Question

`square ABCD` is a parallelogram. The diagonals `AC` and `BD` intersect at point `M`. The length of seg `AC`, `AB` and `AD` is `24`, `22` and `34` respectively. Find the length of seg `BD`.

Answer 1

`square ABCD` is a parallelogram
The diagonals of parallelogram bisect each other
`:. AM=(1)/(2)AC=(1)/(2)xx24=12`
`M` is the midpoint of diagonal `BD`
`:.` seg `AM` is the median of `DeltaABD`
`:.` by Apollonius theorem,
`AB^(2)+AD^(2)=2AM^(2)+2BM^(2)`
`:.(22)^(2)+(34)^(2)=2(12)^(2)+2BM^(2)`
`:.484+1156=2(144)+2BM^(2)`
`:.1640-288=2BM^(2)`
`:.2BM^(2)=1352`
`:.BM^(2)=(1352)/(2)`
`:.BM^(2)=676`
`:.BM=26`
`BM=(1)/(2)BD`.........(Diagonals of paralelogram bisect each other)
`:.26=(1)/(2)xxBD`
`:.BD=52`