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The outcomes of a random experiment are called events connected with the experiment. For example; ‘head’ and ‘tail’ are the outcomes of the random experiment of throwing a coin and hence are events connected with it. Now we can distinguish between two types of events.

Continue reading "Events in Probability |Mutually Exclusive,Impossible,Identical,Certain"

Moving forward to the theoretical probability which is also known as classical probability or priori probability we will first discuss about collecting all possible outcomes and equally likely outcome. When an experiment is done at random we can collect all possible outcomes

Continue reading "Theoretical Probability |Classical or A Priori Probability |Definition"

Worked-out probability questions answers are given here step-by-step to get the clear explanation to the student. 1. Out of 300 students in a school, 95 play cricket only, 120 play football only, 80 play volleyball only and 5 play no games. If one student is chosen at

Continue reading "Probability Questions Answers | Probability Examples | Step-by-Step "

We will discuss here the Circumcircle of a Triangle and the circumcentre of a triangle. A tangent that passes through the three vertices of a triangle is known as the circumcircle of the triangle. When the vertices of a triangle lie on a circle, the sides of the triangle

Continue reading "Circumcircle of a Triangle | Circumcentre of the Triangle | Diagram"

We will discuss here some Examples of Loci Based on Circles Touching Straight Lines or Other Circles. 1. The locus of the centres of circles touching a given line XY at a point M, is the straight line perpendicular to XY at M. Here, PQ is the required locus. 2. The locus of

Continue reading "Examples of Loci Based on Circles Touching Straight Lines | Diagram"

We will solve different type of problems on probability of rolling a die. A die is thrown 200 times and the numbers shown on it are recorded as given below: If the die is thrown at random, what is the probability of getting a (i) 4 (ii) 4 or 5 (iii) Prime number Solution:

Continue reading "Probability of Rolling a Die | Dice Roll Probability |Dice Probability"

Probability in everyday life, we come across statements such as: Most probably it will rain today. Chances are high that the prices of petrol will go up. I doubt that he will win the race. The words ‘most probably’, ‘chances’, ‘doubt’ etc., show the probability of occurrence

Continue reading "Probability |Terms Related to Probability|Tossing a Coin|Coin Probabil"

Here we will learn how to find the probability of tossing three coins. Let us take the experiment of tossing three coins simultaneously: When we toss three coins simultaneously then the possible

Continue reading "Probability of Tossing Three Coins | Tossing or Flipping Three Coins"

Probability for rolling two dice with the six sided dots such as 1, 2, 3, 4, 5 and 6 dots in each die. When two dice are thrown simultaneously, thus number of event can be 6^2 = 36 because each die has 1 to 6 number on its faces. Then the possible outcomes are shown in the

Continue reading "Probability for Rolling Two Dice | Sample Space for Two Dice |Examples"

Problems on coin toss probability are explained here with different examples. When we flip a coin there is always a probability to get a head or a tail is 50 percent. Suppose a coin tossed then we get two possible outcomes either a ‘head’ (H) or a ‘tail’ (T), and it is

Continue reading "Coin Toss Probability | Problems on Coin Toss | Outcomes in an Event"

Definition of Empirical Probability: The experimental probability of occurring of an event is the ratio of the number of trials in which the event occurred to the total number of trials. The empirical probability of the occurrence of an event E is defined as

Continue reading "Empirical Probability | Experiment of Rolling a Die| Tossing Two Coins"

We will discuss here about the definition of probability, terms related to probability, experiment, random experiment and trial. Introduction We often hear sentences like “it will possibly rain today”, “most probably the train will be late”, “most likely Robert will not come

Continue reading "Definition of Probability | Terms Related to Probability | Experiment"

We will discuss here about the Application problems on Area of a circle. 1. The minute hand of a clock is 7 cm long. Find the area traced out by the minute hand of the clock between 4.15 PM to 4.35 PM on a day. Solution: The angle through which the minute hand rotates in 20

Continue reading "Application Problems on Area of a Circle | Shaded Region of a Circle"

