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

Geometry	Chapter 6

Lesson 6-5

Example 1  Proof of Theorem 6.15
Given:	ABCD is a rhombus.	
Prove:	 ⊥ 

Proof:
By the definition of a rhombus,  ≅  ≅  ≅ . A rhombus is a parallelogram and the diagonals of a parallelogram bisect each other, so  bisects  at X. Thus,  ≅ .  ≅  because congruence of segments is reflexive. Thus, △ABX ≅ △CBX  by SSS. ∠BXA ≅ ∠BXC  by CPCTC. ∠BXA and ∠BXC also form a linear pair. Two congruent angles that form a linear pair are right angles. ∠BXA is a right angle, so  ⊥  by the definition of perpendicular lines.

Example 2  Measures of a Rhombus
ALGEBRA  Use rhombus BCDE and the given 	
information to find the value of each variable.
a.	If m∠3 = y^2 + 26, find y.
	m∠3	= 90	The diagonals of a rhombus are perpendicular.
	y^2 + 26	= 90	Substitution
	y^2	= 64	Subtract 26 from each side.
	y	= ± 8	Take the square root of each side.
	The value of y can be 8 or -8.

b.	Find m∠CED if m∠BCD = 38.
	m∠BCD	= m∠BED	Opposite angles are congruent.
	m∠BCD	= 38
	The diagonals of a rhombus bisect the angles. So, m∠CED = 1/2(38) or 19.

Example 3  Squares
COORDINATE GEOMETRY  Determine whether 	
parallelogram WXYZ is a rhombus, a rectangle, or 
a square. List all that apply. Explain
Explore	Plot the vertices on a coordinate plane.

Plan	If the diagonals are perpendicular, then 
WXYZ is either a rhombus or a square.
The diagonals of a rectangle are congruent.
If the diagonals are congruent and
perpendicular, then WXYZ is a square.

Solve	Use the Distance Formula to compare the lengths of the diagonals.

WY	= 
= √(1 + 289)
= √290

XZ	= 
= √(289 + 1)
= √290

	Use slope to determine whether the diagonals are perpendicular.
slope of  = (-7 - 10)/(0 - 1) or 17 | slope of  = (2 - 1)/(-8 - 9) or -1/17

	Since the slope of  is the negative reciprocal of the slope of , the diagonals are perpendicular. The lengths of  and  are the same so the diagonals are congruent. WXYZ is a rhombus, a rectangle, and a square.

Examine	The diagonals are congruent and perpendicular, so WXYZ must be a square. You can verify that WXYZ is a rhombus by finding the slope and length of all four sides. All four sides are congruent and consecutive sides are perpendicular. 

Example 4  Diagonals of a Square
DECORATING  A square picture window 	
with a sun catcher is shown at the right. Is 
the top of sun catcher in the center of the window? 
Explain.
Since a square is a parallelogram, the diagonals bisect
each other. Since a square is a rhombus, the diagonals 
are congruent. Therefore, the distance across the window 
is the same as the distance from the top of the window to
the bottom. Thus, the distance from the top of the window
to its center is 3.54 meters divided by 2 or 1.77 meters.
The distance is longer than the location of the top of the sun
catcher from the top of the window, so it is not in the center
of the window.