# Take any quadrilateral. Let A, B, C and D be its vertices. Let P, Q, R and S be the middle points of its edges.Show that the quadrilateral PQRS is a parallelogram.Hint: Show that vector SP= vector...

Take any quadrilateral. Let A, B, C and D be its vertices. Let P, Q, R and S be the middle points of its edges.

Show that the quadrilateral PQRS is a parallelogram.

Hint: Show that vector SP= vector RQ and that vector SR= vector PQ

You must use vectors operations and/or dot product to solve these questions.

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### 1 Answer

You need to notice that you may compose the following vectors such that:

`bar(AB) + bar(BD) = bar(DA) => bar(BD) = bar(DA) - bar(AB)`

`bar(BD) = bar(DA) + bar(BA)`

Since the problem provides the information that P and S are the midpoints of AB and AD yields:

`bar(AP) + bar(PS) = bar(SA) => bar(PS) = bar(SA) - bar(AP)`

`bar(PS) = bar(SA) + bar(AP)`

You need to remember that `bar(SA) = (1/2)bar(DA)` and `bar(AP) = (1/2)bar(BA)` such that:

`bar(PS) = (1/2)(bar(DA) + bar(BA))`

`bar(PS) = (1/2)(bar(BD))`

You need to use the equation that relates two parallel vectors, such that:

`bar u || bar v <=> bar u = k*bar v`

Reasoning by analogy yields `bar(PS) || bar(BD)` and `bar(QR) || bar(BD)` , hence `bar(PS) || bar(QR) , |bar(PS)| = |bar(QR)|.`

Using the same reasoning yields that `bar(PQ) || bar(RS), |bar(PQ)| = |bar(RS)|.`

**Hence, since `bar(PS) || bar(QR) , |bar(PS)| = |bar(QR)|` and `bar(PQ) || bar(RS), |bar(PQ)| = |bar(RS)|` yields `PQRS` is parallelogram.**