Solving a Ratio for a Triangle

Introduction  of solving a ratio for a triangle:

This topic about the ratio in the mathematical concept.

The ratio of width to height of typical computer displays. In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient

Example: For every Spoon of sugar, you need 2 spoons of flour (1:2)

Solving a Ratio for a Triangle Explanation:

In this explanation the ratio of two triangle of same type is the relative between their measures and determines how many times the one triangle is greater than or less than other triangle.

Thus ratio of x and y is the fraction of x / y, and is generally written as x: y



triangle

Here will explain the ratio of the two triangles Based on the value of the each side.

The triangle one value of each side here given,

First triangle AB=4;      BC=5;      CA=7;

Second triangle value is AB=8;  BC=10;   CA=14, for solving this triangle ratio problem.

The Ratio of the two triangle is = first triangle sides / second triangle sides

AB1/AB2;     BC1/BC2;         CA1/CA2;

Final answer of the two triangle is 4/8;       5/10;           7/14;

Ratio=1/2; 1/2; 1/2;

So, Total ratio of two triangle is ½ or 1:2.

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Solving a Ratio for Triangle Problems:
Ex 1:  To solve this Problem the ratio for the two triangles the values are given below?

The triangle one value of each side here given

First triangle AB=3;     BC=5;        CA=6;

Second triangle value is AB=9;   BC=15;        CA=18;

Sol :  By solving ratio of triangle rule,

The Ratio of the two triangle is = first triangle sides / second triangle sides

This formula used for solving this problems.

AB1/AB2;      BC1/BC2;        CA1/CA2;

Final answer of the two triangle is 3/9;      5/15;         6/18;

Ratio=1/3; 1/3; 1/3;

So, ratio of two triangles is ½ or 1:3

Ex 2: To solve this problem the ratio for the two triangles the values are given below?

The triangle one value of each side here given

First triangle AB=2; BC=3; CA=4;

Second triangle value is AB=8; BC=12; CA=16;

Sol : By solving ratio of triangle rule,

The Ratio of the two triangle is = first triangle value / second triangle

This formula used for solving this problems.

AB1/AB2;   BC1/BC2;      CA1/CA2;

Final answer of the two triangle is 2/8; 3/12; 4/16;

Ratio=1/4; 1/4; 1/4;

So, ratio of two triangles is ½ or 1:4

Numerical Data Table

Introduction of Numerical data table:

Numerical data Table shows the measurement of numerical scale. Statistical methods are using for analyzing the numerical data for numerical data table. Values in the numerical data table is using for charts, histogram and graphs. Numerical data table is associating with collection, organization and interpretation of numerical data. In numerical data table, data should be number.

Description of Numerical Data Table:

Numerical data table is incorporated to statistics. The numerical data table is described with frequency distribution, percent and average. Frequency distribution is used for bar graph, Percent is used for pie chart and Average is used for normal graph.

Ordering data in Numerical Data table:

Ordering the data in the numerical data table will be done by qualitative numerical data and quantitative numerical data.

Ordering qualitative numerical data

It shows the numerical data with some quality parameters.

ordering quantitative data.

It shows the numerical data in terms on quantity.

Qualitative data can be shown in:

Pie charts: Another name of pie chart is circle graph, which is partitioned into sector by sector.

Bar graph: Bar graph consists the rectangular bars, which are representing the values.

Quantitative data can be shown:

Statistical tables: This table contains only  statistical data.

Rank order: This is arranging the data by its rank.

Frequency distribution: Frequency distribution denotes data of  intervals

Understanding numerical data table:

Graphs and number understand the values in the numerical data table. We have already seen the different graphs and number above.Is this topic the prime numbers from 1 to 100 hard for you? Watch out for my coming posts.

Example Problems - Numerical Data Table:

1.compute the average of the numbers, 25, 15, 28, 30, 27and 20 on numerical data table.

Solution:

Mean = 25+15+28+30+27+20 / 6

= 145 / 6

= 24.6             

2.compute the Median of the numbers, 15, 20, 28, 25 and 27 on numerical data table.
Solution:

Median = 15, 20, 28, 25, 27 = 27

3.compute the mode of the numbers, 22, 15, 22, 21, 15 and 20 on numerical data table.

Solution:

Here 15 & 22 repeated twice.

So, the modes are 15, 22.

Delta Mathematical Expression

Introduction for delta mathematical expression:

Delta (uppercase Δ, lowercase δ) is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter Dalet. In the Ancient Greek language, it represented a voiced dental plosive /d/, while in Modern Greek it represents a voiced dental fricative, hence it is pronounced as "th" in "that" or "this". it is Romanized as either "D" or "Dh"

(Source: Wikipedia)

Delta Mathematical Expression - Definition Delta and Symmetric Delta:

Definition for delta:

Delta, Δ, and procedures the rate of change of option value with respect to changes in the fundamental asset's price. Delta is the initial derivative of the value V of the derivative with respect to the fundamental instrument's price S.

