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Triangles, the NCERT chapter that deals with the similarity of triangles. The NCERT Class 10 Maths chapter 6 notes covers an outline of the chapter triangles. The main topics covered triangles Class 10 notes, are properties for two triangles to be similar, how congruency of triangles is different from the similarity of triangles. Class 10 Math’s chapter 6 notes include frequently asked questions about the chapter. These topics can be downloaded from the triangles Class 10 notes pdf download button.
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Also, students can refer,
Triangle is a polygon having three sides and three vertices are known as Triangle.
1. There are three types of Triangles on the basis of the length of the sides.
2. Three types of Triangles on the basis of their angles.
Pythagoras theorem: In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides of the triangle.
1. Centroid of a Triangle
The point at which all the three medians of a triangle intersect is known as the centroid of that Triangle
.
2. Incenter of a Triangle
The incenter of a triangle is the point of intersection of the angle bisectors of the three angles of the triangle. It is the point from where a circle can be inscribed in the Triangle.
3. Circumcenter of the Triangle
The circumcenter of the triangle is the point of intersection of the perpendicular bisectors of the three vertices of the triangle.
Congruent | Similar | |
Angles | When Corresponding angles are the same. | Corresponding angles are the same. |
Sides | When Corresponding sides are the same. | When Corresponding sides are proportional. |
Example | ||
Explanation | Both the triangles have the same magnitude of angles and same magnitude side. | Both the triangles have the same angles but not the same sides. |
Symbols |
Two Triangles will be similar if-
Corresponding angles of the two triangles are the same
The corresponding sides of the two triangles are in the same proportion.
Two Triangles ∆ABC ~ ∆DEF as-
∠A=∠D,∠B=∠E, ∠C=∠F
AB/DE=BC/EF=AC/DF
Thales theorem says if in a given triangle a line is drawn parallel to any of the sides of the triangle so that the other two sides intersect at some distinct point then it divides the two sides in the same ratio.
In ∆KMN, if PQ║MN intersects KM at P and KN at Q, then:
i) KP/PM=KQ/QN
ii) KM/KP=KN/KQ
iii) KM/PM=KN/QN
The converse of Basic Proportionality Theorem
It is converse of basic proportionality theorem, which says if in a given Triangle a straight line divides the two sides of the Triangle in the same ratio then that straight line is parallel to the third side of that Triangle.
In KMN, if KP/PM=KQ/QN then,
PQ||MN
Criteria For The Similarity of Triangles
Three criteria to find the similarity of two Triangles are :
1. AAA (angle-Angle-Angle) criteria of similarity
Two triangles are similar to each other when any two angles of one triangle are equal to any other two angles of the other triangle.
Thus, the two Triangles are similar.
Hence, ∆ABC ~ ∆PQR
2. SSS(Side-Side-Side) criteria of similarity
When in the two triangles, all the sides of one triangle are in the same ratio with respect to the sides of the other triangle, then their corresponding angles are equal. Then the two triangles are similar.
In KMN, if KPPM=KOQN then,
PQ||MN
Criteria For The Similarity of Triangles
Three criteria to find the similarity of two Triangles are :
1. AAA (angle-Angle-Angle) criteria of similarity
Two triangles are similar to each other when any two angles of one triangle are equal to any other two angles of the other triangle.
Thus, the two Triangles are similar.
Hence, ∆ABC ~ ∆PQR
2. SSS(Side-Side-Side) criteria of similarity
When in the two triangles, all the sides of one triangle are in the same ratio with respect to the sides of the other triangle, then their corresponding angles are equal. Then the two triangles are similar.
In ∆ABC and ∆DEF
AB/DE=BC/EF=AC/DF
So ∠A=∠D, ∠B=∠E, ∠C=∠F
Therefore, ∆ABC ~ ∆DEF
3. SAS(Side-Angle-Side) criteria of similarity
When in two triangles, two sides are in the same ratio with the two sides of the other triangle and the angle including those sides is equal then the two triangles are similar.
In ∆ABC and ∆KLM
AB/KL=BC/LM and ∠B=∠L
Hence, ∆ABC ~ ∆KLM
When the two similar triangles are given then the square of the ratio of their corresponding sides will be equal to the ratio of their area.
If ∆ABC ~ ∆PQR, then
ar(ABC)/ar(PQR)=(AB/PQ)^{2}=(BC/QR)^{2}=(AC/PR)^{2}
The similarity of two Triangles made in right angle Triangle
In a right angle triangle, if a perpendicular is drawn from the right angle to the hypotenuse of the Triangle, then the two Triangles formed would be similar to the whole Triangle.
In the above right angle CP is the vertex on the hypotenuse, thus :
∆ACP ~ ∆ACB
∆PCB ~ ∆ACB
∆PCB ~ ∆ACP
Triangles Class 10 notes will help to understand the formulas, statements, rules in detail.
This NCERT Class 10 Math’s chapter 6 notes also contains previous year’s questions and NCERT textbook pdf.
In offline mode, Class 10 Math’s chapter 6 notes pdf download can be used to prepare.
No, all the main derivations are not covered in the notes for Class 10 Math’s chapter 6. This NCERT note is a brief of the main topics and equations covered in the chapter and can be used for revising the triangles .
Students can expect 4 to 5 marks questions from the notes for Class 10 Math’s chapter 46.
As mentioned in NCERT notes for Class 10 Math’s chapter 6.
AAA (angle-Angle-Angle) criteria of similarity
If in two Triangles all the corresponding angles are equal then their corresponding sides are also in proportion.
Thus, the two Triangles are similar.
Hence, ∆ABC ~ ∆PQR
SAS(Side-Angle-Side) criteria of similarity-
If in two triangles, two sides are in the same ratio with the two sides of the other triangle and the angle including those sides is equal then the two triangles are similar.
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