Transcript Applied Geometry

Geometry
Lesson 6 – 5
Rhombi and Squares
Objective:
Recognize and apply the properties of rhombi and squares.
Determine whether quadrilaterals are rectangles, rhombi, or squares.
Rhombus
What is the definition of a rhombus?
A parallelogram with all four sides
congruent.
Properties of Rhombus
Theorem 6.15

If a parallelogram is a rhombus, then its
diagonals are perpendicular.
Properties of Rhombus
Theorem 6.16

If a parallelogram is a rhombus, then each
diagonal bisects a pair of opposite angles.
The diagonals of rhombus FGHJ
intersect at K. Use the given info to
find each value.
98
49
82
49
If GH = x + 9 and JH = 5x – 2, find x.
5x – 2 = x + 9
9y - 5 13
4x = 11
x
+
9
x = 2.75
5
If FK = 5 and FG = 13, find KJ.
6y + 7
(FK)2 + (GK)2 = (FG)2
5x - 2
52 + (GK)2 = 132
(GK)2 = 144 GK = 12
if m  JFK  6 y  7 and m  KFG  9 y  5 , find y
6y + 7 = 9y - 5
12 = 3y
4=y
Square
What is the definition of a square?
A parallelogram with four congruent
sides and four right angles.
Summary: Flow chart
Quadrilateral
Parallelogram
Rectangle
Rhombus
Square
Square has all of the properties of both rectangles and rhombi.
Summary: Venn Diagram
Parallelograms
4 right angles
Squares
Rectangles
Rhombi
4 congruent sides
4 right angles
& 4 congruent sides
Conditions for Rhombi and Squares
Theorem 6.17

If the diagonals of a parallelogram are
perpendicular, then the parallelogram is a
rhombus.
Conditions for Rhombi and Squares
Theorem 6.18

If one diagonal of a parallelogram bisects a
pair of opposite angles, then the
parallelogram is a rhombus.
Conditions for Rhombi and Squares
Theorem 6.19

NEW!
If one pair of consecutive sides of a
parallelogram are congruent, the
parallelogram is a rhombus.
Conditions for Rhombi and Squares
Theorem 6.20

If a quadrilateral is both a rectangle and a
rhombus, then it is a square.
Determine whether parallelogram JKLM with vertices
J (-7, -2) K(0, 4) L (9, 2) and M (2, -4) is a rhombus,
a rectangle, or a square. List all that apply. Explain.
Is the figure a rectangle? Are the diagonals congruent?
2
2
JL    7  9     2  2   272  4 17  16 . 5
KM 
0  2 
2
 4  4  
2
68  2 17  8 . 2
The figure is not a rectangle.
If its not a rectangle, then its not a square.
Is the figure a rhombus?
Can either check that 2 consecutive sides are congruent or that the
slope of the diagonals are perpendicular.
Slope of KM = -4
Slope of JL = 1/4
Parallelogram JKLM is a Rhombus
Given J (5, 0) L (-3, -14) K (8, -11) M (-6, -3),
determine whether parallelogram JKLM is a rhombus,
rectangle, or square. List all that apply. Explain.
Is the figure a rectangle?
2
2
JL  5  3   0  14  
2
2
KM  8  6     11  3  
260  4 65  32 . 2
260  4 165  32 . 2
The figure is a rectangle. Is the figure a square?
Are the diagonals perpendicular?
Slope of JL = 7/4
Slope of KM = -4/7
The figure is a square. Since it’s a square it is also a rhombus.
The figure is a rhombus, rectangle, and a square.
Homework
Pg. 431 1 – 6 all, 8 – 14 E,
22 – 30 E, 48, 52 – 60 E