Solve Converse of Mid-point Theorem Problems

In this article we are going to solve converse of mid-point theorem problems. The straight line which is drawn through the mid point of one side of a triangle is parallel to another side bisects the third side. This is the converse of mid-point theorem. But the mid point theorem states that the line segments joining the mid points of any two sides of a triangle is parallel to the third side and equal to half of it.

Solve Converse of Mid-point Theorem Problems:

Given: In Triangle ABC, D is the midpoint of AB and DE is drawn parallel to BC.

To prove: AE = EC

Construction: Draw CF parallel to BA to meet DB proceed at F.




Proof : 

Statement	Reason
DB || BC BD || CF BCFD is a parallelogram Therefore BD = CF ... ( i ) BD = AD ... ( ii ) Therefore AD = CF In triangle ADE and CFE, we have AD = CF Angle ADE = Angle CFE Angle AED = Angle CEF Therefore triangle ADE = Triangle CFE Therefore AE = EC Therefore DE bisects AC Hence proved	Given By construction Both pairs of opposite sides are parallel Opposite sides of a parallelogram unequal D is the mid point of AB From ( i ) and ( ii ) Just proved Alternate to angle S Vertically opposite angles of S Angle Angle Side criterion C. P. T. C.

Example Problem - Solve Converse of Mid-point Theorem Problems:

ABC of a right angled triangle at B, Here the mid-point of AC is P.

Solve that PB = PA = ½


Solution for the  converse of mid-point theorem problems:

Given:

ABC of a right angled triangle at B, Here the mid-point of AC is P.

To prove:

PB = PA = `1/2` AC.

Construction:

Through P, draw a line parallel to BC, meeting AB at Q.

Proof:

AQP = ABC (corresponding angles)

AQP = 90 degree.

ABC = 90 degree

In D APQ and D BPQ

AQ = BQ

AQP = BQP = 90 degree

PQ = PQ

Therefore in triangle APQ = Triangle BPQ

PA = PB (corresponding parts of congruent triangles)

PA = PB = `1/2` AC

PA = `1/2` AC (given)

Area and Distance Calculator

Area is a number expressing the 2D size of a defined part of a surface, usually a region enclosed by a closed curve. Distance between (x1, y1) and (x2, y2) two points are called as distance.

Area of square = `a^2 `

Distance between two points = `sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]`

The area and distance calculator example problems and practice problems are given below.


Example Problems for Area and Distance Calculator:

Example problem 1:

Find the area of the square, whose side edge is  12.5meters

Solution:

Given:

Side (a) = 12.5 m

Area of square calculator is given below.



Area of square = a2

= 12.5 * 12.5

After simplify this, we get

Area of square = 156.25 square units

Example problem 2:

Find the area of the square, whose side length is 23.4 meters

Solution:

Given:

Side (a) = 23.4 m

Area of square calculator is given below:



Area of square = a2

= 23.4 * 23.4

After simplify this, we get

Area of square = 547.56 square units

Example problem 3:

Find the distance between two points A (4, 1) and B (6, 5)

Solution:

Distance between two points calculator is given below.



The distance between two points are (x1, y1) and (x2, y2)

Here, x1 = 4, x2 = 6, y1 = 1, y2 = 5

By using the following formula

Distance formula = `sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]`

= `sqrt[(6 - 4)^2 + (5 - 1)^2]`

= `sqrt[(2)^2 + (4)^2]`

= `sqrt(4 + 16)`

After simplify this, we get

Distance = `sqrt(20) `

So, the distance between two points A (4, 1) and B (6, 5) is `sqrt(20)`

The above examples are helpful to study of area and distance calculator.


Practice Problems for Area and Distance Calculator:

Practice problem 1:

Find the area of the square, whose side length is 34.5 meters

Answer: Area of square =1190.25

Practice problem 2:

Find the distance between two points A (0.7, 22) and B (0.2, 1.8)

Answer: Distance between two points = `sqrt(408.28)`

Null Set Definition

A set is a collection of well defined objects. What do we mean by well defined objects? What is your favourite subject? Your answer may be math, science or history or geography. If you ask the same question to your friend, then she may say commerce. The answer differs from person to person. It is not well defined. If you ask the following question “Name the days in a week?” to anyone, then the answer will be universally same. It is well defined.

Introduction to Null Set Definition:

Consider the following sets.

The set of vowels in English alphabet = {a, e, i, o, u} --- 1

Set of numbers divisible by 2 = {2, 4, 6, 8, 10…} --- 2

The number of elements in a set is called cardinal number of the set. If A = {a, e, i, o, u}, then cardinal number of set A, denoted as n (A) = 5.