`Delta ` = `del`V / `del`S   

Symmetric difference for delta:

In symmetric distinction of two sets is the set of basics which are in one of the sets, but not in together. This procedure is the set-theoretic kin of the exclusive disjunction (XOR operation) in Boolean logic. The symmetric distinction of the sets A and B is often denoted by

A`Delta` B

Delta mathematical expression - slope and application:

Delta from slope formula:

Slope = (y2 – y1) / (x2 – x1)

= Δy / Δx

Where, Δ changes in difference

Delta mathematical expression – Example problems:

Example 1:

Find the slope between two points (4,1) (7,6)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(4,1) (7,6) – (x1,y1) (x2, y2)

= (6 – 1) / (7 – 4)

= (5) / (3)

Δy / Δx = 5 / 3

Example 2:

Find the slope between two points (5,6) (7,8)

Solution:

Delta from slope = (y2 – y1) / (x2 – x1)

(5,6) (7,8) – (x1,y1) (x2, y2)

= (8 – 6) / (7 – 5)

= (2) / (2)

Δy / Δx = 1

Delta Mathematical Expression - Application of Delta in Quadratic Equation:

In the polynomials of quadratic equations, Δ is referred as the discriminant value.

Δ = b2 – 4ac

Where Δ is the discriminant value.

Delta mathematical expression - Example problems:

Example 1:

Find the discriminant value given b = 8, a = 2, c = 6

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(6)

= 64 – 48

`Delta` = 16

Example 2:

Find the discriminant value given b = 8, a = 2, c = 4

Solution:

Δ = b2 – 4ac

= 82 – 4 (2)(4)

= 64 – 32

`Delta`= 32.

Divide an Angle into Two Equal Angles

Introduction to divide an angle into two equal angles:

Let us assume an angle formed by two rays. The vertex of the angle is the initial points for both the rays. Suppose you need to divide this angle into two equal angles, what is to be done.

One method is to take the help of a protractor to measure the angle. Then you numerically divide that by two, mark a point again using the protractor and join that to the vertex of the angle.

However the better method to divide an angle into two equal angles is by using a compass.

Divide an Angle into Two Equal Angles – Compass Method
The following diagram illustrates the method to divide an angle into two equal angles.



Let OX and OY be the rays making an angle XOY at vertex O.

Set the compass to a convenient radius and strike two arcs from O, one on the ray OX and another on the ray OY. Let the points of intersection of these arcs be A and B respectively. (Fig i)

Now set the compass again for a convenient radius, draw an arc from point A. Without changing the compass setting, draw a similar arc from point B.  Let these arcs intersect at P. Draw a ray connecting the vertex O and the point P. The ray OP divides angle XOY into two  equal angles.  (Fig ii)

The ray OP is called the internal angle bisector of the angle XOY.

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Divide an Angle into Two Equal Angles – Proof for the Method

Join AP and BP (Fig iii)

Considering the triangles  OAO and OBP,

OA = OB  (By construction)

AP = BP  (By construction)

Side OP is common.

Hence the triangles  OAO and OBP are congruent as per SSS postulate.

Therefore , angle AOP = angle BOP (corresponding parts of congruent triangles)

Thus it is proved that the angle XOY is divided into two  equal angles.

Binomial Distribution Normal

Introduction to binomial distribution:
In a specified series of n independent experiments, the discrete probability distribution for the number of successes is called as normal binomial distribution. Bernoulli research is also an experiment of success/failure in normal binomial distribution.Bernoulli experiment is also known as  Bernoulli trial. In normal binomial distribution when n=1, it  is called as Bernoulli distribution.

Mean and Median

Consider the random variable K having the parameters p and n follows the normal binomial distribution. We can write it as K ~ B(n, p).

Probability mass function gives the probability of receiving successes(k) in n trials.

f(k;n,p)=Pr(K=k)=(n k) p^k (1-p)^n-k. for k = 0, 1, 2, ..., n,

where f(k;n,p)=f(n-k;n,1-p)

Mean and variance: If X ~ B(n, p) X  is a binomially distributed variable, then we can say that the expected value of X is E[X]=np. and the variance is Var[X]=np(1-p)

Mode and Median

Normally the mode and the median of a normal binomial B(n,p) distribution is said to be identical to (n + 1)p. However when (n + 1)p is an integer and p is neither 0 nor 1, then the distribution has two modes namely the (n + 1)p and (n + 1)p - 1. However the small special results have been established in the given ratio:

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Mode, median and mean gets coincide for np is the given integer.
Median m should lie in the known interval ?np? = m = ?np?.
For any median m the value cannot recline too far away from the given mean: |m-np| = min {ln2, max {p, 1-p}}.
When p = ½ and n is odd, any m in the given interval ½(n-1) = m = ½(n+1) is said to be a median for the binomial distribution.
If p = ½ and n is the even, then the m = n/2 is an unique median for the given Var(X)=E(X^2)-(E(X))^2. For using the formula need the expected value of X 2:

Geometry Solving Ratios

Introduction to geometry solving ratios

Usually we come across with the ratio of two numbers, if two numbers are x and y then their ratio are given by x : y. Same thing can be extended to geometry of same kind like, between the lengths, areas, perimeters, surface areas, volume etc. These ratios in geometry will help us to know the enlargement or the shrinking of the sizes or shapes of a geometrical figure.

