Chapter 5.5, Problem 25WE

To determine

To Find : The type of quadrilateral given .

Answer to Problem 25WE

Quadrilateral MNOP is a parallelogram.

Explanation of Solution

Given information:

  

Consider the quadrilateral ABCD with midpoint M on AB, N on BC, O on CD, P on AD.

  

Draw diagonals $AC,BD$ and segments $MN,NO,OP,PM$ .

  $MB=\frac{1}{2}AB,NB=\frac{1}{2}CB,OD=\frac{1}{2}CD,PD=\frac{1}{2}AD(M,N,O,P$ are midpoints)

  $\text{Δ}ABC\sim \text{Δ}MBN$ and $\text{Δ}ODP\sim \text{Δ}CDA$ (By Side angle side property)

  $\angle ABC=\angle MNB,\angle OPD=\angle CAD$ (By definition of similar triangles)

  $\angle MBN=\angle ABC,\angle ODP=\angle CDA$ (Equal angles)

  $AC\parallel MN,OP\parallel AC,MN\parallel OP$

From the definition of midpoint:

  $\begin{array}{l}\begin{array}{l}CO=1/2DC,\hfill \\ CN=1/2BC,\hfill \\ AP=1/2AD,\hfill \end{array}\\ AM=1/2AB\end{array}$

Therefore, $\angle DBA=\angle PAM$ and $\angle DCB=\angle OCN$ .

And $\text{Δ}DBA\sim \text{Δ}MBN$ and $\text{Δ}DCB\sim \text{Δ}OCN$ (By side angle side property)

Hence, $\angle DBA=\angle PMA$ and $\angle CDB=\angle CON$ (Corresponding parts of congruent triangles)

Quadrilateral MNOP is a parallelogram.