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In the given figure, D is the midpoint o...

In the given figure, D is the midpoint of side BC and `AE bot BC`. If `BC=a, AC =b, AB= c, ED =x AD=p and AE =h` prove that
`(i) b^(2)=p^(2)+ax +(a^(2))/(4)`
(ii) `c^(2)=p^(2)-ax+(a^(2))/(4)`
(iii) `(b^(2)+c^(2))=2p^(2)+(1)/(2)a^(2)`
`(iv) (b^(2)-c^(2))=2ax`.

Text Solution

Verified by Experts

`(i) AC^(2)=AE^(2)+EC^(2)`
`rArr b^(2)=h^(2)+(x(a)/(3))^(2)=(h^(2)+x^(2))+ax+(a^(2))/(4)`
` =p^(2)-ax+(a^(2))/(4)`

`(ii) AB^(2)=AE^(2)+BE^(2)`
`rArr c^(2)=h^(2)+((a)/(2)-x)^(2)=(h^(2)+x^(2))-ax+(a^(2))/(4)`
`=p^(2)-ax+(a^(2))/(4)`
(iii) and (i) and (ii).
(iv) Subtract (ii) from (ii).
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