Course Content
Chapter 01 – Operations on Sets
The set operations are performed on two or more sets to obtain a combination of elements as per the operation performed on them. In a set theory, there are three major types of operations performed on sets, such as: Union of sets (∪) The intersection of sets (∩) Difference of sets ( – ) In this lesson we will discuss these operations along with their Venn diagram and will learn to verify the following laws: Commutative, Associative, Distributive, and De-Morgans' law.
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Subsets and Super Sets
03:10
Venn Diagrams
03:32
Operations on Set
00:00
Union of Sets
06:11
Intersection of Sets
06:13
Difference of Sets
05:15
Complement of Sets
06:21
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Venn Diagrams (Three or More Sets)
03:21
Let's Try
Commutative Law of Union and Intersection
02:53
Associative Law of Union and Intersection
05:01
Distributive Law over Union and Intersection
04:13
De Morgan’s Law
05:25
Chapter 02 – Real Numbers
All real numbers follow three main rules: they can be measured, valued, and manipulated. Learn about various types of real numbers, like whole numbers, rational numbers, and irrational numbers, and explore their properties. In this chapter, we will learn about Squares and cubes of real numbers and find their roots.
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Irrational Numbers
05:54
Squares
03:04
Square Roots
03:49
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Cubes and Cube Roots
11:23
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Chapter 03 – Number System
The number system or the numeral system is the system of naming or representing numbers. There are different types of number systems in Mathematics like decimal number system, binary number system, octal number system, and hexadecimal number system. In this chapter, we will learn different types and conversion procedures with many number systems.
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Number System
08:13
Conversion – Base 2
00:00
Addition in Binary System
03:02
Subtraction in Binary System
02:32
Multiplication in Binary System
02:43
Conversion – Base 5
00:00
Addition in Base 5
01:55
Subtraction in Base 5
01:38
Multiplication in Base 5
04:17
Conversion – Base 8
08:19
Addition & Subtraction in Base 8
06:07
Multiplication in Base 8
04:39
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Chapter 04 – Financial Arithmetic
Financial mathematics describes the application of mathematics and mathematical modeling to solve financial problems. In this chapter, we will learn about partnership, banking, conversion of currencies, profit/markup, percentage, and income tax.
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Compound Proportion
05:10
Partnership
07:17
Inheritance
09:53
Conversion of Currencies
03:13
Profit or Markup
07:00
Percentage
08:30
Let's Try
Taxes
03:46
Income Tax
02:15
Let's Try
Banking
06:16
Chapter 05 – Polynomials
In algebra, a polynomial equation contains coefficients, exponents, and variables. Learn about forming polynomial equations. In this chapter, we will study the definition and the three restrictions of polynomials, we'll tackle polynomial equations and learn to perform operations on polynomials, and learn to avoid common mistakes.
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Algebraic Expression
02:38
Polynomials
08:19
Addition and Subtraction of Polynomial
15:05
Multiplication of Polynomials
02:42
Division of Polynomial
10:49
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Chapter 06 – Factorization, Simultaneous Equations
In algebra, factoring is a technique to simplify an expression by reversing the multiplication process. Simultaneous Equations are a set of two or more algebraic equations that share variables and are solved simultaneously. In this chapter, we will learn about factoring by grouping, review the three steps, explore splitting the middle term, and work examples to practice verification and what simultaneous equations are with examples. Find out how to solve the equations using the methods of elimination, graphing, and substitution.
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Algebraic Formulas
06:35
Stimulation – Area Model Algebra
00:00
Factorization
00:00
Manipulation of Algebraic Expression
00:00
Stimulation – Equality Explorer
00:00
Linear Equations in One Variable
00:00
Simultaneous Linear Equations
00:00
Stimulation – Equality Explorer (Two Variables)
00:00
Elimination
00:00
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Chapter 07 – Fundamentals of Geometry
Geometry is the study of different types of shapes, figures, and sizes. It is important to know and understand some basic concepts. We will learn about some of the most fundamental concepts in geometry, including lines, polygons, and circles.
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Parallel Lines
05:35
Polygons
04:11
Circles
03:14
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Chapter 08 – Practical Geometry
Geometric construction offers the ability to create accurate drawings and models without the use of numbers. In this chapter, we will discover the methods and tools that will aid in solving math problems as well as constructing quadrilaterals and right-angled triangles.
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Construction of Quadrilaterals
11:16
Construction of a Right Angled Triangle
07:38
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Chapter 09 – Areas and Volumes
The volume and surface area of a sphere can be calculated when the sphere's radius is given. In this chapter, we will learn about the shape sphere and its radius, and understand how to calculate the volume and surface area of a sphere through some practice problems. Also, we will learn to use and apply Pythagoras' theorem and Herons' formula.
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Pythagoras Theorem
00:00
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Math Lab
Heron’s Formula
04:42
Let's Try
Surface Area, and Volume of a Sphere
03:20
Let's Try
Chapter 10 – Demonstrative Geometry
Demonstrative geometry is a branch of mathematics that is used to demonstrate the truth of mathematical statements concerning geometric figures. In this chapter, we will learn about theorems on geometry that are proved through logical reasoning.
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Reasoning
03:34
Axioms
00:00
Postulate
05:01
Theorems
00:00
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Chapter 11 – Trigonometry
Sine and cosine are basic trigonometric functions used to solve the angles and sides of triangles. In this chapter, we will review trigonometry concepts and learn about the mnemonic used for sine, cosine, and tangent functions.
0/7
What is Trigonometry?
00:00
Visualization of Trigonometric Ratios
02:55
Let's Try
Trigonometric Ratios of Complementary Angles
05:21
Simulation – Trig Tour
00:00
Angle of Elevation and Angle of Depression
04:22
Let's Try
Chapter 12 – Information Handling
Frequency distribution, in statistics, is a graph or data set organized to show the frequency of occurrence of each possible outcome of a repeatable event observed many times. Measures of central tendency describe how data sets are clustered in a central value. In this chapter, we will learn to construct the frequency distribution table, and learn more about three measures of central tendency, its importance, and various examples.
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Frequency Distribition
01:03
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Bar Graph
07:49
Histogram
00:00
Pie Chart
04:29
Let's Try
Measures of Central Tendency
00:00
Mean, Median, and Mode
03:17
Let's Try
Mathematics – VIII
About Lesson

