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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 - Geometry

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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.

Concept: Appolonius Theorem
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