Warm-up 2/7/12

Determine whether PQRS is a rectangle.

1. P(2,3) Q(5,9) R (11,6) S( 8,0)

2. P(-1,4) Q(3,6) R (9,-3) S(5,-5)

YES

NO--- not right angles

Homework Answers Rectangles

1. ST= 52. VT=123. VS=144. Perimeter = 345. Area = 606. 7. 8. 9. 10. AT=11,VA=11,AS=1111. (4 – 2)180 = 360

90,90,90,90 VmTmSmRmSTRVRSVT ;

STSRRVSR ;STTVRVTV ;

Remember if LMNO is a rectangle. Opposite sides are

congruent, all angles are 90°, Diagonals are congruent and

bisect each other.

12. 9x = 90, x = 10

13. 3x + 21 = 72, x = 17

14. OZ = ½ LN, 5x = ½ (30), 5x = 15, x = 3

15. LZ = ½ MO, 2(x- 4) = ½ (8), x - 4 = 2, x = 6

16. Slopes of edges: AB = undefined, AD = 0,

CD = undefined, BC = 0

Slope of diagonals: BD = -4/5, AC = 4/5 (diagonals are not

perpendicular)

AB // CD, AD // BC (same slopes)

AB AD and CD BC⊥ ⊥ABCD is a rectangle, because opposites sides are parallel and corner angles are right angles.

17. A = (19)(8) = 15218. A = (6)(12) = 7219. b + 6 = 10 A = (6)(8) = 48 b + 36 = 100

b = 64 b = 8

20. A = 186

 

4.5 Rhombus

Essential Questions:

(1). What are the characteristics of a rhombus?

(2). How can we prove the shape is a rhombus?

Define Rhombus:

Since all the edges of a rhombus are congruent, opposite edges are congruent. A quadrilateral with opposite edges congruent is defined to be a parallelogram.Hence, all __________ are parallelograms.

rhombi

A quadrilateral with four congruent sides.

The five properties of parallelograms also pertain to rhombuses (rhombi):

• Opposite edges of a rhombus are ________.

• Opposite edges of a rhombus are ________.

• Opposite angles of a rhombus are ________.

• Consecutive interior angles of a rhombus are ______________.

• The diagonals of a rhombus ______ each other.

congruent

congruent

supplementary

bisect

parallel

In addition to the five properties above, there are two additional properties.

Investigation: Step 1: Copy ABCD on a sheet of patty paper.Step 2: Fold the patty paper in half so that B overlays D. Crease the patty paper firmly. Unfold.Step 3: Fold the patty paper in half so that A

overlays C. Crease the patty paper firmly. Unfold.

The creases in the patty paper represent the diagonals AC and BD. Are the diagonals perpendicular? YES

Step 4: Fold the patty paper in half again where AC is the crease. Does AC bisect A and C?

YES

Step 5: Fold the patty paper in half again where BD is the crease. Does BD bisect B and D?

YES

From this investigation we can conclude:

• The diagonals of a rhombus are____________.

• The diagonals of a rhombus bisect a pair of _________ angles

opposite

perpendicular

Answer each question using rhombusADCB pictured below.

B D

C

A

(1). If AB = 18 cm, then CD = ___ cm, and BC = _____ cm, and AD = ____cm.

(2). If then and

,60Am ______,Cm_______,Bm ._______Dm

1818 18

60o

120o 120o

AC(3). If and

and mA = 80o, then mBAX = _____, mDAX = ______, mADX = ____, mCDX = ______.

BD intersect at point X, then m AXB = _____, mAXD = _____, and m BXC = _____.

(4). If AC and BD intersect at point X,

90o 90o

90o

40o

40o 50o

50o

B D

A

C

(5). If BX = 2 cm, then BD = __ cm.

(6). If AC = 12 cm, then AX = __ cm.

4

6B D

A

C

(7). BC ║____; CD ║____;

AC ___

AD AB

BD

(9). If AB = 8 ft then the perimeter of the rhombus is ______ ft. 32

(8).

(10). The sum of the measures of the interior angles of the rhombus is ______.

(11). Is the rhombus a quadrilateral? T or F

(12). Is the rhombus a parallelogram? T or F

(13). Is the rhombus a rectangle? T or F

360o

T

T

F

How to prove a quadrilateral is a Rhombus:

1st Prove it is a __________________ (using one of the five ways to prove a quadrilateral is a parallelogram)

 2nd Show (1) Diagonals are _________________  OR (2) Diagonals __________ corner angles  OR (3) 2 Adjacent sides are _______________ (which means all 4 sides will be ______ )

parallelogram

perpendicular

bisect

Congruentequal

(14). The coordinates of the Quadrilateral are W(-3,-3), X(1,-6), Y(5,-3) and Z(1,0).

a. Find the slope of WX 

b. Find the slope of XY 

c. Find the slope of YZ 

d. Find the slope of WZ 

4

3

31

36

WXm

4

3

15

63

XYm

4

3

51

30

YZm

4

3

31

30

WZm

(14). The coordinates of the Quadrilateral are

W(-3,-3), X(1,-6), Y(5,-3) and Z(1,0).

e. Find the slope of WY (the slope of the diagonal)

f. Find the slope of XZ (the slope of the diagonal)

08

0

35

33

WYm

undefinedmXZ

0

6

11

60

We proved that both pairs of opposite sides are parallel, therefore, it is a parallelogram. We also proved thediagonals are perpendicular. So it is a RHOMBUS!

Since a rhombus is a parallelogram, we can use the formula A = bh to find the area of a rhombus. There is an additional formulafor the area of a rhombus as well: A = ½ d1d2

Find the area of each rhombus.

15. A = ________ 1.2 ft 1.2 ft

3.5 ft

3.5 f t

A = ½ d1● d2

A = ½ (7)(2.4)

A = 8.4 ft2

8.4 ft2

16. A = ________

62 + b2 = 102

a2 + b2 = c2

36+b2 = 100

b2 = 64b = 8

10 cm6 cm

A = ½ d1● d2

A = ½ (12)(16)

A = 96 cm2

96 cm2 8cm