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If the diagonals of a quadrilateral are 15cm and 9cm, what is the perimeter of the...

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ukcats11 | eNotes Newbie

Posted January 9, 2012 at 11:01 AM via web

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If the diagonals of a quadrilateral are 15cm and 9cm, what is the perimeter of the quadrilateral formed by connecting the midpoints of the sides?

The diagram shown is a parallelogram with a base of 63 and the heigth is 4x+3.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted January 9, 2012 at 6:03 PM (Answer #2)

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I suggest you to consider the a parallelogram your convex quadrilateral. This quadrilateral cannot be a rectangle because of different values of the lengths of diagonals.

The lengths of diagonals are: AC = 9 cm and BD = 15 cm.

If the vertices of parallelogram are ABCD and the midpoints are MNPQ, then the segment MN denotes the midline of triangle ABD and the segment PQ denotes the midline of triangle CBD.

You need to remember that the midline is one half that of the base, hence MN = PQ = `BD/2`  = `15/2 ` = 7.5 cm.

The segment MQ  and NP are midlines in triangles BAC and DAC, hence:

MQ=NP=`AC/2` =`9/2` =4.5 cm.

The perimeter of rectangle MQNP = 2(7.5 + 4.5) = 24 cm.

Hence, evaluating the perimeter of rectangle MNPQ yields `P_(MNPQ) = 24 cm` .

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