Semicircle Learning

Introduction of semicircle learning :

Semicircle is defined as half of a circle. That is, the angle is 180 degree arc of a circle. A triangle decorated in a semicircle is always called a right triangle.

If two curves or arcs are equal, then both the segments and sectors are similar. This each part of term is called as semicircle region.

Formulas of Semicircle Learning

A semicircle is the area enclosed by a diameter and an arc of the circle joining its two ends. The length of the resulting segment is called the geometric mean, which can be proved using the concept of Pythagorean Theorem.

Formulas:

Area of semicircle (A) =circle /2

A = (pr2)/2

Circumference of semicircle(C) = (2pr)/2

C = pr

A circumference of a semicircle is calculated for the circumference of circle divided by 2.we get,

C = 2pr ==> C/2 = pr

p = 3.14 ( approximately )

A perimeter of a semicircle is the sum of circumference and diameter of a semicircle. We get,

P = pr + 2r = r (p+2)

P = 5.14 r

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Examples of Semicircle Learning:

Semicircle learning Ex 1!:

Find the area of semicircle with radius of 12.5 cm.

Semicircle learning sol :

We can find the area of semicircle by using the following formula,

Area = (pr2)/2

Substitute the values of p and the radius into the above formula. Then we get,

= (3.14*(12.5)2)/2

Squaring the values of radius and multiplying with 3.14 then dividing by the value of 2.

= (3.14*156.25)/2

= (490.625)/2

Then we get the final answer.

=245.3 cm2

Answer: 245.3 cm2

Semicircle learning Ex 2:

Find the perimeter of semicircle with the radius 10 cm.

Semicircle learning sol :

We can find the perimeter of semicircle by using the following formula,

Perimeter = 5.14*r

Substitute the value of r into the above formula,

=5.14*10

=51.4 cm

Answer: 51.4 cm

Semicircle learning Ex 3:

Find the circumference of semicircle with the radius of 7.5 cm.

Semicircle learning sol :

We can find the circumference of semicircle by using the following formula,

Circumference C = (2pr)/2

Circumference C = pr

Substitute the value of p and the radius.

C = 3.14*7.5

C = 23.55 cm2

Answer: C = 23.55 cm2

Measuring Irregular Triangles

Introduction for measuring irregular triangles:

The irregular triangle is nothing but the triangle where the three sides are not equal and the angles present in it also different during its measurement. The only irregular triangle is the scalene triangle. Now we are going to see about the measuring of irregular triangle with some example problems.


About Measuring of Irregular Triangle:
Now we are going to see about the irregular triangles and its measurement. The irregular triangle is nothing but the scalene triangle where the sides are unequal in its length and the angles in it also unequal.

When the two sides and an angle are given and if we want to find the third side of an irregular triangle, then use the formula which is given below,

c2 = a2 + b2 - 2ac * cos ?

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Problems for Measuring Irregular Triangle:

Example 1:

The sides of the triangle are 5cm and 10 cm and the angle measuring is 40. Determine the third side of the irregular triangle.

Solution:

The third side of the triangle can be calculated by using the formula,

c2 = a2 + b2 - 2ac * cos ?

Now substitute the values in the formula we get,

c2 = 52 + 102  2(5)(10) * cos 40

Now square the values which are substituted in the formula as follows,

= 25 + 100  100 cos 40

c2 = 48.39

c = `sqrt 48.39`

c = 6.95

The value can be rounded as 7cm.

Example 2:

Find the third side of the triangle whose measurements of the triangle are 7cm, 8cm and angle measuring is about 50 degree.

Solution:

The third side of the triangle can be calculated by using the formula,

c2 = a2 + b2 - 2ac * cos ?

Now substitute the values in the formula we get,

c2 = 72 + 82  2(7)(8) * cos 40

= 49 + 64  112 cos 50

c2 = 41

c = ` sqrt 41`

c = 6.4

The value can be rounded as 6cm.

Non Coplanar Definition

Introduction to non-coplanar points:
The points which do not lie in the same plane or geometrical plane are called as non-coplanar points. Any 3 points can be enclosed by one plane or geometrical plane but four or more points cannot be enclosed by one. The points belong to the same plane are called as coplanar points. In this article we shall be discussing the non-coplanar points. Now we know what non-coplanar point is and we shall see some examples of the non-coplanar points and solve it for the same.

