Last updated at May 29, 2018 by Teachoo

Transcript

Example 12 (Method 1) Show that the points A(2 + ), B( 3 5 ) , C(3 4 4 ) are the vertices of a right angled triangle. A(2 + ), B( 3 5 ) C(3 4 4 ) We know that two vectors are perpendicular to each other, i.e have an angle of 90 between them , if their scalar product is zero. = ( 3 5 ) (2 + ) = 1 3 5 2 + 1 1 = (1 2) + ( 3 + 1) + ( 5 1) = 1 2 6 = (3 4 4 ) ( 3 5 ) = 3 4 4 1 + 3 + 5 = (3 1) + ( 4 + 3) + ( 4 + 5) = 2 1 + 1 = (2 + ) (3 4 4 ) = 2 1 + 1 3 + 4 + 4 = (2 3) + ( 1 + 4) + (1 + 4) = 1 + 3 + 5 Now, . = (2 1 + 1 ) . (-1 + 3 + 5 ) = (2 1) + ( 1 3) + (1 5) = ( 2) + ( 3) + 5 = 5 + 5 = 0 Since, . = 0 Therefore, is perpendicular to . Hence ABC is a right angled triangle Example 12 (Method 2) Show that the points A(2 + ), B( 3 5 ) , C(3 4 4 ) are the vertices of a right angled triangle. A(2 + ), B( 3 5 ) C(3 4 4 ) Considering ABC as a right angled triangle, By Pythagoras theorem, AB2 = BC2 + CA2 or AB 2 = BC 2 + CA 2 = ( 3 5 ) (2 + ) = 1 3 5 2 + 1 1 = (1 2) + ( 3 + 1) + ( 5 1) = 1 2 6 = (3 4 4 ) ( 3 5 ) = 3 4 4 1 + 3 + 5 = (3 1) + ( 4 + 3) + ( 4 + 5) = 2 1 + 1 = (2 + ) (3 4 4 ) = 2 1 + 1 3 + 4 + 4 = (2 3) + ( 1 + 4) + (1 + 4) = 1 + 3 + 5 Now, Magnitude of = 1 2+ 2 2+ 6 2 = 1+4+36 = 41 Magnitude of = 22+ 1 2+1 = 4+1+1 = 6 Magnitude of = ( 1)2+32+52 = 1+9+25 = 35 Now, 2 + 2 = ( 6 )2 + ( 35 )2 = 6 + 35 = 41 = ( 41 )2 = 2 Thus, 2 = 2 + 2 So, ABC is a right angled triangle.

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Example 12 You are here

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Example 29 Important

Example 30 Important

Example 26 (Supplementary NCERT)

Example 27 (Supplementary NCERT)

Example 28 (Supplementary NCERT)

Example 29 (Supplementary NCERT)

Example 30 (Supplementary NCERT)

Example 31 (Supplementary NCERT)

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.