Q32 Prove Theorem 5-12.... [FREE SOLUTION]

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 5: Q32 (page 188)

Prove Theorem 5-12.

Expert verified

Theorem 5-12 is proved by substitution, property, segment addition postulate.

Diagonals of the rectangle are congruent for rectangle $Q R S T$ , it is to be shown that $\overset{炉}{Q S} 鈮� \overset{炉}{R T}$ .

The given statement can be proved as below:

Statement	Reasons
Rectangle $Q R S T$ .	Given.
is the midpoint $\overset{炉}{R T}$ and $\overset{炉}{Q S}$ .	Theorem 5-3 diagonals of a parallelogram bisect each other.
In right $螖 T Q R$ , O is the midpoint $\overset{炉}{R T}$ .	Proved above.
$\overset{炉}{O T} 鈮� \overset{炉}{O Q} 鈮� \overset{炉}{O R}$ .	Theorem 5-15 the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
In right $螖 S R Q$ , O is the midpoint $\overset{炉}{Q S}$ .	Proved above.
$\overset{炉}{O S} 鈮� \overset{炉}{O Q} 鈮� \overset{炉}{O R}$ .	Theorem 5-15 the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
$\overset{炉}{O S} 鈮� \overset{炉}{O T} 鈮� \overset{炉}{O Q} 鈮� \overset{炉}{O R}$ .	Substitution property.
$\begin{array}{l} \overset{炉}{O S} 鈮� \overset{炉}{O Q} 鈮� \overset{炉}{O R} \\ \overset{炉}{R T} = \overset{炉}{O T} + \overset{炉}{O R} \end{array}$	Segment addition postulate.
$\begin{array}{l} \overset{炉}{Q S} = 2 \overset{炉}{O Q} \\ \overset{炉}{R T} = 2 \overset{炉}{O Q} \end{array}$	Substitution property.
$\overset{炉}{Q S} = \overset{炉}{R T}$	Substitution Property.