Example for Non Co-planar Points
1)      From the below shown figure the points are non coplanar points as they do not lie on the same plane it lies in different planes.

2)      We can see four planes with the help of four non co-planar points.

3)      Plane is the two dimensional geometrical object.

Non co-planar points

Non co-planar points

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Solved Examples for Non Co-planar Points:

Ex 1:  Check whether the following lines are co-planar or not.

3x+6y+9 = 0 and 4x+4y+11 = 0

Sol :  The given equations are  3x+6y+9 = 0 and 4x+4y+11 = 0

The slope intercept form can be given as y = mx+b

Where m indicates slope.

Comparing the above equation with the given equation, we get:

6y = -2x-9

Dividing by 6 on both sides we get:

We get,

--- (1)

The slope intercept form can be given as y = nx+b

Where n indicates slope.

Comparing the above equation with the given equation, we get:

4y = -4x-11

Dividing by 4 on both sides we get:

y = -1

We get n = -1--------- (2)

Equation (1) (2), that is m n

That is the slopes of the two equations are not equal and therefore the points lie on the two lines are non co-planar points.

Ex 2:   Check whether the following lines are co-planar or not.

x+5y+9 = 0 and 2x+10y+11 = 0

Sol :  The given equations are  x+5y+9 = 0 and 2x+10y+11 = 0

The slope intercept form can be given as y = mx+b

Where m indicates slope.

Comparing the above equation with the given equation, we get:

5y = -x-9

Dividing by 5 on both sides we get:

We get,

------- (1)

The slope intercept form can be given as y = nx+b

Where n indicates slope.

Comparing the above equation with the given equation, we get:

10y = -2x-11

Dividing by 10 on both sides we get:

We get

--------- (2)

Equation (1) =(2), that is m = n

That is the slopes of the two equations are equal and therefore the points lie on the two lines are co-planar points.

Quadrilaterals Shapes Learning

The word Quadrilateral may look difficult but it is not so. It just means a four sided figure with four vertices (corners). Absolutely, for any figure the number of sides equals the number of angles. It has four angles too. Based on the sides and the angles, we name the figures. The word quadrilateral formed from the words quad (meaning "four") and lateral (meaning "of sides").

The word Quadrilateral doesn't rhyme with the triangles (which has one fewer side and angle) and the pentagons (which has one more side and angle). Hence, we can say Quadrangle by analogy with the Triangle. Similarly, we can also call as "Tetragon" by analogy with the Pentagon.The sides of a quadrilateral may be congruent or non-congruent, may be parallel or perpendicular. The sum of the angles in a quadrilateral is found out by assuming the right angles for each angle. So, the sum of the angles = 90 + 90 + 90 + 90 = 360 degrees.
Lets have a look on the quadrilaterals shapes.

Quadrilaterals Shapes
The quadrilaterals are mainly divided by two types "Regular" and "Irregular". If all the sides and the angles are congruent, it is a Regular Quadrilateral and if they are non-congruent, then they are Irregular Quadrilaterals.Only the Square belongs to Regular quadrilateral, whereas all the other quadrilaterals are Irregular.Now, you may think that Rhombus whose all sides are congruent, why is it not included in Regular quadrilaterals. Well, I agree that the Rhombus has all sides congruent, but the angles ? Are all angles congruent too ? No, the angles in a Rhombus is not congruent.

Making all Quadrilaterals from a Square
If all the sides are equal and all the angles same in a quadrilateral, then it is a Square. Right ?
Let me show you to make all types of quadrilaterals using just a single square.
Draw a square. If you drag one side far away, there becomes a Rectangle. If you drag any two adjacent vertices to any one direction, there becomes a Parallelogram. We can also call a  Parallelogram as a Squashed Rectangle. Pull in one side of a Parallelogram inside the figure (provided the horizontal lines should not get changed), it makes a Trapezoid. If you make all the sides slant in a square, there becomes Rhombus. If you drag one vertex of a Rhombus far away, there becomes Kite.

