Right Triangle Similarity Theorem

The right triangle similarity theorem is otherwise called as Pythagoras theorem. This right angle theorem was introduced by Philosopher and Greek Mathematician, Pythagoras. The right triangle is a part of geometrical figures. This theorem is used for finding the length of any sides of a right triangle. The right triangles are special triangles that contain only one right angle. Here the right triangle is a triangle that measuring an angle `90^o`

Statement for Right Triangle Theorem:

In terms of area, we can define this theorem. In any right triangle, the area of the square whose side is hypotenuse is equal to the sum of the areas of the other two sides. Here the hypotenuse refers to the side that opposite to the right triangle. The other two sides of a triangle meet at right angle.



Here Hypotenuse is the side opposite to the right angle. an adjacent side is the side adjacent to the given angle and the opposite side is the side that opposite to given  .

According to this theorem, the equation can be given as

(Opposite)2+ (Adjacent)2 = (Hypotenuse)2

Here, an angle  value lies between 0 and 90 degree. Here, 90 degree is one of the interior angle and the other two interior angles are complementary. Complementary means the angle value less than 90 degree. The other two interior angle sum should be equal to 90 degree.  This right triangle similarity theorem can also be used in trigonometric functions. This right triangle forms six possible ratios in trigonometry.



Other Forms of the Equation:



Here consider a is opposite side, b is an adjacent side and c is hypotenuse. Therefore, an equation can be written as a2 + b2 = c2. To find c, an equation can be written as c= `sqrt(a^(2)+b^(2))` . If c is known value, the length of one side is given; the following equations can be used;

a =`sqrt(c^(2)+b^(2))`   (or)       b = `sqrt(c^(2)-c^(2))`

Proof:

This theorem proof is based on the proportionality of two similar triangles. It depends on the ratio of any two corresponding sides of similar triangles.



Let ABC represent a right triangle, with the right angle located at B. Here, H is the altitude drawn from B and it also intersects AC. The point H divides the hypotenuse c into two parts d and e. The new triangle ABH is similar to the triangle ABC because both the triangles have a right angle. Similarly, the triangle BCH is also similar to the triangle ABC. Thus the proof of similarity of the right triangles requires the Triangle postulate i.e., the sum of the angles in a triangle is 2 right angles, and is equivalent to the parallel postulate. Similarity of the right angle triangles leads to the equality of ratios of their corresponding sides:

a/c =e/a and b/c= d/b

In this the first result is equal to cosine angle and the second result is equal to sine angle.

These ratios can be written as:

a2= c*e and b2=c*d

Summing these two equations we get:

a2 + b2 = (c*e) + (c*d) = c* (e + d) = c*c = c2

Therefore a2+ b2 = c2

Hence, right triangle similarity theorem is proved.

Graphing Calculator Polar Coordinates

In this article, we are going to study about graphing calculator polar coordinates. It is very simple to work on the polar coordinates graphing calculator. For example, if we want to graphing radians r = 3 and `theta=45^o` on the polar graph paper, we have to just enter `3` in the radians text box and `45^o` in angles in degrees text box and then click the enter button, Now the calculator automatically plot the polar coordinates on the display of the polar graph paper. These process are shown in the below figure.




Graphing Calculator Polar Coordinates - Example 1:

Study and graphing of the given polar coordinates.

(2, 50o), (3, 210o), and (-2, 135o).

Solution:

Graphing the given polar coordinates are on the below polar graph paper:



Steps for plotting polar coordinates (2, 50o):

1:   The radian 2 is marked from the origin (pole) on the positive side of the x axis.

2:   The given angle 50o is rotated in anticlockwise directions and then plot the required polar coordinates (2, 50o) as shown in figure.

Steps for plotting polar coordinates (3, 210o):

1:   The radian 3 is marked from the origin (pole) on the positive side of the x axis.

2:   The given angle 210o is rotated in counterclockwise directions and then plot the polar coordinate (3, 210o) as shown in figure.

Steps for plotting negative polar coordinates (-2, 135o):

1:   The radian 2 is marked from the origin on the positive side of the x axis.

2:   The given angle 135o , is rotated in anticlockwise directions and it is determined.