If we could write the cardinal number of the set then it is called finite set. Set 1 is a finite set. We could not count the number of elements in set 2. So it is called as infinite set.

Definition of null set:

A null set is a set whose cardinal number is zero. In other words, a null set is a set which has no elements. Null set is also called as empty set or void set. The empty set is denoted by the symbol {} or φ

Problems on Null Set:

Ex :   Let A = {x: 2< x <3, x is a whole number}. Then A is the empty set, because there is no natural number between 2 and 3.

Sol : Let B = {x: x² – 5 = 0 and x is a natural number}. Then B is an empty set because the equation x² – 5 = 0 is not satisfied by ant rational value of x.

Let C = {x: x² = 25, x is even}. Then C is an empty set, because the equation x² = 25 is not satisfied by any even value of x.

Solve for Y in Function Table

Function tables playing an important role in mathematics. Function is defined as the set of order pairs (x, y).  For example {(1,2) (4,3) }. A common rule is followed in the function that is y = 3x +1 (or)    y = x2. In this article we shall discuss about how to solve the function table with suitable example problems.


Example Problem on Function Table:

Solve for y in the function tale y = 3x +5.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = 3x + 5
0	y = 3 (0) + 5 = 5
1	y = 3(1) +5 = 8
2	y = 3(2) +5 = 11
-1	y = 3(-1) +5 = 2
-2	y = 3(-2) +5 = -1

Hence the order pair for the above function is (0, 5) (1, 8) (2, 11) (-1, 2)(-2, -1)

Example Problem on Function table:

Solve for y in the function tale y = x + 4.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = x + 4
0	y = (0) + 4 = 4
1	y = (1) +4 = 5
2	y = (2) +4 = 6
-1	y = (-1) +4 = 3
-2	y = (-2) +4 = 2

Hence the order pair for the above function is (0, 4) (1, 5) (2, 6) (-1, 3)(-2, 2)

Example Problem on Function Table:

Solve for y in the function tale y = x2 + 2.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = x2 + 2
0	y = (0)2 + 2 = 2
1	y = (1)2 +2 = 3
2	y = (2)2+2 = 6
-1	y = (-1)2 +2 = 3
-2	y = (-2)2 +2 = 6

Hence the order pair for the above function is (0, 2) (1, 3) (2, 6) (-1, 3)(-2, 6)

Example Problem on Function table:

Solve for y in the function tale y = 2x2 + 3.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = 2x2 + 3
0	y = 2(0)2 + 3 = 3
1	y = 2(1)2 +3 = 5
2	y = 2(2)2+3 = 11
-1	y = 2(-1)2 +3 = 5
-2	y = 2(-2)2 +3 = 11

Hence the order pair for the above function is (0, 3) (1, 5) (2, 11) (-1, 5)(-2, 11)

Sine Theorem

In a triangle we can explore a number of relationships. These relation ships help us to solve a triangle. That is by knowing minimal quantities out of all the three angles and all the three sides, the remaining quantities can be figured out.

 When two sides and one angle (not the included angle) or two angles and any side are known the remaining parameters can be found. The sine theorem of triangle helps us in that.

Let us see what a sine theorem is.

The Statement of Sine Theorem-



Let ABC be any scalene type of triangle.

As per sine theorem, the following ratios are equal in any triangle.

$\frac{a}{sin A}$ = $\frac{b}{sin B}$ = $\frac{c}{sin C}$

This ratio is equal to the diameter of the circumscribing circle of the triangle.

Let us illustrate with an example.

The measures of two angles of a triangle are 30 degrees and 45 degrees. The measure of one side is 30 cm. Solve the triangle.

Let us assume A = 30o B = 45o and c = 30cm.

Using the property of sum of angles of a triangle,

C = 180o – (30o + 45o) = 105o

As per the sine theorem,

$\frac{a}{sin 30}$ = $\frac{b}{sin 45}$ = $\frac{30}{sin 105}$

Therefore,

a = 30 $\frac{sin 30}{sin 105}$ = 30 (0.52) = 15.6 cm (approximately)

b = 30 $\frac{sin 45}{sin 105}$ = 30 (0.74) = 22.2 cm (approximately)

Sine Theorem- Derivation



In the same diagram shown earlier, draw a perpendicular CD on AB and a perpendicular BE on AC.