Conic sections

The conic sections are curves obtained by making sections, or cuts, at particular angles through a cone. First we will discuss about history of conic sections. Conics are amongst the oldest of the curves, and are the oldest math subject studied systematically and thoroughly. The conics had been discovered by Menaechmus, the tutor to the great Alexander. These conics were perceived in order to solve the three problems (a) trisecting an angle, (b) duplicating the cube, and (c) squaring the circle.

If we cut a cone at different angles, then we will obtain different types of conics. There are four different types of conics that we can obtain. That is circle, where the cone is cut at right angle to its axis, ellipse, where the cone is cut at an oblique angle, parabola, where the cone is cut parallel to the generator and finally hyperbola, where a double cone is cut at an angle steeper than the generator.

Let’s discuss about conic sections formulas and conic sections equations. First is circle, the standard formula of circle is  (X^2+y^2=r^2),  where centre is (0,0) and radius is r. second is ellipse, the standard formula of a ellipse is (x^2/a^2+y^2/b^2 =1 a = 1/2) length major axis,b = 1/2 length minor axis, third one is parabola, the standard formula of parabola is 4px=y^2, where p=distance from vertex to focus. And the last one is hyperbola, the standard formula of hyperbola is (x^2/a^2-y^2/b^2=1), where a = 1/2 length major axis, b = 1/2 length minor axis. The general equation for all conic section is (Ax^2+Bxy+Cy^2+Dx+Ey+F=0). And by using the quadratic formula

If (B^2-4AC) <0 circle="circle" curve.="curve." curve="curve" ellipse="ellipse" is="is" no="no" or="or" p="p" point="point" the="the" then="then">If (B^2-4AC) =0, then the curve is a parabola or two parallel lines or a single line or no curve.
If (B^2-4AC)>0, then the curve is hyperbola or two intersecting lines.

The procedure of graphing conic sections, in this part first we will focus on graphing circle. If we cut a circular cone with a plane which is perpendicular to the symmetric axis of cone, then a circle is formed. The intersection line is parallel to the plane generating circle of the cone. A circle means from its center all points are at equal distance.

Now graphing of ellipse, in ellipse long axis is major axis and short axis is minor axis. Intersection point of two axis called vertices. The vertices along horizontal and vertical axis form points. This point along with center will provide method to graph ellipse in standard form.

Introduction to geometry word problems

Geometry is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially the body of the practical knowledge is concerning with the lengths, areas, and volumes. (Source:wikipedia)

In this article we shall discuss about geometry word problems.

Geometry Word Problems Involving Area:

A rectangle is 4 times as extended as it is wide. If the measurement lengthwise is more than 4 inches and the width is less than 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle?

Solution:

Let x = original width of rectangle

Area of the rectangle is

A = l w

Plug in the values from the question

60 = (4x + 4) (x –1)

Use distributive property to remove brackets

60 = 4x2 – 4x + 4x – 4

Put in quadratic formula 4x2 – 4 – 60 = 0

4x2 – 64 = 0

This quadratic written as a difference of two squares

(2x) 2 – (8)2 = 0

Factorize Difference of two squares are

(2x) 2 – (8)2 = 0

(2x – 8)(2x + 8) = 0

Therefore the values for x is

Since x is a dimension, it would be positive.

So, we take x = 4

The question wants the dimensions of the original rectangle.

The width of the given rectangle is 4.

The length is 4 times the width = 4 × 4 = 16

Answer: The dimensions of the rectangle are 4 and 16.

Geometry Word Problems Involving Perimeter:

 A triangle has a perimeter of 50. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?

Solution:

Let x = length of the equal side

The formula for perimeter of rectangle

P = sum of the three sides

Plug in the values from the question

50 = x + x + x+ 5

Combine like terms

50 = 3x + 5

Isolate variable x

3x = 50 – 5

3x = 45

x =15

The question requires length of the third side.