Now let us see few problems of this kind in the following examples.

Example Problems on Geometry Solving Ratios:
Ex 1: The ratio of two supplementary angles are in the ratio 3 : 7. Find the measure of each angle.

Soln: Let the angles be given by 3x : 7x.

Since their sum is supplementary, then 3x + 7x = 180.

Therefore, 10x = 180

`=>` x = 18.

Therefore the angles are 3(18) and 7(18). Hence, they are 540 and 1260.

Ex 2: If the lengths of the sides of a triangle are in the ratio 3 : 4 : 5. Find the area of the triangle if its perimeter is 144cm.

Soln: Given: Perimeter = 144 cm.

The ratios are 3x + 4x + 5x = 144

`=>` 12x = 144

x = 12.

Therefore, the sides of the triangle are given by 3(12), 4(12), 5(12). That is, 36cm, 48cm, 60cm.

To find the area, let us use the Heron’s formula.

Area = `sqrt[s(s-a)(s-b)(s-c)]` . The values of a, b, c can be taken as 36, 48, 60 respectively.

Here S = `[a+b+c]/2`

=`[36 + 48 + 60]/2`

= `144/2` = 72.

Therefore, the area is given by:

A = `sqrt(72(72-36)(72-48)(72-60))`

= 864 cm2.


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More Example Problem on Geometry Solving Ratios:

Ex 3: The total surface area of solid cylinder is 616 cm2. if the ratio between its curved surface area and its total surface area is 1 : 2, find volume of the cylinder. [Take `pi = 22/7` ]

Soln: Given: Total surface area = 2`pi` r(r + h)

= 616 cm2 -----------------(1)

Also, the ratios of curved surface area : total surface area = 1 : 2.

`=>` 2`pi` rh : 2`pi` r(r+h) = 1 : 2

`h/(r+h) = 1/2`

2h = h + r

`=>` h = r.

Therefore from (1), we get, 2`pi` r(r+r) = 616

4 `pi` r2 = 616

r2 = 49

r = h = 7.

Therefore, the volume of the cylinder = `pir^2h` = `22/7 xx 7^2 xx 7` = 1078 cm3.                                                

Perimeter of a Right Angle Triangle

Introduction about right angle triangle:

A right triangle or right-angled triangle is a triangle in which one angle is a right angle (that is, a 90 degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The length of boundary is called perimeter triangle. We can calculate the perimeter of right angle triangle by adding the all three sides. In this article we shall discus how to calculate the perimeter of right triangle.

Formula to Find the Perimeter of the Right Triangle:

Perimeter of triangle (P) = a + b + c units

a, b and c are the names of three sides.

Example Problems to Find the Perimeter of the Right Angle Triangle:

1.      Find the perimeter of right angle triangle whose sides are 3 cm, 4 cm and 5 cm.

Solution:

Step 1:  Given:

Side a = 3 cm

Side b = 4 cm

Side c = 5 cm

Step 2:       Perimeter of right angle triangle (P) = a + b + c units

= 3+ 4 + 5

Step 3:        Perimeter of right angle triangle (P) = 12 cm

2.      Find the perimeter of right angle triangle whose sides are 5 cm, 12 cm and 13 cm.

Solution:

Step 1:  Given:

Side a = 5 cm

Side b = 12 cm

Side c = 13 cm

Step 2:     Perimeter of right angle triangle (P) = a + b + c units

= 5 + 12 + 13

Step 3:    Perimeter of right angle triangle (P) = 30 cm

3.      Find the perimeter of right angle triangle whose sides are 9 cm, 12 cm and 15 cm.

Solution:

Step 1:  Given:

Side a = 9 cm

Side b = 12 cm

Side c = 15 cm

Step 2:    Perimeter of right angle triangle (P) = a + b + c units

= 9 + 12 + 15

Step 3:  Perimeter of right angle triangle (P) = 36 cm

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Practice problems to find the perimeter of the right angle triangle:

4.      Find the perimeter of right angle triangle whose sides are 6 cm, 8 cm and 10 cm.

Answer: perimeter of right angle triangle (P) = 24 cm

5.      Find the perimeter of right angle triangle whose sides are 4 cm, 2 cm and 4.47 cm.

Answer: perimeter of right angle triangle (P) = 10.47 cm

6.      Find the perimeter of right angle triangle whose sides are 5 cm, 10 cm and 11.18 cm.

Answer: perimeter of right angle triangle (P) = 26.18 cm