We will discuss about the important properties of transverse common tangents. I. The two transverse common tangents drawn to two circles are equal in length. Given: WX and YZ are two transverse common tangents drawn to the two given circles with centres O and P. WX and YZ

Continue reading "Important Properties of Transverse Common Tangents |Proof with Diagram"

Here we will learn how to find the area of the shaded region. To find the area of the shaded region of a combined geometrical shape, subtract the area of the smaller geometrical shape from the area of the larger geometrical shape. 1.A regular hexagon is inscribed in a circle

Continue reading "Find the Area of the Shaded Region | The Area of a Composite Figure"

We will learn how to find the Area of the shaded region of combined figures. To find the area of the shaded region of a combined geometrical shape, subtract the area of the smaller geometrical shape from the area of the larger geometrical shape. Solved Examples on Area of

Continue reading "Area of the Shaded Region | Shaded Areas | Area of Combined Figures"

A combined figure is a geometrical shape that is the combination of many simple geometrical shapes. To find the area of combined figures we will follow the steps: Step I: First we divide the combined figure into its simple geometrical shapes. Step II:Then calculate the

Continue reading "Area of Combined Figures | Area of Composite Shapes | Irregular Shapes"

We will learn how to find the Area and perimeter of a semicircle and Quadrant of a circle. Area of a semicircle = 1/2 ∙ πr^2 Perimeter of a semicircle = (π + 2)r. because a semicircle is a sector of sectorial angle 180°. Area of a quadrant of a circle = 1/4 ∙ πr^2.

Continue reading "Area and Perimeter of a Semicircle and Quadrant of a Circle | Examples"

10th grade math topics are planned and covered all the lessons in different segments. 10th grade math help is provided for the 10th grade students in all segments to cover all the math lesson plans which are categorized into Arithmetic, Algebra, Geometry, Mensuration and

Continue reading "10th Grade Math | Tenth Grade Math Lessons Plan | Practice Problems"

We will discuss the Area and perimeter of a sector of a circle. Problems on Area and perimeter of a sector of a circle 1. A plot of land is in the shape of a sector of a circle of radius 28 m. If the sectorial angle (central angle) is 60°, find the area and the perimeter

Continue reading "Area and Perimeter of a Sector of a Circle |Area of Sector of a Circle"

We will discuss the Area and Perimeter of a Circle. The area (A) of a circle (or circular region) is given by A = πr^2 where r is the radius and, by definition, π = Circumference/Diameter = 22/7 (Approximately). The circumference (P) of a circle, or the perimeter of a circle

Continue reading "Area and Perimeter of a Circle | Solved Examples | Diagram"

Here we will solve different types of problems on common tangents to two circles. 1.There are two circles touch each other externally. Radius of the first circle with centre O is 8 cm. Radius of the second circle with centre A is 4 cm Find the length of their common tangent

Continue reading "Problems on Common Tangents to Two Circles | Transverse Common Tangent"

Here we will solve different types of Problems on relation between tangent and secant. 1. XP is a secant and PT is a tangent to a circle. If PT = 15 cm and XY = 8YP, find XP. Solution: XP = XY + YP = 8YP + YP = 9YP. Let YP = x. Then XP = 9x. Now, XP × YP = PT^2, as the

Continue reading "Problems on Relation Between Tangent and Secant | Square of Tangent"

We will prove that, PQR is an equilateral triangle inscribed in a circle. The tangents at P, Q and R form the triangle P’Q’R’. Prove that P’Q’R’ is also an equilateral triangle. Solution: Given: PQR is an equilateral triangle inscribed in a circle whose centre is O.

