1.

If a parallelogram ABCD, |AB|=a, |AD|=b and |AC|=C, then DA. AB is equal to

Answer»

`(1)/(2)(a^(2)+b^(2)+c^(2))`
`(1)/(2)(a^(2)-b^(2)+c^(2))`
`(1)/(4)(a^(2)+b^(2)-c^(2))`
`(1)/(3)(b^(2)+c^(2)-a^(2))`

SOLUTION :We have, `|AB|=a, |AD|=b, |AC|=c`

In `Delta ABC,""AB + BC = AC `
`rArr AB+AD=AC "" [ because BC=AD]`
`rArr AB-DA=AC "" [ because AD=-DA]`
`rArr |AB-DA|=|AC|`
`rArr |AB-DA|^(2)=|AC|^(2)`
`rArr |AB|^(2)+|DA|^(2)-2AB*DA=(AC)^(2)`
`rArr a^(2)+b^(2)-2AB*DA=c^(2) "" [ because |AD|=|DA|=b]`
`rArr 2DA *AB=a^(2)+b^(2)-c^(2)`
`rArr DA*AB=(1)/(2)[a^(2)+b^(2)-c^(2)]`


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