YES NO--- not right angles Determine whether PQRS is a rectangle. 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

ST= 5 VT=12 VS=14 Perimeter = 34 Area = 60 7. 8. 9. 10. AT=11,VA=11,AS=11 11. (4 – 2)180 = 360

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) = 152 18. A = (6)(12) = 72 19. 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: A quadrilateral with four congruent sides. 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

The five properties of parallelograms also pertain to rhombuses (rhombi): 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. parallel congruent congruent supplementary bisect

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 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 perpendicular opposite

Answer each question using rhombus ADCB pictured below. (1). If AB = 18 cm, then CD = ___ cm, and BC = _____ cm, and AD = ____cm. (2). If then and C 18 18 18 60o 120o 120o

intersect at point X, then m AXB = _____, m 90o 90o AXD = _____, (3). If and intersect at point X, then m AXB = _____, m 90o 90o AXD = _____, and m BXC = _____. 90o (4). If and intersect at point X, and mA = 80o, then mBAX = _____, mDAX = ______, mADX = ____, mCDX = ______. 40o 40o 50o 50o

4 6 (5). If BX = 2 cm, then BD = __ cm. A 4 (5). If BX = 2 cm, then BD = __ cm. (6). If AC = 12 cm, then AX = __ cm. D B 6 C (7). ║____; ║____; (8).  ___ (9). If AB = 8 ft then the perimeter of the rhombus is ______ ft. 32

360o (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: parallelogram 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 ______ ) perpendicular bisect Congruent equal

(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

(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) We proved that both pairs of opposite sides are parallel, therefore, it is a parallelogram. We also proved the diagonals 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 formula for the area of a rhombus as well: A = ½ d1d2 Find the area of each rhombus. 8.4 ft2 3.5 ft 15. A = ________ 1.2 ft 1.2 ft 3.5 f t A = ½ d1● d2 A = ½ (7)(2.4) A = 8.4 ft2

96 cm2 16. A = ________ a2 + b2 = c2 62 + b2 = 102 36+b2 = 100 A = ½ d1● d2 b2 = 64 A = ½ (12)(16) b = 8 A = 96 cm2