Question

A skew quadrilateral SKEW is shown. M, N, O, and P are the midpoints of $\overline{SK},\overline{KE},\overline{WE}$and $\overline{SW}$ . Explain why PMNO is a parallelogram.

Short answer

It is shown that PMNO is a parallelogram.

Step-by-step solution

Step 1. Consider the diagram.

Here, SKEW is a skew quadrilateral.

M, N, O and P are the midpoints of$\overline{SK},\overline{KE},\overline{WE}$ and $\overline{SW}$.

Step 2. Show the proof.

In $\Delta SKW$, as M and P are midpoints of SK and SW respectively,

That implies, MP is parallel to KW …… (i)

And $MP=\frac{1}{2}KW$ …… (ii)

In $\Delta EKW$, as N and O are midpoints of KE and WE respectively,

That implies, NO is parallel to KW …… (iii)

And $NO=\frac{1}{2}KW$ …… (iv)

Therefore, from (i) and (iii),

MP is parallel to KW is parallel to NO.

That implies, MP is parallel to NO …… (v)

From (ii) and (iv),

$MP=\frac{1}{2}KW=ON$

That is $MP=ON$ …… (vi)

When opposite sides of a quadrilateral are congruent and parallel, the quadrilateral is a parallelogram.

From (v) and (vi),

PMNO is a parallelogram.

Step 3. State the conclusion.

Therefore, PMNO is a parallelogram (proved).