Triangle Problem and Solution

Question: Prove that the points A(-5, 4), B(-1, -2) and C(5, 2) are the vertices of an isosceles right angled triangle.

Given that ABC is a triangle.



If ABC is an isosceles D le, then AB = BC.













Hence ABC is an isosceles triangle.

If it is a right angled triangle then,

AC² = AB² + BC²







AC² = 104 units





= 2 x 52

= 104 units

Hence AC² = AB² + BC² and AB = BC

is an isosceles right angled triangle.

In the above Cartesian system problem solving, we can see while calculating the Pythagoras theorem is used.

Defination of Matrix

Definition of a Matrix:

A rectangular array of entries is called a Matrix. The entries may be real, complex or functions.

The entries are also called as the elements of the matrix.

The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. Matrices are denoted by capital letters.

Example:

(i)

Note that the entries in a given matrix need not be distinct.

(ii)

The entries in this matrix are function of x.

A matrix having m rows and n columns is called as matrix of order mxn. Such a matrix has mn elements.

In general, an mxn matrix is in the form

Where a_ij represents the element in i^th column.

The above matrix may be denoted as A = [a_ij]_mxn.

Introduction to algebra with integers:

Introduction to algebra with integers:

The set of integers, Z, consists of the whole numbers and their negative counterparts. Z = { …, -3, -2, -1, 0, 1, 2, 3, … }
The absolute value of a number is the distance between the number and zero on a number
It is defined by the formula: x = x, if x ≥ 0 − x, if x <>
The algebra with integers is the set of integers that has whole number and their negative counterparts. The algebra with integers include different operations as addition, subtraction, multiplication, division of algebra with integers.Let us see the algebra with integers concepts and example problems.

Types of Integers

Types of Integers:

There are two types of Integers:

1. Positive Integers:
Positive integers are whole numbers, which are greater than zero. For example, 25, 27, 103, 758…etc.
2. Negative Integers:
Negative integers are the opposites of the whole numbers. For example, -5, -22, -38, -504, -4585…etc. Negative numbers indicated by the sign (-). Zero is neither positive nor negative.

Rules for Dividing Integers:

The rules for solving dividing integers is explained below:
Consider this division example: 24 ÷ 4 = 6.

In division each number is referred by a special name.

Here, 24 is dividend, 4 is divisor, and 6 is quotient

quotient × divisor = dividend

dividend ÷ divisor = quotient

dividend ÷ quotient = divisor

Rules for Solving Dividing Integers

1) Positive ÷ Positive = Positive

Example: 28 ÷ 7 = 4

28, 7, and 4 are positive.

2) Negative ÷ Negative = Positive

Example: -28 ÷ -7 = 4

28 and 7 are negative, but 4 is positive.

3) Negative ÷ Positive = Negative

Example: -28 ÷ 7 = -4

28 is negative, 7 is positive, but 4 is negative.

4) Positive ÷ Negative = Negative
Example: 28 ÷ -7 = -4

28 is positive, 7 is negative, and 4 is negative.

Geometrical Interpretation - Scalar Triple Product Proof

Suppose there exists a parallelepiped with vectors a, b and c along sides OA, OB and OC respectively.

Height = OA

= a cos

where angle which the height OA makes with the base of the parallelepiped is

Area of base = area of parallelogram OBDC

= | b * c | (from definition of cross product)

= |b| |c| sin
where angle between OB and OC is
Volume of parallelepiped = Area of base * Height
= Area of parallelogram OBDC * OA
= (|b| |c| sin theta) * ( |a| cos alpha)
= ( |b| * |c| ) a cos= a . ( b * c)

Definition- Calculate Ratio Math

Definition- Calculate Ratio Math:

The ration of two numbers r and s(s≠0) is the section of the numbers. The numbers r and s are called the terms of the ratio.

Concept - calculate ratio math:

The numeric ratio of two numbers r and s(s≠0) is the section of the numbers. The numbers r and s are called the conditions of the numeric ratio.

Types of ratio- calculate ratio math:

1. Compounded ratio in math.
2. Duplicate ratio in math.
3. Triplicate ratio in math.

Perfect square of a trinomial

Perfect square of a trinomial:

If all the terms of the polynomial have a common factor, we take out the common factor and factorise.
If the polynomial can be expressed as the difference of two squares,
we use a² - b² = (a + b) (a - b).

• If all the terms of the polynomial have a common factor, we take out the common factor and factorise .
• If all the terms of the polynomial have a common factor, we take out the common factor and factorise .

• If the polynomial can be expressed as the difference of two squares,

we use a² - b² = (a + b) (a - b)

• Quadratic trinomials of the form x² + ax + b can be factorised using the identity. (x + a) (x + b) = x² + x(a + b) + ab.

• When the trinomial is ax² + bx + c and , we follow the following steps. We find two factors whose sum is b, and whose product is a x c.

We split the middle term using these two factors and factorise by grouping the terms.

• If the polynomial can be expressed as the sum or difference of two cubes we use the following identities.

a³ + b³ = (a + b) (a² - ab + b²)
a³ - b³ = (a - b) (a² + ab + b²)

What are equal and parallel line?

What are equal and parallel line?


If two parallel lines are cut by a transverse, the alternate angles are equal.

and
These are two pairs of alternate angles.



A transversal intersects two lines. If the alternate angles are equal, then the lines are parallel.
If or then AB is parallel to CD.