The length of third side is = 15 + 5 =20

Answer: The length of third side is 20

Practice Geometry Word Problems:

A triangle has a perimeter of 80. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?
                   Answer: The length of third side is 30

A triangle has a perimeter of 95. If 2 of its sides are equivalent and the third side is five more than the equivalent sides, what is the length of the third side?
                   Answer: The length of third side is 35

Various types of conics sections

Definition of conic sections: We can define conic section as follows: Consider a double cone. If we have a plane that cuts this double cone, the cross section thus obtained at the intersection of the plane and the cone is called a conic section.

Depending on the angle at which the plane cuts the cone and the position of the plane, we can have mainly 4 different conics: circle, ellipse, parabola and hyperbola.

Conic s have been studied for over 2000 years. Greek mathematician Apollonius studied them intensely. He wrote a book ‘the conic’ that remained a standard work on the topic for eighteen centuries. In the sixteenth century, Galileo declared that the trajectory of a projectile was a parabola. The reflectors in head lights of a car, the speakers in a sound system and the mirrors in a telescope are all in the shape of a parabola.

Parabolic mirrors are used to harness solar energy. In the seventeenth century, Kepler declared that planets revolved around the sun in elliptical orbits. It is because of our knowledge of ellipses that precise predictions of time and place of solar and lunar eclipses is possible. The path of motion of a comet in the solar system is also in the shape of a parabola, ellipse or a hyperbola. Knowledge of conics is extremely useful in such terrestrial sciences as architecture and bridge building.

Thus, the study of conics has been proved very useful in space sciences, mechanics, optics, engineering, architecture and other fields.

How to graph conic sections?
To be able to graph conics, let us look at the following concept. Suppose line l is a fixed vertical line and another line m intersects l in the point V and makes an angle of measure a(0 < a < pi/2) with it. If m is made to rotate around V in such a way that a remains constant, then the surface generated is called a double cone. The point V is the vertex and the line m is a generator of the double cone. The line l is the axis of the double cone.  Then the plane cutting this double cone defines the four conics as follows: (see picture below)

Solving Solid Geometry

In solid geometry we study three dimensional geometry (3-D geometry).
For examples: Cube, cuboid, cylinder, cone, sphere, Pyramids, Prisms etc. Dimensions are the terms as length, width, height, thickness etc. A three dimensional figure must have length, width and height.

Cube : A three dimensional shape having equal length(a), width(a) and height (a).
Cuboid: A three dimensional shape having different length(l), width(w) and height(h).
Cylinder: A three dimensional shape having two circular faces of radius(r) at two ends and a curved surface of height(h).
Cone: A three dimensional shape having a circular face at one end and a curved surface of height(h).
Sphere: A three dimensional shape of radius r. For example: a ball.

Formulas for Solving Solid Geometry
(1) Cube :           Lateral Surface Area ( Area of four sides i.e. front, back, left, right ) = 4a2
                           Total Surface Area( Area of all six faces) = 6a2
                            Volume = a x a x a = a3
(2) Cuboid:        Lateral Surface Area ( Area of four sides i.e. front, back, left, right ) = 2h(l+w)
                            Total Surface Area( Area of all six faces) = 2( lw + wh + hl )
                             Volume = lwh
(3) Cylinder:      Curved surface area = 2Ï€rh
                            Total surface area (including two circles on both ends) = 2Ï€r(r+h)
                            Volume = 2Ï€r2h
(4) Cone :          Curved surface area = Ï€rl where l is the slant height of the cone l = `sqrt(h^2 + r^2)`
                            Total surface area (including a circles on the base) = Ï€r(r+l)
                            Volume = 1/3 Ï€r2h
(5) Sphere :      Surface area = 4Ï€r2
                            Volume = 4/3 Ï€r3

How to Solve Problems for Solid Geometry
Step 1) Make a figure of solid given in the problem.
Step 2) Write the dimensions of the solid e.g. length, width, height, radius etc.
Step 3) Apply the formula for particular solid geometry figure.
Step 4) Write the unit of the particular physical quantity e.g. square meters, cubic centimeters etc.

Circles and semi circles

Important definitions related to circles:

1. Circle: A circle is a simple closed curve all of whose points are at a constant distance from a fixed point in the same plane. The fixed point is called the centre of the circle.

