Chapter 5.1, Problem 38WE

To determine

To prove: that the segment is bisected by the diagonal which passes through the midpoint of a parallelogram.

Answer to Problem 38WE

The segment is bisected by the diagonal while passes through the midpoint of a parallelogram.

Explanation of Solution

Given information:

A segment whose endpoints lie on the opposite sides of a parallelogram passes through the midpoint of a diagonal.

Proof:

Consider the following diagram:

  

Consider a parallelogram $ABCD$

The ray $BD$ is a diagonal of that parallelogram.

The ray $EF$ is the line segment which bisects the diagonal.

To prove $EG=GF$

Consider the triangles $DFG$ and triangle $BEG$ .

Since $ABCD$ is a parallelogram

And in parallelogram opposite sides are equal

And $BD$ is a transversal

Thus, alternate interior angles are equal

That is;

  $\angle DGF=\angle BGE$

Since, $EF$ is the line segment which bisects the diagonal.

Side $DG=$ Side $GB$

  $\angle FDG=\angle EBG$

By half of the opposite angles of a parallelogram

Therefore, $△DFG$ is congruent to $△BEG$

By corresponding parts of congruent triangles

  $EG=GF$

Thus, it is concluded that the segment is bisected by the diagonal.