Divisibility rules

Divisibility rules:

• Numbers ending with 0‚ 2‚ 4‚ 6 or 8 are divisible by 2
• If the sum of the digits of a given number is divisible by 3, the number is divisible by 3.
• If the number formed by the end two digits of a given number is divisible by 4‚ then the number will be divisible by 4.
• Numbers ending with 0 or 5 are divisible by 5
• If the number formed by the end three digits of a given number is divisible by 8‚ then the number will be divisible by 8.
• If the sum of the digits of a number is divisible by 9, the number is divisible by 9.
• Numbers ending with 0 are divisible by 10
• If the difference of the sums of the digits in alternate places is divisible by 11, the number is divisible by 11.

Addition Rule of Probability

The Addition Rule:

Probability is the probably that event will happen – how likely the event will happen. The addition rule for probability: a statistical property that states the probability of one and/or two events occurring at the same time is equal to the probability of the first event occurring, plus the probability of the second event occurring, minus the probability that both events occur at the same time.
If events A and B are mutually exclusive or disjoint, then P(A U B) = P(A) + P(B)
Otherwise, P(A U B) = P(A) + P(B) – P(A ∩ B).

Quadrants

Quadrants:




Let X'OX and Y'OY perpendicular coplanar lines intersecting each other at O. We refer X'OX as x-axis and Y'OY as y-axis. It is clear from the adjoining figure, that these two lines divide the plane into four equal parts, each part is called a Quadrant.
The four Quadrants are:
XOY - first Quadrant
YOX' - second Quadrant
X'OY' - third Quadrant
Y'OX - fourth Quadrant

Trigonometry Heights and Distances

Trigonometry Heights and Distances:

• The word "Trigonometry" is derived from the two Greek words meaning measurements or solution of triangles. Trigonometry which is a branch of mathematics that deals with the ratio between the sides of a right triangle and its angles.
• Trigonometry is the study about relationships between the sides and angles of a triangle.
• Trigonometry is used in surveying and to determine Heights and Distances, in navigation it is to determine the location and the distances, and in the fields like nondestructive testing for determining things such as the angle for reflection or refraction of an ultrasound wave.

Types of Angles

Types of angles:

Algebra

Algebra is one of the main branches of mathematics. It explains the relations and properties of quantity by means of letters and other symbols. Algebra is used to simplify the long and very complicated statements. The basic algebra has the following subtopics are

1. Variables
2. Expressions
3. Terms
4. Polynomials
5. Equations.

Theorem in Mathematics

Theorem in Mathematics:

Congruent Supplements Theorem: When two angles are supplementary to the congruent angles or to the same angle, then the given angles are congruent.
Vertical Angle Theorem: Two angles are said as congruent even when they are vertical angles.
Alternate Interior Angle Theorem:Pairs of alternative interior angles are said as congruent when two parallel lines are cut transversely.
Consecutive Interior Angle Theorem:Pairs of consecutive interior angle are said as supplementary when two parallel lines are cut transversely.
Alternative Exterior Angle Theorem: Pairs of alternative exterior angle are said as congruent when two parallel lines are cut transversely.

Parabola equation

X intercept and Y intercept for Parabola equation:
The general form of parabola equation is
y = ax2 + bx + c where a,b and c are parts of the parabola.
Here, x intercepts are the roots of the equation of parabola. The x intercepts are the roots of the equation 0 = ax2 + bx + c. The very common methods to solve the equation are by factoring or by quadratic formula. The y intercept is (0 , c) for parabola equation.

Solve the equations Graphically

Solve the equations Graphically:

Check whether the pair of equations x + 3y = 6 and 2x – 3y = 12 is consistent. If so, solve them graphically.
Solution : Let us draw the graphs of the Equations (1) and (2). For this, we find two
solutions of each of the equations, which are given in Table

Plot the points A(0, 2), B(6, 0), P(0, – 4) and Q(3, – 2) on graph paper, and join the points to form the lines AB and PQ as shown in Fig below We observe that there is a point B (6, 0) common to both the lines AB and PQ. So, the solution of the pair of linear equations is x = 6 and y = 0, i.e., the given pair of equations is consistent.

General Equation of a Line

General Equation of a Line:
Equation of a Straight line is also called a Linear Equation.

• A straight line is represented by an equation of the first degree in two variables (x and y). Conversely locus of an equation of the first degree in two variables is a straight line.

• A straight line is completely determined by its slope (direction) and a point is given through which the line must pass.

Graph of the equation Ax + By + C = 0 is always a straight line
Therefore, any equation of the form Ax + By + C = 0, where A and B are not zero
simultaneously is called general linear equation or general equation of a line.

Find the area of a square

Problem:
Find the area of the shaded design in below fig where ABCD is a square of side 10 cm a semicircles are drawn with each side of the square as diameter. (Use π = 3.14).


Solution : Let us mark the four unshaded regions as I, II, III and IV .Area of I + Area of III = Area of ABCD – Areas of two semicircles of each of radius 5 cm
(10×10-2×1/2×π×52)cm2= (100 – 3.14 × 25) cm2
= (100 – 78.5) cm2 = 21.5 cm2
Similarly, Area of II + Area of IV = 21.5 cm2
So, area of the shaded design = Area of ABCD – Area of (I + II + III + IV)
= (100 – 2 × 21.5) cm2 = (100 – 43) cm2 = 57 cm2