Continue reading "Equilateral Triangle Inscribed | Sum of Three Angles of a Triangle"

We will prove that, in the figure ABCD is a cyclic quadrilateral and the tangent to the circle at A is the line XY. If ∠CAY : ∠CAX = 2 : 1 and AD bisects the angle CAX while AB bisects ∠CAY then find the measure of the angles of the cyclic quadrilateral. Also, prove that DB

Continue reading "Measure of the Angles of the Cyclic Quadrilateral | Proof with Diagram"

We will prove that, A tangent, DE, to a circle at A is parallel to a chord BC of the circle. Prove that A is equidistant from the extremities of the chord. Solution: Proof: Statement 1. ∠DAB = ∠ACB 2. ∠DAB = ∠ABC 3. ∠ACB = ∠ABC

Continue reading "Tangent is Parallel to a Chord of a Circle | Extremities of the Chord"

In Online Math Quiz on Progressions we will complete 10 multiple choice questions on Progressions. If a, b, c, d ∈ N and they are four consecutive terms of an AP then the ath, bth, cth, dth terms of a GP are in (i) AP (ii) GP (iii) HP (iv) None of these

Continue reading "Online Math Quiz on Progressions | 10 Multiple Choice Questions | Ans"

Here we will prove that two circles with centres X and Y touch externally at T. A straight line is drawn through T to cut the circles at M and N. Proved that XM is parallel to YN. Solution: Given: Two circles with centres X and Y touch externally at T. A straight line is

Continue reading "Contact of Two Circles | Point of Contact of Two Circles | Proof "

Here we will prove that two parallel tangents of a circle meet a third tangent at points A and B. Prove that AB subtends a right angle at the centre. Solution: Given: CA, AB and EB are tangents to a circle with centre O. CA ∥ EB. To prove: ∠AOB = 90°. Proof: Statement

Continue reading "Two Parallel Tangents of a Circle Meet a Third Tangent | Proof"

Kids have fun while enjoying math coloring pages. Select your favorite math coloring pages and print out the coloring sheets you like best and let’s start coloring. You can make your coloring colorful

Continue reading "Math Coloring Pages | Math Coloring Sheets |Free Coloring Pages |Enjoy"

We will prove that the tangents MX and MY are drawn to a circle with centre O from an external point M. Prove that ∠XMY = 2∠OXY. Solution: Proof: Statement 1. In ∆MXY, MX = MY. 2. ∠MXY = ∠MYX = x°. 3. ∠XMY = 180° - x°. 4. OX ⊥ XM, i.e., ∠OXM = 90°. 5. ∠OXY = 90° - ∠MXY

Continue reading "Two Tangents are Drawn to a Circle from an External Point | Proof"

A rectangular array of mn elements aij into m rows and n columns, where the elements aij belongs to field F, is said to be a matrix of order m × n (or an m × n matrix) over the field F. Definition of a Matrix: A matrix is a rectangular arrangement or array of numbers

Continue reading "Matrix | Definition of a Matrix | Examples of a Matrix | Elements"

The solved examples on the basic properties of tangents will help us to understand how to solve different type problems on properties of triangle. 1. Two concentric circles have their centres at O. OM = 4 cm and ON = 5 cm. XY is a chord of the outer circle and a tangent to

Continue reading "Solved Examples on the Basic Properties of Tangents | Tangent Circle"

We will solve some Problems on two tangents to a circle from an external point. 1. If OX any OY are radii and PX and PY are tangents to the circle, assign a special name to the quadrilateral OXPY and justify your answer. Solution: OX = OY, are radii of a circle are equal.

Continue reading "Problems on Two Tangents to a Circle from an External Point | Diagram"

We will discuss circumcentre and incentre of a triangle. In general, the incentre and the circumcentre of a triangle are two distinct points. Here in the triangle XYZ, the incentre is at P and the circumcentre is at O. A special case: an equilateral triangle, the bisector

Continue reading "Circumcentre and Incentre of a Triangle | Radius of Circumcircle"

We will discuss here the Incircle of a triangle and the incentre of the triangle. The circle that lies inside a triangle and touches all the three sides of the triangle is known as the incircle of the triangle. If all the three sides of a triangle touch a circle then the

Continue reading "Incircle of a Triangle |Incentre of the Triangle|Point of Intersection"

