Rectangles, Squares and Rhombuses
Rectangles A rectangle is a parallelogram with four interior right angles. Here are the properties of a rectangle: Properties All the interior angles are right angles. Diagonals are equal. Diagonals bisect each other into four equal parts. All properties of a parallelogram A B C D AC = BD Abbreviation Property of rectangle
(property of rectangle) In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at O. A B C D a O b 30 ∵ ABCD is a rectangle. ∴ BCD = 90 ∴ OCD = 90 – 30 = 60 ∵ OC = OD ∴ a = OCD (property of rectangle) = BCD – BCA (property of rectangle) (base s, isos. △)
In the figure, ABCD is a rectangle In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at O. A B C D a O b 30 Consider △OCD. ∴ a + b + OCD = 180 60 + b + 60 = 180 b = 60 ( sum of △)
Follow-up question In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at E. Find the unknowns. b cm A B C D E a cm 15 cm 8.5 cm Solution ∵ CE = BE ∴ a = 8.5 ∵ DE = BE = 8.5 cm ∴ BD = 2 × 8.5 cm = 17 cm (property of rectangle) (property of rectangle)
Follow-up question (cont’d) In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at E. Find the unknowns. b cm A B C D E a cm 15 cm 8.5 cm Solution Consider △BCD. ∵ BCD = 90 ∴ (property of rectangle) 2 - = BC BD CD (Pyth. theorem) 15 17 2 - = b 64 = 8 =
Squares A square is a parallelogram with four equal interior right angles and four equal sides. Here are the properties of a square: Properties All sides are equal. Diagonals are perpendicular to each other. Angle between each diagonal and a side is 45. All properties of a rectangle 45 45° Property of square Abbreviation
In the figure, ABCD is a square. ∵ AC is a diagonal. 60 a b In the figure, ABCD is a square. ∵ AC is a diagonal. ∴ BAO = 45 (property of square) Consider △ABO. BAO + ABO + AOB = 180 ( sum of △) 45 + 60 + a = 180 a = 75 ABC = 90 (property of square) 60 + b = 90 b = 30
Follow-up question A B C D 4 cm a cm b c O d cm In the figure, ABCD is a square. The diagonals AC and BD intersect at O. Find the unknowns. Solution ∵ ABCD is a square. ∴ CD = AD a = 4 (property of square) ∵ AC is a diagonal. ∴ b = 45 (property of square)
Follow-up question (cont’d) A B C D 4 cm a cm b c O d cm In the figure, ABCD is a square. The diagonals AC and BD intersect at O. Find the unknowns. Solution ∵ The diagonals of a square are perpendicular to each other. ∴ c = 90 (property of square) Consider △BCD. ∵ ∠BCD = 90 (property of square) 2 + = CD BC BD (Pyth. theorem) 4 2 + = cm 32 = cm 2 4 (or cm)
Follow-up question (cont’d) A B C D 4 cm a cm b c O d cm In the figure, ABCD is a square. The diagonals AC and BD intersect at O. Find the unknowns. Solution 2 1 = BD (property of square) ∴ OD d 2 (or ) 32 =
Rhombuses A rhombus is a parallelogram with four equal sides. Here are the properties of a rhombus: Properties All sides are equal. Diagonals are perpendicular to each other. Interior angles are bisected by the diagonals. All properties of a parallelogram Property of rhombus Abbreviation
∵ BC = AD (property of rhombus) ∴ a = 3 In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O. 3 cm b c O 25 d a cm ∵ BC = AD (property of rhombus) ∴ a = 3 ∵ ABO = ADO (property of rhombus) ∴ c = 25 Consider △ABO. AOB = 90 (property of rhombus) b + c + AOB = 180 ( sum of △) b + 25 + 90 = 180 b = 65
d = b (property of rhombus) = 65 A B C D In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O. 3 cm b c O 25 d a cm d = b (property of rhombus) = 65
Follow-up question In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O. Find the unknowns. (p – 2) cm A B C D O (10 – p ) cm 2q 3q Solution AD = CD (property of rhombus) 10 – p = p – 2 2p = 12 p = 6
Follow-up question (cont’d) In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O. Find the unknowns. (p – 2) cm A B C D O (10 – p ) cm 2q 3q Solution Consider △AOD. AOD = 90 (property of rhombus) AOD + 3q + 2q = 180 ( sum of △) 5q + 90 = 180 5q = 90 q = 18