3:   Now we can extend the 135o degree line in opposite direction (315o) where we get  the radian 2 in negative sign and then plot the polar coordinates (-2, 135o).

Graphing Calculator Polar Coordinates - Example 2:

Study and graphing of the given polar coordinates by using the above calculator.

(2.5, 60o), (-1, 75o), and (3, -30o).

Solution:
Polar coordinates are on the below polar graph paper:



Steps for plotting polar coordinates (2.5, 60o):

Step 1:   The radian 2.5 is marked from the origin (pole) on the positive side of the x axis.

Step 2:   The given angle 60o is rotated in anticlockwise directions and then plot the required polar coordinates (2,.5 60o) as shown in figure.

Steps for plotting polar coordinates (3, -30o):

Step 1:   The radian 3 is marked from the origin (pole) on the positive side of the x axis.

Step 2:   The given angle 30o is rotated in clockwise directions and then plot the polar coordinate (3, -30o) as shown in figure.

Steps for plotting negative polar coordinates (-1, 75o):

Step 1:   The radian 1 is marked from the origin on the positive side of the x axis.

Step 2:   The given angle 75o , is rotated in anticlockwise directions and it is determined.

Step 3:   Now we can extend the 75o degree line in opposite direction (255o) where we get  the radian 1 in negative sign and then plot the polar coordinates (-1, 75o).

Definition Non Linear

In math, a nonlinear system is one of the system which is not linear, that is, a system which does not assure the superposition principle, or whose result is not directly proportional to its input. The equation used to solve the non linear problem is f(x)=C. This function is called as linear if f(x) is a non linear. This article gives some of the examples about the non linear by using this definition.


Examples of Definition Non Linear:

By using the definition of non linear solve the following examples.

Example problem1:

Use the following function rule to find f(7).

`f(x)=2x^2`

Solution:

Given `f(x)=2x^2` ; Find f(7).

Plug x = 7 into the function and simplify.

`f(x) = 2x^2`

`f(7) = 2(7)^2`            Plug in x = 7

f(7) = 2(49)                  Square

f(7) = 98                       Multiply

Which is the required solution.

Example problem2:

Use the following function rule to find f(8).

`f(x) = (x - 4)^2`

Solution:

Given:

`f(x) = (x - 4)^2`

Find: f(8)

Plug x = 8 into the function and simplify.

`f(x) =(x - 4)^2`

`f(8) = ( 8 - 4)^2`            Plug in x = 8

`f(8) = (4)^2`                  Subtract

f(8) = 16                          Square

Which is the required solution.

Example problem3:

Use the following function rule to find f(1).

f(x) = 5 – 4|x|

Solution:

Given: f(x) = 5 – 4|x|; Find f(1)

Plug x = 1 into the function and simplify.

f(x) = 5 –4|x|

f(1) =5 – 4|1|                  Plug in x = 1

f(1) =5 – 4(1)                 Take the absolute value

f(1) =5 – 4                       Multiply

f(1) = 1                            Subtract

Which is the required solution.

Practice Problems of Definition Non Linear:

Problem 1:

Use the following function rule to find f(7).

f(x) = -9|x| + 4

Solution:

f(7) = -59

Problem 2:

Use the following function rule to find f(-8).

`f(x) = 3x^2`

Solution:

f(-8) = 192

Relevance and Application of Exponential Functions in Real Life Situations

In this article relevance and application of exponential functions in real life situations , we will see how exponential function used in real life. Exponential function is used in more real life application like compound interest, problem based on population, problem based on radioactive decay, mortgage problems. Let us work out some problem to make understand that application of an exponential function.

Exponential Function Growth:

Exponential function growth`g=c(p)^t where,`

c-Number at initial

p-growth factor (p =1+r,here r is given growth percentage)

t-time

Example 1 -  Relevance and application of exponential functions in real life situations

A group of 1000 people increase by 5% in an hour near to accident place. How many people will be in the crowd after 3 hour?

