, SAS.
Basic Proportionality Theorem (Thales’ Theorem) is essential for solving problems involving proportional sides.
Pythagoras Theorem is vital for right-angled triangles, and its converse helps in proving if a triangle is right-angled.

These notes summarize the key points from Chapter 5: Triangles of Class 10 NCERT Maths. If you need further explanations, detailed examples, or specific theorem proofs, let me know!

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