Quadrilateral

A quadrilateral refers to a four-sided polygon that has four angles. There are many types of quadrilaterals. The five most common types of quadrilaterals are the parallelogram, the rectangle, the square, the trapezoid, and the rhombus.

To determine geometric designs four important tools of geometry are-compass, straightedge, protractor, and ruler are usually used.

  • Compass– helps to draw the different angles.
  • Straightedge – it has two set squares, used to draw symmetric lines.
  • Protractor – for determining and drawing correct and accurate angles.
  • Ruler – Used to measure and draw lines to construct the figures.

Construction of Quadrilaterals

The following types of quadrilateral will be constructed:

  • If four sides and one diagonal of a quadrilateral are given.
  • If two diagonals and three sides of a quadrilateral are given.
  • If two adjacent sides and three angles of a quadrilateral are given.
  • If three sides and two included angles of a quadrilateral are given.
  • If other special properties of a quadrilateral are known.

If four sides and one diagonal of a quadrilateral are given:

Follow the following steps to draw the quadrilateral:

Construct a quadrilateral PQRS where PQ= 5 cm, QR = 7 cm, RS = 6 cm, PS = 6.5 cm and PR = 8 cm.

Construct a rough sketch for reference first. Then follow the following:

Step 1: As we can see in the rough sketch, it is easy to see that ∆PQR can be constructed using SSS construction conditions. Draw ∆PQR.

Step 2: After this, we have to locate the fourth point S. This is on the side opposite to Q with reference to PR. For that, we have two measurements.

S is 6 cm away from point P. So, with P as centre an arc is drawn of radius 6 cm.

Step 3: S is 6.5 cm away from R. So with R as centre, draw an arc of radius 6.5cm

Step 4: S lies on the point of intersection of two arcs. Mark S and complete PQRS. PQRS is the required quadrilateral.

If two diagonals and three sides of a quadrilateral are given:

Follow the following steps to draw the quadrilateral:

Construct a quadrilateral ABCD, given that BC = 5.5 cm, AD = 6.5 cm, CD = 5 cm the diagonal AC = 6 cm and diagonal BD = 8 cm.

First, draw a rough sketch of the quadrilateral ABCD.

Step 1: Draw ∆ACD using SSS construction.

Step 2: With D as centre we draw an arc of radius 7 cm. With C as centre we draw an arc of radius 4.5 cm.

Step 3: With C as centre we draw an arc of radius 4.5 cm.

Step 4: Since B lies on both the arcs, B is the point intersection of the two arcs. ABCD is the required quadrilateral.

If two adjacent sides and three angles of a quadrilateral are given:

Follow the following steps to draw the quadrilateral:

Construct a quadrilateral MIST where MI = 3.5 cm, IS = 6.5 cm, ∠M = 75°, ∠I = 105° and ∠S = 120°.

