Theorems

Chapter 6 Class 10 Triangles
Serial order wise

Transcript

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: βABC right angle at B To Prove: γπ΄πΆγ^2= γπ΄π΅γ^2+γπ΅πΆγ^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 βΌ Ξ ABC Since, sides of similar triangles are in the same ratio, β π΄π·/π΄π΅= π΄π΅/π΄πΆ βAD . AC= γπ΄π΅γ^2 Ξ BDC βΌ Ξ ABC Since, sides of similar triangles are in the same ratio β πΆπ·/π΅πΆ= π΅πΆ/π΄πΆ βCD . AC= γπ΅πΆγ^2 Adding (1) and (2) AD . AC + CD . AC = γπ΄π΅γ^2 + γπ΅πΆγ^2 AC (AD + CD) = γπ΄π΅γ^2 + γπ΅πΆγ^2 AC Γ AC = γπ΄π΅γ^2 + γπ΅πΆγ^2 γπ΄πΆγ^2 = γπ΄π΅γ^2 + γπ΅πΆγ^2 Hence Proved