NCERT CBSE Class 10th Mathematics Chapter 6: Triangles

Safalta Expert Published by: Sylvester Updated Wed, 22 Jun 2022 06:45 PM IST

Highlights

NCERT CBSE Class 10th Mathematics Chapter 6: Triangles

The sixth chapter in Mathematics textbook is 'Triangles'.


Similar Figures

  • Two figures having the same shape but not necessary the same size are called similar figures.
  • All congruent figures are similar but all similar figures are not congruent.


Similar Polygons

Two polygons are said to be similar to each other, if:
(i) their corresponding angles are equal, and
(ii) the lengths of their corresponding sides are proportional


Criterion for Similarity of Triangles

Two triangles are similar if either of the following three criterion’s are satisfied:
  • AAA similarity Criterion. If two triangles are equiangular, then they are similar.
  • Corollary(AA similarity). If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
  • SSS Similarity Criterion. If the corresponding sides of two triangles are proportional, then they are similar.
  • SAS Similarity Criterion. If in two triangles, one pair of corresponding sides are proportional and the included angles are equal, then the two triangles are similar.


Results in Similar Triangles based on Similarity Criterion:

  1. Ratio of corresponding sides = Ratio of corresponding perimeters

  2. Ratio of corresponding sides = Ratio of corresponding medians

  3. Ratio of corresponding sides = Ratio of corresponding altitudes

  4. Ratio of corresponding sides = Ratio of corresponding angle bisector segments.

 

The topics discussed in this chapter are as follows:

  • Types of Triangles
  • Basic Proportionality Theorem
  • Areas of Similar Triangles
  • Proof of Pythagoras Theorem

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Which triangles have the same side lengths?

Equilateral

Area of an equilateral triangle with side length a is equal to

√3/4 a2

If perimeter of a triangle is 100 cm and the length of two sides are 30 cm and 40 cm, the length of third side will be

30 cm

If triangles ABC and DEF are similar and AB=4 cm, DE=6 cm, EF=9 cm and FD=12 cm, the perimeter of triangle is

ABC ~ DEF

AB=4 cm, DE=6 cm, EF=9 cm and FD=12 cm

AB/DE = BC/EF = AC/DF

4/6 = BC/9 = AC/12

BC = (4.9)/6 = 6 cm

AC = (12.4)/6 = 8 cm

Perimeter = AB+BC+AC

= 4+6+8

=18 cm

Sides of two similar triangles are in the ratio 4: 9. Areas of these triangles are in the ratio

Let ABC and DEF are two similar triangles, such that,

ΔABC ~ ΔDEF

And AB/DE = AC/DF = BC/EF = 4/9

As the ratio of the areas of these triangles will be equal to the square of the ratio of the corresponding sides,

∴ Area(ΔABC)/Area(ΔDEF) = AB2/DE2

∴ Area(ΔABC)/Area(ΔDEF) = (4/9)2 = 16/81 = 16: 81

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