Q. 384.2( 19 Votes )

If a line is draw

Answer :

Given: In Δ ABC, AB which intersects AB and AC at D and F respectively.


To Prove:



Construction:


Join B, E and C, D and then draw DM AC and EN AB.



Proof:


Area of Δ ADE


Area of Δ BDE


So



… (2)


Again, area of Δ ADE


Area of Δ CDE


So



…(3)


Observe that Δ BDE and Δ CDE are on the same base DE and between same parallels BC and DE.


So, ar(Δ BDE) = ar(Δ CDE) … (3)


From (1), (2) and (3), we have



Hence, proved.


OR


Statement: In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.


Given: A right triangle ABC right angled at B.


To Prove: AC2 = AB2 + BC2


Construction: Draw BD AC



Proof:


We know that if a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.


Δ ADB ~ Δ ABC


So,


Or AD × AC = AB2 … (1)


Also, Δ BDC ~ Δ ABC


So,


Or CD × AC = BC2 … (2)


Adding (1) and (2),


AD × AC + CD × AC = AB2 + BC2


AC (AD + CD) = AB2 + BC2


AC2 = AB2 + BC2


Hence, proved.


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