Title: Example 1 Geometric Mean
Author: Tim Heppner
Last saved by: MHE
Company: MCGRAW-HILL

Geometry	Chapter 6

Lesson 6-4

Example 1  Diagonals of a Rectangle
ALGEBRA  Quadrilateral ABCD is a rectangle.	
If AC = 4x - 13 and BD = 2x + 14, find x and 
the length of. 
The diagonals of a rectangle are congruent,
so  ≅ .

BD = 2x + 14	
	= 2+ 14	Substitution
	= 41	Simplify. 

		≅ 	Diagonals of a rectangle are ≅.
	AC	= BD	Definition of congruent segments.
	4x - 13	= 2x + 14	Substitution
	2x - 13	= 14	Subtract 2x from each side.
	2x	= 27	Add 13 to each side.
	x	= 27/2	Divide each side by 2.

Example 2  Angles of a Rectangle
ALGEBRA  Quadrilateral PQRS is a rectangle.	
a.	Find x.
	∠QPS is a right angle, so m∠QPS = 90.

	m∠QPR + m∠RPS	= m∠QPS	Angle Addition Postulate
	70 - 4x + 18x - 8	= 90	Substitution
	62 + 14x	= 90	Simplify.
	14x	= 28	Subtract 62 from each side.
	x	= 2	Divide each side by 14.

b.	Find y.
	Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent.

	∠PQS	≅ ∠QSR	Alternate Interior Angles Theorem
	m∠PQS	= m∠QSR	Definition of congruent angles
	7y + 6	= y^2 - 2	Substitution
	0	= y^2 - 7y - 8	Subtract 7y and 6 from each side.
	0	= (y -8)(y + 1)	Factor.

	y - 8	= 0	y + 1	= 0
	 y	= 8	y	= -1	Disregard y = -1 because it yields angle measures less than 0.

Example 3  Diagonals of a Parallelogram
CONSTRUCTION  The Owens family is building a deck in their back yard. Mrs. Owens has laid out stakes where the corners of the deck will be. She has made sure that the opposite sides are congruent. If she measures the diagonals and they are congruent, how can Mrs. Owens be sure that the deck will be a rectangle? Explain.
Draw a diagram. We know that	
 ≅ ,  ≅ , and  ≅ .
Because  ≅  and  ≅ , 
MNOP is a parallelogram.
and are diagonals and they are 
congruent. A parallelogram with congruent 
diagonals is a rectangle. 
So, Mrs. Owens can be sure that the deck is a rectangle.

Example 4  Rectangle on a Coordinate Plane
COORDINATE GEOMETRY  Quadrilateral ABCD has 	
vertices A(-6, 9), B(4, 7), C(3, 2), and D(-7, 4).
Determine whether ABCD is a rectangle.
Method 1: Use the Slope Formula, m =  , to see if 
opposite sides are parallel and consecutive sides are perpendicular.
slope of  = (7 - 9)/(4 - (-6)) or -1/5
slope of  = (2 - 4)/(3 - (-7)) or -1/5
slope of  = (4 - 9)/(-7 - (-6)) or 5
slope of  = (2 - 7)/(3 - 4) or 5

║ and ║. Quadrilateral ABCD is a parallelogram. 

The product of the slopes of consecutive sides is -1. This means that, , , and . The perpendicular segments create four right angles. Therefore, by definition ABCD is a rectangle.

Method 2: Use the Distance Formula, d = , to determine whether opposite sides are congruent.

First, we must show that quadrilateral ABCD is a parallelogram. 
AB	= 	DC	= 
	= √(100 + 4)		= √(100 + 4)
	= √104 or 2√26		= √104 or 2√26

AD	= 	BC	= 
	= √(1 + 25)		= √(1 + 25)
	= √26		= √26

Since each pair of opposite sides of the quadrilateral have the same measure, they are congruent. Quadrilateral ABCD is a parallelogram.

AC	= 	BD	= 
	= √(81 + 49)		= √(121 + 9)
	= √130		= √130

The length of each diagonal is √130. Since the diagonals are congruent, ABCD is a rectangle by Theorem 6.14.