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Question
Adjacent sides of a parallelogram are 11 cm and 17 cm. If the length of one of its diagonal is 26 cm, find the length of the other.
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Solution
Let □PQRS be a parallelogram.
Then, PQ = 17 cm, QR = 11 cm and diagonal PR = 26 cm The diagonals of a parallelogram bisect each other. Point M is the point of intersection of diagonals PR and QS.
`therefore PM=MR=1/2PR=1/2xx26`
`therefore PM=MR=13 cm ......(1)`
`QM=1/2QS`
`thereforeQS=2QM......(2)`
In ΔPQR, QM is the median.
`PQ^2+QR^2=2PM^2+2QM^2` .......(By Apollonius theorem)
`(17)^2+(11)^2=2(13)^2+2QM^2`
`therefore 289+121=2(169)+2QM^2`
`therefore 410=2(169)+2QM^2`
Diving by 2, we get
`205 =169 + QM^2`
`therefore QM^2= 205 -169=36`
`therefore QM=6`
`therefore QS=2QM=2xx6=12cm`
Thus, the length of the other diagonal is 12 cm.
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