Q. 145.0( 2 Votes )
In ΔRST, ∠S is right angle. The mid- points of tow sides RS and ST are X and Y respectively; let us prove that, RY2 + XT2 = 5XY2
Given: ΔRST is a right angled triangle with ∠S as right angle.
X is midpoint of RS.
Y is midpoint of ST.
To prove: RY2 + XT2 = 5XY2
RY2 = RS2 + SY2 [∵ H2 = P2 + B2]
⇒ RY2 = (2XS)2 + SY2 [∵ X is midpoint of RS]
⇒ RY2 = 4XS2 + SY2 ……………… (1)
XT2 = XS2 + ST2 [∵ H2 = P2 + B2]
⇒ XT2 = XS2 + (2SY)2 [∵ Y is midpoint of ST]
⇒ XT2 = XS2 + 4SY2 ……………… (2)
XY2 = XS2 + SY2 …………… (3) [∵ H2 = P2 + B2]
Now, adding eqn. (1) and (2) gives,
RY2 + XT2 = 4XS2 + SY2 + XS2 + 4SY2
⇒ RY2 + XT2 = 5XS2 + 5SY2
⇒ RY2 + XT2 = 5(XS2 + SY2)
Substituting from (3) gives,
RY2 + XT2 = 5XY2
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