In triangle ACD, CD = ACsin A = bsin A

In triangle BCD, CD = BCsinB = asin B

Therefore, bsinA = asinB

or,    $\frac{a}{sin A}$ = $\frac{b}{sin B}$

Similarly in triangle ABE, BE = ABsin A = csin A

and in triangle BCE, CE = BCsinC= asin C

Therefore, csinA = asinC

or,    $\frac{a}{sin A}$ = $\frac{c}{sin C}$

Combining both the results,

 $\frac{a}{sin A}$ = $\frac{b}{sin B}$ = $\frac{c}{sin C}$

Precalculus Function and Graph

Precalculus is one of the most important and interesting branch of mathematics.
Functions are basically the mappings by which elements of a given set are uniquely related to the elements of the other set. Each element on the domain have a unique image onto its co domain.
Graphs are the representation of functions on to the 2-d space . It desribes the nature of the functions . Graph can be possibly different for various classes of functions .

In this article we are going to deal with the functions and its graphs.

Precalculus Function and Graph : Examples

Example 1  : Make the graph for the precalculus function y = `e^x`

Solution : 

The `e^x` is the exponentail function . The domain is the set of all the real numbers while the range is the set of all positive real numbers.

For making the graph , we have to find the plotting points.

By putting x = 0 in the given equation , we get

y = `e^0`   =  1

By putting x = 1 in the given equation , we get

y= `e^ 1` = e = 2.71

By putting x = 0 in the given equation , we get

y = `e^2` =  7.38

x	0	1	2	3
f(x)	1	2.71	7.38	20.08

Graph is as shown :

            

Example 2  : Make the graph for the precalculus function y = ln x

Solution : 

The ln x is the natural logarithmic  function . The domain is the set of all the positive real numbers while the range is the set of all real numbers.

For making the graph , we have to find the plotting points.

By putting x = 1 in the given equation , we get

y = ln (1)   =  0

By putting x = 2 in the given equation , we get

y = ln (2)  = 0.69

By putting x = 3 in the given equation , we get

y = ln (3) = 1.09

x	0.5	1	2	3
f(x)	-0.69	0	0.69	1.09

Graph is as shown :

            

Precalculus Function and Graph : Practice Problems

Problem 1  : Make the graph for the precalculus function y = `|x|`

Problem 2  : Make the graph for the precalculus function y = sgn (x)

Standard Deviation Frequency Table

Standard deviation is an important study in Statistics. Standard deviation is the square root of the mean of the squared deviations divided by number of data. The notation of Standard deviation is σ.

The formula for calculating standard deviation is

Standard deviation(σ) =` sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

Here, n = `sum` f  number of data

In this article, we discuss about calculating standard deviation from frequency table.

Steps to Calculate Standard Deviation from Frequency Table:

Step 1: In frequency table, we calculate sum of fx2.

Step 2: Then we calculate mean bar(x) from frequency table.

Mean `bar(x)` = `(sum fx)/"n"`

Step 3: Calculate standard deviation using formula.

Let us see example problems for calculating standard deviation.

Example 1:

Calculate the standard deviation from the frequency table.

X = Weight(kg)	40	50	60	80
F = Frequency	6	8	10	16

Solution: 

The frequency table is,

Weight (x)	Frequency (f)	fx	x2	fx2
40	6	240	1600	9600
50	8	400	2500	20000
60	10	600	3600	36000
80	16	1280	6400	102400
Total	40	2520		168000

Now we are going to calculate mean bar(x) using  formula,

Mean `bar(x)` = `(sum fx)/n`

From frequency table, we know that,

`bar(x)` = `2520/40`

`bar(x)` = 63

Now we are going to calculate standard deviation from formula.

Standard deviation(σ) = `sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

From frequency table, we know that,

Standard deviation(σ) = `sqrt((168000/40)-63^2)`

Standard deviation =  `sqrt(4200-3969)`

Standard deviation = `sqrt(231)`

Standard deviation = 15.2

Therefore, standard deviation of frequency table is 15.2.

Another Example Problem for Calculating Standard Deviation from Frequency Table:

Example 2:

Calculate the standard deviation from the frequency table.

X = Marks	30	40	50	60
F = Frequency	5	7	9	15

Solution: 

The frequency table is,

 

Marks (x)	Frequency (f)	fx	x2	fx2
10	10	100	100	1000
20	11	220	400	4400
30	14	420	900	12600
40	16	640	1600	25600
Total	51	1380		43600

Now we are going to calculate mean bar(x) using  formula,

Mean `bar(x)` = `(sum fx)/n`

From frequency table, we know that,

`bar(x)` = `1380/51`

`bar(x)` = 27.1

Now we are going to calculate standard deviation from formula.

Standard deviation(σ) = `sqrt((sum_(i=1)^n (fx^2))-bar(x)^2)/n `

From frequency table, we know that,

Standard deviation(σ) =` sqrt((43600/51)-(27.1)^2)`

Standard deviation =  `sqrt(854.9-734.4)`

Standard deviation = `sqrt(120.5)`

Standard deviation = 10.9

Therefore, standard deviation of frequency table is 10.9.