Understand construction of an equilateral Triangle in a Given circle. Click on the link to Watch the VIDEO explanation: Watch Video
To Construct an Equilateral Triangle in a Given Circle
Draw a circle with centre O.
Draw 2 radii OA and OB such that
Angle AOB is equals to 120 degrees i.e. 360 degrees divided by 3. Since we constructing a triangle.
Using a compass measure arc AB and cuts arc AC and BC to the same measurement.
Now join AB, BC and AC.
You can see a triangle ABC is the required equilateral triangle.
Understand to calculate Sides of the Triangle. Click on the link to Watch the VIDEO explanation: Watch Video
Numerical
The sides of a right-angled triangle containing the right angle are 5x and 3x-1. If the area of the triangle is 60 cm2, find the sides of the triangle.
Solution:
Given: Area of triangle 60 cm2
1 by 2 *3x-1 *5x = 60
Therefore 15x2 - 5x - 120 = 0
Solving the equation 3x2 - x - 24 = 0 we get x = -8 by 3 and x = 3
Since the length of the side of the triangle cannot be negative discard the negative value x = -8 by 3 hence x = 3.
Taking the value X = 3 and find the values of the sides of the triangle
From triangle ABC we have AC2 = AB2 + BC2
Key in the values for the sides AB and BC and find the values of AC.
Hence, the sides of the triangle are 15 cm,8 cm and 17 cm.
Understand multiplication of matrices. Click on the Link to Watch the VIDEO explanation:
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Multiplications Of Matrices - Numerical
Let us solve the numerical on multiplication of matrices.
Problem:
If P = matrix 5 2 2 1 and Q = 1 0 0 1 find PQ.
Let us see if the two matrices P and Q are multipliable the condition for multiplying two matrices is. two matrices A and B can be multiplied if the number of columns of A = number of rows of B. The resulted matrix will be of the order number of rows A = number of columns of B . i.e. if A be a matrix of order m by n and b be a matrix of order n by q. then A and B can be multiplied and there product will be a matrix of order m by q.
Here, in this example matrix B is of the order 2 by 2 and the matrix Q is of the order 2 by 2 so they can multiplied since the no. of columns in matrix B is equal the no. of rows in matrix Q i.e. 2.
Therefore the order of matrix PQ will be 2 by 2. Now let us understand how to multiply the matrix P and matrix Q. Matrices P and Q can be multiplied by using the row column multiplication technique as illustrated below:
Multiply the first row first column element of matrix P with a first row first column element of matrix Q then multiply the first row second column element of matrix P with the second row first column element of matrix Q in the matrix PQ you have now got 5*1 + 2*0 this will be first row first column element of matrix PQ continue the process to get the remaining element of the matrix PQ. Thus, we get PQ = matrix is 5 2 2 1 which is of the order 2 by 2. Now that matrix PQ = matrix P.
Hence matrix Q is an identity matrix we called a definition of the identity matrix a square matrix in which every diagonal element is 1 and every non diagonal element is 0 is called an identity or unit matrix and the property of an identity matrix is when a matrix is multiplied by an identity matrix we get the same matrix i.e. A*I = A = I*A.
Click on the exercise button at the bottom right of your screen and solve the problem.
Understand to add and subtract matrices. Click on the link to Watch the VIDEO explanation: Watch Video
Addition and Subtraction of Matrices - Numerical
Let us learn how to add and subtract the matrices.
Problem:
Two Matrices A and B of order 3 by 3 are given.
Find A + 3B and 2A - 3B.
Since there are two matrices A and B are of equal order they can be added by adding there corresponding elements. We are require to add matrix A and 3 times of matrix B. So first multiplying of each element of matrix B by 3 and obtain matrix 3B. An illustration is shown to obtain the first row first column element of matrix 3B. Go ahead keep the values and obtain the remaining element of the matrix.
click on check to verify your answer. Else click on the solve button to get the matrix.
Now find matrix A + 3B
Good!
Therefore A + 3B = Matrix 7 12 16 -6 14 11 -17 -3 15
Now try and solve the 2A - 3B. click on the next button.
Understand Three Dimensional Geometry. Click on the link to Watch the VIDEO explanation: Watch Video
Three Dimensional Geometry
CO - ORDINATES OF A POINT IN SPACE
Let O be the Origin and let OX, OY and OZ be three mutually perpendicular lines taken as x - axis, y- axis, z- axis in such a way that they form a right - handed system.
The 3 mutually perpendicular lines determine 3 mutually perpendicular planes XOY, YOZ, ZOX known as co-ordinate planes.
Plane XOY is xy-plane, YOZ is called yz-plane and ZOX is called xz-plane.
The 3 co-ordinates planes XOY, YOZ, ZOX divide the space into eight compartments known as octants.
Let P be a point in space. Through P draw planes parallel to coordinate planes meeting the axes OX, OY, OZ in points respectively. Complete the parallelopiped whose coterminous edges are OA, OB, OC.
Let OA = x, OB = y, OZ = z.