Given:

c=1000

p =1+r =1+0.05=1.05

t =3 hour

Solution:

Exponential function growth g=c(p)^t

Substitute the given data in the formulae

g=1000(1.05)^3

g=1000(1.1576)

g=1157.6

g=1158 people

Exponential Function Decay:

Exponential function decay `d=c(p)^t`

where,

c-Number at initial

p-growth factor

(p=1-r,here r is given decay percentage)

t-time

Example 2: Relevance and application of exponential functions in real life situations

The price of a violin is  $1,000 which decreases at a rate of interest of 3%.What is the price of a violin after 2 year?

Given:

c=1,000

p=1-r=1-0.03=0.97

t=2 year

Solution:

Exponential function decay `d=c(p)^t`

`d=1,000(0.97)^2`

d=1,000(0.9409)

d=940.9

d=941

The price of a violin after 2 year=$941

Example 3: Relevance and application of exponential functions in real life situations

John invests $50,000 for 2 year with the interest of 4% compounded half yearly .Find out the compound interest for his investment at a given rate of interest?

Given:

P=$50,000

R=4%

N=2 year

Solution:

Compound interest =` P[1+(R/2)/100]^(2n)`

`=50,000[1+(4/2)/(100)]4`

`=50,000[1+(2/100)]4`

`=50,000xx(102/100)xx(102/100) xx(102/100) xx(102/100)`

=$50,000(1.0824)

=$54121.6

Compound interest=Total amount-principle

=54,121.6-50,000

=$4,121.6

Example 4: Relevance and application of exponential functions in real life situations

Calculate how long blood clotting cell will take to produce 20,400

Solution:

We need to consider  assume function to calculate population based on problems

`f(t)=2^t`

`20,400=2^t`

Find the natural logarithm on both side

`ln(20,400)=ln(2^t)`

ln(20,400)= t ln(2)

`t= ln(20,400)/ ln(2)`

`t=(9.923)/0.693=14.29`

After 14.32 min the blood clotting cell can produce 20,400  cell.

Angle Addition Property

In this article we are going to discuss Angle Addition Property , the terms related to Angle Addition Property and some solved problems on Angle Addition Property .

Introduction to angle addition property :

The angle addition postulate states that if a point is within an angle and you add the two angles that are made by drawing a line through the point that the total will equal the large angle. ...




Angle Addition Property or postulate says that if there is a line segment SV lies in the interior of the angle TSR then Angle TSV + Angle VSR = Angle TSR

For example if Angle TSR = 40 degree and Angle TSV = 15 degree then the other angle will be of 25 degree.

The same property can be applied if there are two or more lines lies in the interior of an angle.

There are some important terms that are associated with Angle Addition Property

If two or more angle sums to 90 degree they are called complementary angles.



If two or more angles are lies in a straight line then definitely sums to 180 degree and called Supplementary Angle.



Two or more angles sharing same side are called Adjacent Angles.



Having problem with ------ Read my upcoming post, i will try to help you.

Lets Learn more about Angle Addition Property in Supplementary Angles



Supplementary Angles

What is the measure of Angle B if angle A is 120 degree?

Angle B + Angle A = 180

Angle B + 120 = 180

Angle B = 60 degree

Problem Based on Angle Addition Property.

Lets learn more about Angle Addition Property



Suppose Angle CAD is a complementary angle
‹CAB = 40 degree

‹EAD = 40 degree

‹BAE =  ?

Solution Angle CAB + Angle EAD + Angle BAE = Angle CAD
40 degree +40 degree + Angle BAE = 90
Angle BAE = 10 degree
What if I remove the line AB
calculate ‹CAE

Solution

‹CAE = 50 degree (‹CAB + ‹ BAE)

What if the three of the angle are of equal

Solution :Suppose each angle = x degree
X+x+x = 90
3x= 90
x= 30 degree
If Angle EAD is thrice to angle CAB

Angle BAE is 10 more to Angle CAB

Measure each angle

Let Angle CAB = x degree
Angle BAE = x+10 degree
Angle EAD = 3x
x+x+10+3x = 90
5x+10= 90
5x=80
x=16
Therefore Angle CAB ,BAE ,EAD are 16 ,26 ,48 degree respectively.

