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Q28

Expert-verifiedFound in: Page 32

Book edition
Student Edition

Author(s)
Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages
227 pages

ISBN
9780395977279

Tell whether each statement is true or false. Then write the converse and tell whether it is true or false.

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

The statement "*p* is the midpoint of $\overline{GH}$ implies that $GH=2PG$” is **true**.

The **converse of the given statement “if $GH=2PG$, then P is the midpoint of $\overline{GH}$ .” is true.**

The converse of the conditional is obtained by interchanging the hypothesis and the conclusion. If *p* represents the hypothesis and *q* represents the conclusion, then converse of the statement “If *p*, then *q*” will be “If *q*, then *p*”.

The statement "*P* is the midpoint of $\overline{GH}$ implies that $GH=2PG$”, *P* is the midpoint of $\overline{GH}$

That is:

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

Therefore, the statement is true.

The converse is of the form "if *q*, then *p*”.

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

Therefore, the converse of the given statement is – “If $GH=2PG$ then, *P* is the midpoint of $\overline{GH}$.”

Here $GH=2PG$

$GH=2PG$

$=PG+PG$

$=GP+PH$

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

Therefore, the converse of the given statement is true.

Prove the following statement by filling in the blanks.

If *A* and *B* have coordinated *a* and *b*, with $b>a$, and the midpoint *M* of $\overline{AB}$ has coordinate *x*, then prove $x=\frac{a+b}{2}$.

Proof:

Statement | Reasons |

1. | 1. ? |

2. $AM=x-a\text{};\text{}MB=b-x$ | 2. ? |

3. | 3. ? |

4. $\overline{AM}\cong \overline{MB}$, or $AM=MB$ | 4. ? |

5. $x-a=b-x$ | 5. ? |

6. $2x=$? | 6. ? |

7. $x=\frac{a+b}{2}$ | 7. ? |

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