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Mar 31, 2020

Events in Probability |Mutually Exclusive,Impossible,Identical,Certain

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.

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Mar 30, 2020

Theoretical Probability |Classical or A Priori Probability |Definition

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

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Mar 30, 2020

Probability Questions Answers | Probability Examples | Step-by-Step

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

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Mar 30, 2020

Circumcircle of a Triangle | Circumcentre of the Triangle | Diagram

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

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Mar 29, 2020

Examples of Loci Based on Circles Touching Straight Lines | 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

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Mar 28, 2020

Probability of Rolling a Die | Dice Roll Probability |Dice Probability

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:

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Mar 27, 2020

Probability |Terms Related to Probability|Tossing a Coin|Coin Probabil

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

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Mar 27, 2020

Probability of Tossing Three Coins | Tossing or Flipping Three Coins

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

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Mar 26, 2020

Probability for Rolling Two Dice | Sample Space for Two Dice |Examples

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

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Mar 26, 2020

Coin Toss Probability | Problems on Coin Toss | Outcomes in an Event

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

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Mar 25, 2020

Empirical Probability | Experiment of Rolling a Die| Tossing Two Coins

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

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Mar 24, 2020

Definition of Probability | Terms Related to Probability | Experiment

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

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Mar 23, 2020

Application Problems on Area of a Circle | Shaded Region of a Circle

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

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Mar 21, 2020

Important Properties of Transverse Common Tangents |Proof with Diagram

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

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Mar 20, 2020

Find the Area of the Shaded Region | The Area of a Composite Figure

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

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Mar 19, 2020

Area of the Shaded Region | Shaded Areas | Area of Combined Figures

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

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Mar 18, 2020

Area of Combined Figures | Area of Composite Shapes | Irregular Shapes

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

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Mar 17, 2020

Area and Perimeter of a Semicircle and Quadrant of a Circle | Examples

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.

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Mar 16, 2020

10th Grade Math | Tenth Grade Math Lessons Plan | Practice Problems

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

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Mar 16, 2020

Area and Perimeter of a Sector of a Circle |Area of Sector of a Circle

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

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Mar 15, 2020

Area and Perimeter of a Circle | Solved Examples | Diagram

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

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Mar 13, 2020

Problems on Common Tangents to Two Circles | Transverse Common Tangent

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

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Mar 11, 2020

Problems on Relation Between Tangent and Secant | Square of 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

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Mar 11, 2020

Equilateral Triangle Inscribed | Sum of Three Angles of a Triangle

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.

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Mar 08, 2020

Measure of the Angles of the Cyclic Quadrilateral | Proof with Diagram

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

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Mar 07, 2020

Tangent is Parallel to a Chord of a Circle | Extremities of the Chord

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

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Mar 07, 2020

Online Math Quiz on Progressions | 10 Multiple Choice Questions | Ans

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

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Mar 04, 2020

Contact of Two Circles | Point of Contact of Two Circles | Proof

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

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Mar 01, 2020

Two Parallel Tangents of a Circle Meet a Third Tangent | 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

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Mar 01, 2020

Math Coloring Pages | Math Coloring Sheets |Free Coloring Pages |Enjoy

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

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Feb 28, 2020

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

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

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Feb 27, 2020

Matrix | Definition of a Matrix | Examples of a Matrix | Elements

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

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Feb 27, 2020

Solved Examples on the Basic Properties of Tangents | Tangent Circle

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

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Feb 27, 2020

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

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.

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Feb 25, 2020

Circumcentre and Incentre of a Triangle | Radius of Circumcircle

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

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Feb 24, 2020

Incircle of a Triangle |Incentre of the Triangle|Point of Intersection

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

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Feb 20, 2020

Transverse Common Tangents | Circles lie on Opposite Sides | Diagram

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

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Feb 18, 2020

Important Properties of Direct Common Tangents |Explained With 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

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Feb 18, 2020

Direct Common Tangents | Common Tangent | Three Different Cases

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

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Feb 18, 2020

Chord and Tangent Intersect Externally | The Length of the Tangent

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

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Feb 18, 2020

Problems on Properties of Tangents | Tangents of Circles and Angles

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

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Feb 18, 2020

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

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

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Feb 17, 2020

Two Tangents from an External Point | Two Tangents are equal in Length

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

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Feb 17, 2020

Secant and Tangent | Properties of Tangents | Definition & Properties

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

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Feb 13, 2020

Two Circles Touch each Other | Common Tangent | Tangent of a Circle

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:

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Feb 03, 2020

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

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

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Feb 02, 2020

Common Tangents to Two Circles | Transverse Common Tangents

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

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Jan 26, 2020

Compound Interest Definition | Some Basic Terms |Principal, Time, Rate

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

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Jan 20, 2020

Worksheet on Matrix | Solving Matrix Equations Worksheet | Answers

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.

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