2. Circumference of a circle: The distance right around the circle is called its circumference. It is the perimeter of the circle. The traditional method to measure this perimeter of a circle was using a thread or a rope long the circumference. However this method is not too practical for very large circular fields or pieces of land. Therefore for all practical purposes, the following formula was derived by mathematicians for circumference of a circle.
 C = pi*d. Where, C = circumference of the circle, d = diameter of the circle and pi = ratio of the circumference of a circle to its diameter. The value of the Greek letter pi (read as pi) was experimentally calculated by mathematicians. It is an irrational number. A decimal of nonrecurring type. It is a constant. Indian mathematician Ramanujan gave two approximations for the value of pi in the year 1914. Generally all mathematicians have accepted the value of this constant as 3.141 592 653 589 793....

3. Semi circle: A diameter divides a circle into two equal parts which are called semi circles. The length of the curved portion of a semi circle is equal to half the circumference of the circle. The total perimeter of a semi circle is equal to sum of half the circumference and diameter. So putting that mathematically, perimeter of a semi circle = P,
P = C/2 + d, where C = circumference of a circle with diameter d.
From any point on the semi circle if we draw two lines that meet both the ends of the diameter, the angle so formed is called angle in a semi circle. This angle in a semi circle is always a right angle.

4. Unit of a circle (or unit circle): A circle which has a unit radius is called a unit circle. In other words a circle with radius = 1 and diameter = 2 is called a unit circle.

5. Intersecting circles: If there are two circles in a plane then any of the following three possibilities are there:
(a) The circles do not touch or intersect each other at all.
(b) The circles touch each other in exactly one point.
(c)  The circles intersect each other in exactly two points.

Beginners Guide to Geometry of Circle

A branch of Mathematics, Geometry is a study of the size, shape and position of two and three dimensional figures. Geometry of a Circle is a study of a circle, its parts and its properties. A math circle is an important and special figure and as such its parts have special names. Circle in Geometry is a planar figure in which all points are equidistant from a fixed point. This fixed point is called the centre of the circle. A segment with one endpoint at the centre of the circle and the other endpoint on the curve of the circle is a radius; the plural of radius is radii.

A segment whose endpoints lie on the circle is called the chord. A chord that passes through the centre of the circle is called the diameter of the circle. Let us learn more about Geometry circle, there are special lines and line segments in a circle; like secant, tangent and point of tangency.  Any line that contains a chord is called a secant. A tangent of a circle is a line in the plane of the circle that intersects the circle in exactly one point. The point where the tangent intersects a circle is called the point of tangency. Let us now learn a bit about tangents of a circle, a line that is tangent to two circles in the same plane is a common tangent. A common tangent that intersects the segment joining the centers of two circles is an external common tangent.

Now that we have a brief introduction to a circle and its parts, let us learn about the geometry circle formula. The major formulas in circles are as given below:
• Diameter is twice the radius. d=2r
• Circumference of a circle is the distance around the outer edge. It is like the perimeter of a circle. It is calculated using the formula 2 pi r, where r is the radius and pi is taken as 3.14
• Area of a circle is given by pi r2, r is the radius and pi value is taken as 3.14
• An arc is a part of circle. Length of an arc can be calculated using the formula, (a/360)x 2 pi r
• A sector is a portion of a circle bounded by two radii and the arc joining the radii. Area of a sector in degrees is, (sector angle/360) pi r2 and in radians it is (sector angle/2) r2

Know more about the solid geometry, Math Homework Help. This article gives basic information about geometry circle. Next article will cover more Geometryconcept and its advantages,problems and many more. Please share your comments.

Different kinds of graph

Lets learn kinds of graph below. We have learn t what is a bar graph a few days back.

Look below for the kinds of graph

Kind 1: Pictograph.

Kind 2: Bar graph.

Kind 3: Line graph.

Kind 4: Scatter plot.

These are some important graphs in mathematics, we will learn

Absolute values - Inequation and complex number

The absolute value of an integer is the numerical value of  the integer regardless of its sign .the absolute value of any integer say , a is denoted by |a|.On the number line the absolute value of an integer is regarded as the distance from  a point irrespective of its sign. The absolute value of a integer is always positive .
Some Absolute value examples are |-5| = 15 , |13|= 13.