A common tangent is called a transverse common tangent if the circles lie on opposite sides of it. In the figure, WX is a transverse common tangent as the circle with centre O lies below it and the circle with P lie above it. YZ is the other transverse common tangent as the

Continue reading "Transverse Common Tangents | Circles lie on Opposite Sides | Diagram"

Important Properties of Direct common tangents. The two direct common tangents drawn to two circles are equal in length. The point of intersection of the direct common tangents and the centres of the circles are collinear. The length of a direct common tangent to two circles

Continue reading "Important Properties of Direct Common Tangents |Explained With Diagram"

A common tangent is called a direct common tangent if both the circles lie on the same side of it. The figures given below shows common tangents in three different cases, that is when the circles are apart, as in (i); when they are touching each other as in (ii); and when

Continue reading "Direct Common Tangents | Common Tangent | Three Different Cases"

Here we will prove that if a chord and a tangent intersect externally then the product of the lengths of the segments of the chord is equal to the square of the length of the tangent from the point of contact to the point of intersection. Given: XY is a chord of a circle and

Continue reading "Chord and Tangent Intersect Externally | The Length of the Tangent"

Here we will solve different types of Problems on properties of tangents. 1. A tangent, PQ, to a circle touches it at Y. XY is a chord such that ∠XYQ = 65°. Find ∠XOY, where O is the centre of the circle. Solution: Let Z be any point on the circumference in the segment

Continue reading "Problems on Properties of Tangents | Tangents of Circles and Angles"

Here we will prove that if a line touches a circle and from the point of contact a chord is down, the angles between the tangent and the chord are respectively equal to the angles in the corresponding alternate segments. Given: A circle with centre O. Tangent XY touches

Continue reading "Angles between the Tangent and the Chord | A Line Touches a Circle "

Here we will prove that from any point outside a circle two tangents can be drawn to it and they are equal in length. Given: O is the centre of a circle and T is a point outside the circle. Construction: Join O and T. Draw a circle with TO as diameter which cuts the given

Continue reading "Two Tangents from an External Point | Two Tangents are equal in Length"

Here we will learn about Secant and Tangent. Let a circle with centre O and a straight line AB be drawn on the same plane. Then only one of the following cases is possible: (i) The straight line AB does not touch or cut the circle at any point. (ii) The straight line AB cuts

Continue reading "Secant and Tangent | Properties of Tangents | Definition & Properties"

Here we will prove that if two circles touch each other, the point of contact lies on the straight line joining their centres. Case 1: When the two circles touch each other externally. Given: Two circles with centres O and P touch each other externally at T. To prove:

Continue reading "Two Circles Touch each Other | Common Tangent | Tangent of a Circle"

Here we will prove that the tangent at any point of a circle and the radius through the point are perpendicular to each other. Given: A circle with centre O in which OP is a radius. XPY is a tangent drawn to the circle at the point P. To prove: OP ⊥ XY. Construction: On XY

Continue reading "Tangent - Perpendicular to Radius | The tangent at any point of a ...."

We will discuss here about common tangents to two circles. 1. If one circle line completely inside another circle without cutting or touching it at any point then the circles will have no common tangent. 2. If two circles touch each other internally at one point, they will

Continue reading "Common Tangents to Two Circles | Transverse Common Tangents"

Principal: The money lent by the lender (or received by the borrower) is the principal. It is denoted by P. Time: The duration for which money is lent (or borrowed) is the time. It is denoted by T (or t). Interest: The difference of the money paid back and the money borrower

Continue reading "Compound Interest Definition | Some Basic Terms |Principal, Time, Rate"

In Worksheet on matrix the questions are based on finding unknown elements and matrices from matrix equation. (i) Find the matrix C(B – A). (ii) Find A(B + C). (iii) Prove that A(B + C) = AB + AC. 2. Show that 6X – X^2 = 9I, where I is the unit matrix.

Continue reading "Worksheet on Matrix | Solving Matrix Equations Worksheet | Answers"

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