Determine the Slope and Y Intercept

Slope:

The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is




Y intercept:

In the coordinate system, the y intercept of a line is a point at which the line cuts the Y-axis. The y-intercept of a line is denoted as (0, y)




Example Problems to Determine the Slope and Y Intercept:

Example problem 1:

Determine the slope and y intercept of y = 7x - 12

Solution:

Step 1: Given equation

y = 7x - 12

Step 2: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 3: Compare the given equation with the slope intercept form

By comparing the given equation with the slope intercept form, we get

m = 7     and b = - 12

Step 4: Solution

Therefore,  Slope = 7

y intercept = - 12

Example problem 2:

Determine the slope and y intercept of 2x - 5y = 4

Solution:

Step 1: Given equation

2x - 5y = 4 ................... (1)

Step 2: Subtract 2x on both sides of the equation 2x - 5y = 4

2x - 5y - 2x  = 4 - 2x

- 5y = 4 - 2x

Step 3: Divide by (-5) on both sides of the equation

Therefore,

y = `2/5` x - `4/5`

y = 0.4x - 0.8 ............. (2)

Step 4: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 5: Compare the equation (2) with the slope intercept form of a line equation

By comparing the equation (2) with the slope intercept form, we get

m = 0.4     and b = - 0.8

Step 6: Solution

Therefore,  Slope = 0.4

y intercept = - 0.8

Example problem 3:

Determine the slope and y intercept of 9x + 3y = 0

Solution:

Step 1: Given equation

9x + 3y = 0 ............ (1)

Step 2: Subtract 9x on both sides of the equation 9x + 3y = 0

9x + 3y - 9x = - 9x

3y = - 9x

Step 3: Divide by 3 on both sides of the equation

Therefore,

y = - 3x ............... (2)

Step 2: The slope intercept form of a line equation is given by

y = mx + b

Where,

m → slope

b → y intercept

Step 3: Compare the equation (2) with the slope intercept form of a line equation

By comparing the equation (2) with the slope intercept form, we get

m = - 3     and b = 0

Step 4: Solution

Therefore,  Slope = - 3

y intercept = 0

Practice Problems to Determine the Slope and Y Intercept:

1) Determine the slope and y intercept of y = 4x - 13

2) Determine the slope and y intercept of 5x + y = 5

3) Determine the slope and y intercept of 3/2 + 2y = 4

Solutions:

1) Slope = 4; y intercept = - 13

2) Slope = - 5; y intercept = 5

3) Slope = -0.75; y intercept = 2

Non Consecutive Vertices

Non consecutive vertices mean the vertices which are not in the adjacent. Here we are going to learn about the non consecutive vertices of the planes. Generally we can say opposite vertices of the polygons are called non consecutive vertices. We will see some examples for non consecutive vertices. This will help us to understand the non consecutive vertices.  Basically a vertex in a polygon refers the intersection point of two sides.

Non Consecutive Vertices:

Basically non consecutive vertices of a polygon mean the vertex which does not lies next to the other. In other words we can say the opposite vertices or the other vertices which is not lies very next to the vertex. The distance between the two vertices will give us the height or width of the polygon. If we find the distance we can calculate the area and volume of the polygon.

If we are having a polygon with n number of vertices then each polygons vertex is having n – 3 non consecutive vertices. For example if we take a square it is having 4 sides. So for each vertex it has n – 3 = 4 – 3 = 1 non consecutive vertices. We can calculate the diagonals of the square using the non consecutive vertices.




Examples for Non Consecutive Vertices:

Example 1 for non consecutive vertices:

Find the number of non consecutive vertices in the following polygon.



Solution:

In the above diagram we are having the number of sides as 6.

We know for a single vertex we have n – 3 numbers of vertices.

So for a six vertices we have 6 (n – 3) vertices = 6 (6 – 3) = 6 x 3 =18 non consecutive vertices.


Example 2 for non consecutive vertices:

Find the number of non consecutive vertices in the following polygon.



Solution:

In the above diagram we are having the number of sides as 8.

We know for a single vertex we have n – 3 numbers of vertices.

So for a six vertices we have 8 (n – 3) vertices = 8 (8 – 3) = 8 x 5 =40 non consecutive vertices.