Draw a rough sketch first.

Step 1: We first locate the points.

Step 2: Construct ∠IST = 120° at S

Step 3: Construct ∠IMT = 75° at M.

Step 4: We get the required quadrilateral MIST.

If three sides and two included angles of a quadrilateral are given:

Follow the following steps to draw the quadrilateral:

Construct a quadrilateral ABCD, where ST = 5 cm, TE = 6 cm, EP = 7.5 cm and ∠T = 105° and ∠E = 80°.

A rough sketch is drawn first

Step 1: Start with taking TE = 6 cm on Draw an angle of 105° along with TX.

Locate a point 5 cm away on this. We now have T, E, and S

Step 2: The fourth point P is on EY which is inclined at 80° to TE. So make ∠TEY = 80° at C on TE.

Step 3: D is at a distance of 7.5 cm on EY. With E as centre, draw an arc of length 7.5 cm. It cuts EY at P.

Step 4: Complete the quadrilateral STEP.

Step 5: STEP is the required quadrilateral.

Question 1: Draw a square with side 6cm.

Answer: 

We can see that only the side is given but we know that square is a special figure with all the angles equal to right angle. We can use this information and draw the square.

Step 1: Draw a rough figure.

Step 2: Taking 6cm as base draw a line AB.

Step 3: Draw right angles at vertex A and B.

Step 4: Cut 6cm on the rays drawn from A and B respectively.

Step 5: Name the cut points C and D respectively.

Step 6: Join CD.

Step 7: ABCD is the required square.

Question 2: Construct a rhombus ABCD where AC = 6 cm and BD = 7 cm.

Answer: 

The measurement of two diagonals is given. As we know,

The diagonals of a rhombus are perpendicular bisectors of each other.

Step 1: Firstly we will draw AC= 6cm and then we will draw its perpendicular bisector.

Step 2: Let them meet at 0.

Step 3: Cut off 3 cm lengths on either side of the drawn Bisector.

Step 4: We get B and D.

Step 5: Join the lines and we get the required rhombus ABCD.

Construct quadrilaterals when four sides and one diagonal is given.

Construct a quadrilateral PQRS where PQ = 4 cm, QR = 6 cm, RS = 5 cm, PS = 5.5 cm and PR= 7 cm

Quadrilaterals

  • Draw Δ PQR  using SSS construction condition.
  • With P as the centre, draw an arc of radius 5.5 cm.
  • With R as the centre, draw an arc of radius 5 cm.
  • S is the point of intersection of the two arcs. Also, mark S and complete PQRS.
  • PQRS is the required quadrilateral.

Construct a quadrilateral ABCD where BC = 4.5 cm, AD = 5.5 cm, CD = 5cm and the diagonal AC = 5.5 cm, diagonal BD = 7 cm

Quadrilaterals

  • Draw ΔACD using SSS construction condition
  • Taking D as the centre, draw an arc of radius 7 cm.
  • Now let C  be the centre, draw an arc of radius 4.5 cm
  • Since B lies on both the arcs, B is the point intersection of the two arcs.
  • Mark B and complete ABCD.
  • ABCD is the required quadrilateral.

Construct a parallelogram when two consecutive sides and the included angle are given.

Construct a parallelogram ABCD with sides AB = 4 cm and AD = 5 cm and ∠A = 60

Quadrilaterals

  • First, construct a line segment AB = 4 cm and construct a 60 angle at point A.
  • Now construct a line segment AD = 5 cm on the other arm of the angle. Then, place the sharp point of the compasses at B and make an arc 5 cm above B.
  • Stretch your compasses to 4 cm, place the sharp end at D and draw an arc to intersect the arc drawn in step 2.
  • Label the intersecting point C. Join C to D and B to C to form the parallelogram ABCD.

Construct a parallelogram when two consecutive sides and a diagonal are given.

Construct a parallelogram ABCD in which AB = 6cm, BC = 4.5cm and diagonal AC = 6.8 cm.

Quadrilaterals

  • Draw AB = 6 cm.
  • With A as centre and radius 6.8 cm, draw an arc.
  • With B as centre and radius, 4.5 cm draw another arc, cutting the previous arc at C.
  • Join BC and AC.
  • With A as centre and radius 4.5 cm, draw an arc.
  • With C as centre and radius, 6 cm draw another arc, cutting the previously drawn arc at D.
  • Join DA and DC
  • ABCD is the required parallelogram.
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