Maths quadrilateral presentation
- 2. What is a Quadrilateral ？ Ø It is a four-sided polygon with four angles Ø The sum of interior angles is 360
- 3. Types of Quadrilateral Square Rectangle Parallelogram Rhombus Kite TrapeziumCyclic Quadrilateral
- 5. Rectangle What is a rectangle? A quadrilateral where opposite sides are parallel.
- 6. Properties of a Rectangle Ø Opposite sides are congruent Ø Opposite sides are parallel Ø Internal angles are congruent Ø All internal angles are right angled (90 degrees)
- 7. Perimeter of a Rectangle Step 1: Add up the sides DONE Example: Perimeter: x + y + x + y OR 2x + 2y
- 8. Area of a Rectangle How? Multiply the length with the width Example: Area = x . y
- 9. The Diagonal of a Rectangle To find the length of diagonal on a rectangle: Let diagonal = D D squared = x squared + y squared D
- 10. Properties of the Diagonals on Rectangles Ø Diagonals do not intersect at right angles Ø Angles at the intersection can differ Ø Opposite angles at intersection are congruent
- 11. Square What is a square? A quadrilateral with sides of equal length.
- 12. Properties of a Square Ø The sides are congruent Ø Angles are congruent Ø Total internal angle is 360 degrees Ø All internal angles are right-angled (90 degrees) Ø Opposite angles are congruent Ø Opposite sides are congruent Ø Opposite sides are parallel
- 13. Perimeter of a Square Step 1: Add up all the sides DONE Example: Perimeter: a + a + a + a OR 4a
- 14. Area of a Square How? Multiply two of the sides together or just “SQUARE” the length of one side Example: Area = a x a OR a squared
- 15. Diagonal of a Square To find the length of the diagonal of the square, multiply the length of one side with the square root of 2. Example: d = a
- 16. Properties of the Diagonals in Squares Ø The diagonals intersect at 90 degree angles (right-angled) Ø Diagonals are perpendicular Ø Diagonals are congruent
- 18. Parallelogram Opposite Interior Angles: Ø Equal Ø A = C B = D D + C = 180 Known as supplementary angles.
- 19. Parallelogram The diagonals: Ø Bisect each other Ø Intersect each other at the half way point Each diagonal separates it into 2 congruent triangles.
- 20. Perimeter of Parallelogram Perimeter = 2(a+b) a b
- 21. Area of Parallelogram Area = Base x Height
- 22. Area of Parallelogram (Example) Solution: 180o – 135o = 45o sin 45o = h / 15 h = 10.6 18 Area = Base x Height = 18 x 10.6 = 190.8
- 23. Rhombus A flat shape with 4 equal straight sides that looks like a diamond.
- 24. Properties of Rhombus 1 - All sides are congruent (equal lengths). length AB = length BC = length CD = length DA = a. 2 - Opposite sides are parallel. AD is parallel to BC and AB is parallel to DC. 3 - The two diagonals are perpendicular. AC is perpendicular to BD. 4 - Opposite internal angles are congruent (equal sizes). internal angle A = internal angle C and internal angle B = internal angle D. 5 - Any two consecutive internal angles are supplementary : they add up to 180 degrees. angle A + angle B = 180 degrees angle B + angle C = 180 degrees angle C + angle D = 180 degrees angle D + angle A = 180 degrees
- 25. Area of Rhombus
- 27. Example : Question : The lengths of the diagonals of a rhombus are 20 and 48 meters. Find the perimeter of the rhombus. Solution : • Below is shown a rhombus with the given diagonals. Consider the right triangle BOC and apply Pythagora's theorem as follows • BC 2 = 10 2 + 24 2 • and evaluate BC • BC = 26 meters. • We now evaluate the perimeter P as follows: • P = 4 * 26 = 104 meters.
- 28. CYCLIC QUADRILATERAL A cyclic quadrilateral is a quadrilateral when there is a circle passing through all its four vertices.
- 29. Theorem 1: Sum of the opposite angles of a cyclic quadrilateral is 180°. Example: ∠P + ∠R=180° and ∠S + ∠Q=180° Theorem 2: Sum of all the angles of a cyclic quadrilateral is 360°. Example: ∠P+∠Q+∠R+∠S = 360°
- 30. Proving Cyclic Quadrilateral Theorem
- 31. Area of Cyclic Quadrilateral The area of the cyclic quadrilateral with sides a,b,c and d, and perimeter S= (a+b+c+d)/2 is given by Brahmagupta’s Formula.
- 32. Kite Ø Two pairs of equal length - a & a, b & b, are adjacent to each other. Ø Diagonals are perpendicular to each other.
- 33. Perimeter = AB +BC + CD + DA Area = ½ x d1 x d2 PERIMETER AREA
- 34. Area = ½ x d1 x d2 = ½ x 4.8 x 10 = 24cm2 Area = ½ x d1 x d2 = ½ x (4+9) x (3+3) = 39m2 EXAMPLE
- 35. Find the length of the diagonal of a kite whose area is 168 cm2 and one diagonal is 14 cm. Solution: Given: Area of the kite (A) = 168 cm2 and one diagonal (d1) = 14 cm. Area of Kite = ½ x d1 x d2 168 = ½ x 14 x d2 d2 = 168/7 d2 = 24cm EXAMPLE
- 36. Trapezium Properties: Only one pair of opposite side is parallel.
- 37. Area
- 39. Example
- 41. Class Activity
- 42. Question 1 • Only one pair of opposite side is parallel.
- 43. Question 2 • Opposite sides are parallel. • All sides are congruent (equal lengths). • Opposite internal angles are congruent (equal sizes). • The two diagonals are perpendicular. • Any two consecutive internal angles are supplementary : they add up to 180 degrees.
- 44. Question 3 Opposite sides: • Parallel • Equal in length The diagonals: • Bisect each other • Intersect each other at the half way point Each diagonal separates it into 2 congruent triangles
- 45. Question 4 • Two pairs of equal length - a & a, b & b, are adjacent to each other. • Diagonals are perpendicular to each other.
- 46. End of Presentation Thank You.