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What is a Quadrilateral ?
Ø It is a four-sided polygon with four angles Ø The sum of interior angles is 360
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)
The Diagonal of a Rectangle
To find the length of diagonal on a rectangle:
Let diagonal = D
D squared = x squared + y squared
D
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
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
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
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
Properties of the Diagonals in Squares
Ø The diagonals intersect at
90 degree angles
(right-angled)
Ø Diagonals are
perpendicular
Ø Diagonals are congruent
Parallelogram
Opposite Interior Angles: Ø Equal Ø A = C
B = D
D + C = 180 Known as supplementary angles.
Parallelogram
The diagonals: Ø Bisect each other Ø Intersect each other at the half way point Each diagonal separates it into 2 congruent triangles.
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
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
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.
CYCLIC QUADRILATERAL
A cyclic quadrilateral is a quadrilateral when there is a circle passing through all its four vertices.
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°
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.
Kite
Ø Two pairs of equal length - a & a, b & b, are adjacent to each other.
Ø Diagonals are perpendicular to each other.
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
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.
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
Question 4 • Two pairs of equal length - a & a, b & b, are adjacent
to each other. • Diagonals are perpendicular to each other.