Maths Quadrilaterals
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Transcript of Maths Quadrilaterals
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Quadrilateral
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What is a Quadrilateral ? It is a four-sided polygon with four angles The sum of interior angles is 360
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Types of Quadrilateral
Square Rectangle Parallelogram Rhombus
Kite TrapeziumCyclic Quadrilateral
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Rectangles and Squares
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RectangleWhat is a rectangle?
A quadrilateral where opposite sides are parallel.
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Properties of a Rectangle Opposite sides are congruent Opposite sides are parallel Internal angles are congruent All internal angles are right angled (90 degrees)
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Perimeter of a RectangleStep 1: Add up
the sides
DONEExample:
Perimeter: x + y + x + y
OR
2x + 2y
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Area of a RectangleHow?
Multiply the length with the width
Example:
Area = x . y
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The Diagonal of a Rectangle
To find the length of diagonal on a rectangle:
Let diagonal = D
D squared = x squared + y squared
D
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Properties of the Diagonals on Rectangles
Diagonals do not intersect at right angles
Angles at the intersection can differ
Opposite angles at intersection are congruent
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SquareWhat is a square?
A quadrilateral with sides of equal length.
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Properties of a Square
The sides are congruent Angles are congruent Total internal angle is 360 degrees All internal angles are right-angled (90 degrees) Opposite angles are congruent Opposite sides are congruent Opposite sides are parallel
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Perimeter of a SquareStep 1:
Add up all the sides
DONEExample:
Perimeter:a + a + a + a
OR
4a
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Area of a SquareHow?
Multiply two of the sides together or just “SQUARE” the length of one side
Example:
Area = a x a OR a squared
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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
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Properties of the Diagonals in Squares
The diagonals intersect at 90 degree angles (right-angled) Diagonals are perpendicular Diagonals are congruent
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Parallelogram
Opposite sides: Parallel Equal in length
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Parallelogram
Opposite Interior Angles: Equal
A = C
B = D
D + C = 180
Known as supplementary angles.
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Parallelogram
The diagonals: Bisect each other Intersect each other at the half way point
Each diagonal separates it into 2 congruent triangles.
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Perimeter of Parallelogram
Perimeter = 2(a+b)
a
b
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Area of Parallelogram
Area = Base x Height
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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
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RhombusA flat shape with 4 equal straight sides
that looks like a diamond.
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Properties of Rhombus1 - 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
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Area of Rhombus
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Perimeter of Rhombus
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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.
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CYCLIC QUADRILATERAL
A cyclic quadrilateral is a quadrilateral when there is a circle passing through all its four
vertices.
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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°
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Proving Cyclic Quadrilateral Theorem
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Area of Cyclic Quadrilateral
The area of the cyclic quadrilateral with sides a,b,c and d,
and perimeter S= a+b+c+d is given by Brahmagupta’s Formula.
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Kite
Two pairs of equal length - a & a, b & b, are adjacent to each other.
Diagonals are perpendicular to each other.
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Perimeter = AB +BC + CD + DA
Area = ½ x d1 x d2
PERIMETER
AREA
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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
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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
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Trapezium
Properties:Only one pair of opposite side is parallel.
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Area
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Example
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Class Activity
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Question 1
• Only one pair of opposite side is parallel.
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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.
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Question 3Opposite 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
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Question 4• Two pairs of equal length - a & a, b & b, are
adjacent to each other.• Diagonals are perpendicular to each other.