Complex number

Absolute value inequality
To understand absolute value inequality , we will take few examples.

Example1 : |3x| ≤ 6
To  solve absolute inequality , here  we will use the absolute inequality results
|x|≤ a  =>  -a ≤  x ≤  a

=>  |3x|≤ 6
=>  -6 ≤  3x ≤ 6
=> Divide both sides by 3, we have
=>  -2 ≤  x ≤  2

Absolute value of complex number.

If a , b are two real number , then a number  a+ ib is  called as complex number.

Real and imaginary part of complex number : if z = a+ib  is a complex number , then a is called the real part of z and b is known as the imaginary part  of z . The real part of z is denoted by Re(z) and imaginary part is denoted by Im(Z).

Complex Number

The plane in which we represent a  complex number  geometrically  is known  as complex plane or argand plane or the Gaussian plane The point Z plotted on the argand plane is called the argand diagram.The length of the line segmemt 0z is called the absolute  of  complex number z and is denoted by |z|.thus |z|=√x²+y ²

= √(Re(z))²+(Im(z))²  In the above given figure z = 3+ 3i , so absolute of z , |z| = √3²+3 ²

=√18 = 3√2

Absolute value equations and inequalities

Now let us solve absolute value equations

Example1: Solve absolute value equation|x+ 5|= 4

Solution : For solving absolute value equations we will consider two cases

=> x + 5 = 4 or x + 5  = -4

=> x= -1 or x =-9 ans

Example 2: Solve the absolute inequality |x-2| ≥ 5

Solution : For solving absolute inequality , we will use the result

|x-a|≥ r => x ≤ a-r  or x ≥ a+r

=> |x-2| ≥ 5 => x ≤ 2-5  or x ≥ 2+5

=> X ≤ 3  or x ≥ 7

=> X ∈ ( -∞ , -3] or x ∈ [ 7 ,∞)

The solution set  of absolute inequality  is ( -∞ , -3] ∪ [ 7 ,∞)

Adding Fractions

Hello friends, in today's post we will study the concept of adding fractions. There are two types of fractions: like fractions and unlike fractions and the method for adding both is different. Below are the ways shown:

• Adding fractions with like denominators
• Adding fractions with unlike denominators

While adding like fractions, simple addition is done by adding the numerators alone. On the other when we adding unlike fractions, first we need to find out the lcm and then take a common denominator and then addition should be performed.

For more help, get it from online. You can also avail to free algebra tutoring as well.

Do post your comments.

A learning on Equivalent Fractions

Moving further with fractions learning, let's understand the concept of equivalent fractions in today's post. As mentioned earlier fractions are nothing but rational numbers having a numerator and a denominator. Now let's understand what are equivalent fractions.

Two fractions are equivalent when the values of both are similar after simplification. As for example there are two fractions given: 3/4 and 9/16. Fist step is simplifying fractions: 9/16 = 3/4. Therefore, both the fractions are equivalent fractions. Thus, small or large any fractions can be examined in this way whether they are equivalent fractions or not.

For more help connect to an online tutor and avail your help. Not just fractions but you can avail to geometry tutoring and trigonometry tutoring as well.

Next time, we will learn some other concept of fractions. Till then, enjoy learning this concept. Also do comeback with your feedback.

Learn how to subtract fractions with me

Let's start today's learning with fractions and specifically studying the area of how to subtract fractions. To start with it, we should understand what is a fraction? In simple words, Fraction is a rational number having a numerator and a denominator. As for example: 2/4, 5/8, 8/9 and so on.

Now moving on to subtracting fractions, when the denominators are similar subtraction with fraction is very easy. All you need to do is take the denominator as common and subtract the numerators. But subtracting fractions having different denominators has a slightly different method. At first the LCM of both the denominators has to be found out and then taking a common denominator for both the fractions, the subtraction should be performed.

With mixed fractions, first it has to be converted to an improper fraction and then similar way the fractions should be subtracted. For more help in this topic one can also avail to online help. One can connect with a free math tutor online tutor and learn the subject with one on one conversation.

Next session we will move on at learning some other concepts of fractions. Till then enjoy this session, do practice problems and improve your knowledge over fractions.