NCERT Solutions for Class 10 Maths Chapter 6 Triangles (Ex 6.6) Exercise 6.6
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We had studied the congruent triangles. They have the same shape and size. We had also studied the property of congruency. But there are some triangles which may have the same shape but not the same size. These are known as similar triangles. The study of similarity of triangles has helped us in finding the distance of distant objects like the distance to the moon or the height of mountains. This would not have been possible with the normal measurements by a tape.
The chapter discusses some concepts in detail which are listed as follows:
Similarities of triangles
Criteria for the similarity of triangles
Areas of similar triangles
Let's study these contents in detail.
Consider 2 circles with different radii. Are they even same in size? No. Are they congruent? No. But since both are of the same shape that is a circle, they are similar. Consider a circle and a square. Are they similar? No. You can answer this merely by looking at the figures. They cannot be similar because their shapes are different.
All congruent are similar, but all similar figures need not be congruent.
Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion) as mentioned in the NCERT textbook of Class 10 Maths Chapter 6.
Satisfying only one condition is not sufficient. Both the conditions should be satisfied for 2 polygons to be similar.
Exercise 6.1 deals with 3 questions based on similar and non-similar figures.
2. Similarities of triangles
Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. If the corresponding angles of 2 triangles are similar then they are equiangular triangle. Also, the ratio of any two corresponding sides in two equiangular triangles is always the same
Theorem 6.1 states that If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Theorem 6.2 states that If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Exercise 6.2 covers 10 questions based on the above Basic Proportionality Theorem.
3. Criteria for the similarity of triangles
We learnt this earlier, two triangles are similar if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion).
This statement can be proved by the following Theorem.
Theorem 6.3 indicates that If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.
If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This is known as the AAA test of similarity.
Theorem 6.4 illustrated If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar.
This will be the SSS test of similarity.
Theorem 6.5 explains 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.
This is the SAS test for similarity.
Exercise 6.3 deals with 26 questions based on the above tests.
4. Areas of similar triangles
Theorem 6.6 explains the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Now we shall study the Pythagoras Theorem and its converse in detail.
5. Pythagoras Theorem
Theorem 6.8 proves if, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Theorem 6.9 describes in a triangle, if 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.
Exercise 6.6 is a complete worksheet of the above chapter Triangles. For its detailed solution, you may download our PDF for NCERT solutions for Class 10 Maths Chapter 6 Triangles.
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