Then (x,y,z) are the co-ordinates of P.
x = distance of P from yz-plane.
y = distance of P from xz-plane.
z = distance of P from xy-plane.
Every point in yz plane has x co-ordinate 0.
Every point in xy plane has z co-ordinate 0.
every point in xz plane has y co-ordinate 0.
Any point on x- axis is of the form X,0,0.
Any point on y- axis is of the form 0,Y,0.
Any point on z- axis is of the form 0,0,Z.
Understand compound interest with the help of a numerical. click on the link to Watch the VIDEO explanation: Watch Video
Numerical
Find the compound interest on Rs. 1500 for 5 years at 9.2% p.a. when C.I. is compounded quarterly.
Solution:
The given data is
The principal amount P = RS. 1500
the rate of interest R = 9.2% p.a.
The time period = 5 years.
Mass over on Formula to know the formula to calculating the amount. Enter the values to calculating the amount in the boxes.
Solve for the solution
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That's right!
Click Solve for the solution
Try again
That's Right.
Taking logs on both sides we get log of A = log of 1500 and 20 log of (1.023)
using the log tables find the log returns of the numbers thousand 500 and 1.023 now enter the correct values in the respective boxes.
Solve for the solution
Try Again
That's right!
Wrong
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That's Right.
Find the anti log of 3.3721 using anti log tables.
Solve for the solution
Wrong
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That's Right.
Mouse Over on the formula button to know the formula to calculating the compound interest key in the values for the amount and principal and calculate the values for the compound interest.
Solve for the solution
Wrong
Try again
That's Right.
You have solved the problem correctly the compound interest is found to be RS. 856.
Understand irrational number on number line. Click on the Link to Watch the VIDEO explanation:
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Representation Of Irrational Numbers On The Number Line
We should now learn the method of representing root 2 and root 5. Take up one of the links.
To represent root 2 on the number line.
Solution:
Draw a number line.
Let OA = 1 unit.
Draw a perpendicular at A.
Mark AB = 1 unit.
Join OB.
Since angle OAB = 900, triangle OAB is right- angled at A.
In the right-angled triangle OAB,
OB square = OA square + AB square by pythagoras theorem.
Therefore OB square = 2
Therefore OB = root 2
with O as centre and compass radius as OB, cut an arc to get point P on the number line.
Then, the point P represents root 2 on the number line.
Click the home button to return to the main menu.
To represent root 5 on the number line.
Solution:
Draw a number line.
Let OA = 2 units.
Draw a perpendicular at A.
Mark AB = 1 unit.
Join OB.
Since angle OAB = 900, triangle OAB is right- angled triangle.
In the right-angled triangle OAB,
OB square = OA square + AB square by pythagoras theorem.
Therefore OB square = 5
Therefore OB = root 5
with O as centre and compass radius as OB, cut an arc to get point P on the number line.
Then, the point P represents root 5 on the number line.
Understand Tangent, Secant and Chord of a circle. Click on the link to Watch the VIDEO explanation:
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Tangent, Secant and Chord of a circle
Definition of Tangent, Secant and Chord of a circle
A line in the plane of a circle and having one and only one point common with it is known as tangent of the circle.
The common point is called the point of contact.
Here, O is the center of the circle, line l is the tangent and A is the point of Contact.
A line which intersects a circle at two points is called a secant to the circle.
Line m is the secant and it intersects the circle at A and B.
A segment whose endpoints lie on the circle is called a chord of the circle.
AB is the chord to the circle.
Understanding the theorem on arcs of circles. Click on the Link to Watch the VIDEO explanation:
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Theorem on Arcs of a circle.
In equal circles or in a same circles, if two arcs are subtend equal angles at the centers, they are equal.
Given AXB and CYD are equal circles with centres P and Q; arcs AMB and CND subtend equal angles angle APB and angle CQD.
To prove is that arc AMB is equal to arc CND
Proof is as follows:
Apply circle CYD to circle AXB so that centre Q falls on the centre P and QC along PA and D on the same side as B. Therefore, circle CYD overlaps circle AXB.
Therefore, C falls on A. Since circles are equal
Angle APB is equal to angle CQD. Given
Therefore, QD falls along PB.
Understand the meaning of Scalars and Vectors. click on the link to Watch the VIDEO explanation:
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Scalars and Vectors
A scalar quantity or briefly a scalar has a magnitude but is not related to any direction in space.
The fig. shows the liquids filled in a beaker and a bowl. The volume that this liquid occupies is a scalar as it has only magnitude and there is no direction.
Here in this fig. we see a thermometer is placed in a glass fill with liquid. With temperature of the liquid is shown in the thermometer is a scalar.
Mass is a scalar Quantity. Every real no is a scalar.
A vector is a physical Quantity which requires both magnitude and direction for its representation. Velocity is a vector.
Here we see motion of cars in different directions with common speed they are the examples of the vector.
Displacement is another example of a vector. Here we see the triangle ABC is shifted to a new position A dash, B dash, C dash. Movement of triangle ABC to new position A dash B dash C dash is an example of a vector.
