Circle

A circle is a simple shape of Euclidean Geometry consisting of those points in a plane which are equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius.

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Formula to learn circles:

Formula to find Area of the Circle:

-->Area of the circle = ‘pi’ * r² Where, r is the radius of the circle.
Formula to find Diameter of the Circle:
Diameter of circle = 2 * r Where, r is the radius of the circle

Formula to find Circumference of the circle:
Circumference of circle = 2 * ‘pi’ * r (or) ‘pi’ * d Where, r is the radius of the circle and d is the diameter of the circle.
Let us now solve one problem in each of the above.
Problem on Area of Circle.
Find the Area of a circle with radius of 8 cm.

Area of the circle = 'pi' * r²
Here 'pi' is equal to 3.14
= (3.14)*8*8
=(3.14)*64
=200.96
The Area of the circle is 200.96 cm.

Problem on Diameter of a Circle.
Find the Diameter of a circle with radius of 8 cm.

Diameter of the circle = 2*r
=2 * 8
= 16
The Diameter of the circle is 16 cm.

Problem on Circumference of a Circle.

Find the Circumference of a circle with radius of 8 cm.
Circumference of the circle = 2 * 'pi'
= 2 * 3.14
= 6.28
The Circumference of the circle is 6.28 cm.

Lines and Types of Lines

Line is a straight curve. Lines are used to represent direct objects. It has negligible width and height. It can be defined in many ways. Line can be infinite and straight. A line is a collection of continuous points I should be extended indefinitely in either of its direction. Line is entirely differ from Line segment.

Line segment is a part of a line. It should be bounded by two distinct end points and contains every point on the line between two end points. Line segment is defined as finite length of two end points. These line segments may have some of the same relationships as lines, such as being parallel, intersecting, or skew.

Types of Lines:

There are different types of lines as follows:

• Parallel Line: Consider two lines in a plane. They either have one point in common, or they have no common point. When they have a common point, they are called intersecting lines. When they have no common point, they are called Parallel Lines. The distance between two parallel lines is the same at all points.


• Perpendicular line: A line is perpendicular to another line if it meets or crosses it at right angles (90 degrees). The two lines which intersect to form a 90 degree angle are called Perpendicular lines.


• Intersect line: If two or more lines that meet at a point are called Intersecting lines. Each line should meet at one central point. That point would be on each of these lines.

Math Word Problems

Word problems are a series of expressions that fits into an equation.

Steps to solve math word problems:

1. Read the problem entirely and understand what does the question ask for.

2. List all the data given like the variables along with their units must be listed.

3. Look for key words like more than, less than, times, reciprocal etc.

4. Translate the given data into equations which combine expressions.

5. solve the equation and get the unknown variable or data asked for.

In many word problems, key words play an important role in our calculation to approach answer, There is the list of key words that are very commonly used.

Math word problems can be categorized in to various areas:

• Number word problems
• Algebra word problems
• Geometry word problems
• Calculus word problems
• Trigonometry word problems
• Probability and Statistics word problem

Let us understand this by one solved example of a math word problem.

One gardener can water the entire garden in twelve hours, and the second gardener takes eight hours. How long would it take the two gardeners together to water garden?

solution:

hours to complete job:

first gardener = 12 hrs

second gardener = 8 hrs

together = t hrs

completed per hr :

first gardener completes = 1/12 th part

second gardener completes = 1/ 8 th part

together = 1/ t th part

1/ 12 + 1/ 7 = 1/ t

5/24 = 1/ t

t = 24/5

so they can complete the work together in 4.8 hrs.

Quadratic Equation

An equation of the form ax²+bx+c = 0 where a, b, c are real numbers and where “a” does not equal to zero (0).
The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term divided by the leading coefficient.
The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.

Example of a Quadratic Equation:

Factor the equation x2 – 5x -36 = 0 using quadratic equation solver.

Solution:

We can split as add of the roots and the multiply for the roots.

Product of the roots – 36 = -9*4

Sum of the roots -5 = -9 +4

X2 +(-9x+4x) -36 = 0

(x2-9x) + (4x -36) = 0

X(x-9) + 4(x-9) = 0 now take the common term

(x-9)(x+4) = 0

x-9 = 0 and x+4 = 0

x= 9 and x= -4

The factors of the given quadratic equation are 9, -4.

For word problems on Quadratic Equations click here

Hyperbola and Parabola

Hyperbola:

A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed distinct points is a given positive constant that is less than the distance between the fixed points.

The locus of a point from a fixed point is at a constant ratio and it is greater than one of its distance from a fixed line is called a Hyperbola.

Standard form of a Hyperbola

x²/a² – y²/b² = 1

Parabola:

A parabola is a conic section with set of all points in the plane which are equidistant from a fixed line and a fixed point not on the line. The line is called directrix of the parabola.

General equation of Parabola.

y² = 4ax

To be continued........................................