Chapter 2.1, Problem 28WE

To determine

To state: The converse of statement and state whether it is true or false

Answer to Problem 28WE

The statement is true.

The converse of the given statement is true.

Explanation of Solution

Given information:

The given statement is:

  $″P$ is the midpoint of $\overline{GH}$ implies that $GH=2PG″$

  $″P$ is the midpoint of $\overline{GH}$ implies that $GH=2PG″$

Here $P$ is the midpoint of $\overline{GH}$

That is

  $\begin{array}{l}GH=GP+PH\\ =2PG\end{array}$

Therefore, the statement is true.

Further, here $p$ is $″P$ is the midpoint of $\overline{GH}″$ and $q$ is $″GH=2PG″$

The converse is of the form "if $q$ ,then $p″$

Therefore, the converse of the given statement is;

If $GH=2PG,$ then $P$ is the midpoint of $\overline{GH}$

Here $GH=2PG$

  $\begin{array}{l}GH=2PG\\ =PG+PG\\ =GP+PH\end{array}$

Then $P$ is the midpoint of $\overline{GH}$

Therefore, the converse of the given statement is true.