Theorem 6.8 (Pythagoras Theorem) : If a right triangle, the square of the hypotenuse is equal to the sum of the squares of other two sides. Given: βABC right angle at B
To Prove: γπ΄πΆγ^2= γπ΄π΅γ^2+γπ΅πΆγ^2
Construction: Draw BD β₯ AC
Proof: Since BD β₯ AC
Using Theorem 6.7: If a perpendicular is drawn from the vertex of the right angle of the a right triangle to the hypotenuse then triangle on both side of the perpendicular are similar to whole triangle and to each other.
Ξ ADB βΌ Ξ ABC
Since, sides of similar triangles are in the same ratio,
β π΄π·/π΄π΅= π΄π΅/π΄πΆ
βAD . AC= γπ΄π΅γ^2
Ξ BDC βΌ Ξ ABC
Since, sides of similar triangles are in the same ratio
β πΆπ·/π΅πΆ= π΅πΆ/π΄πΆ
βCD . AC= γπ΅πΆγ^2
Adding (1) and (2)
AD . AC + CD . AC = γπ΄π΅γ^2 + γπ΅πΆγ^2
AC (AD + CD) = γπ΄π΅γ^2 + γπ΅πΆγ^2
AC Γ AC = γπ΄π΅γ^2 + γπ΅πΆγ^2
γπ΄πΆγ^2 = γπ΄π΅γ^2 + γπ΅πΆγ^2
Hence Proved

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.