a telegraph pole is supported by a wire, the wire is attached to the pole 6m above the ground and to a point on the ground 2.5m from the foot of the pole calculate the total length of the wire needed, if an extra 0.8m is needed for the attachment.​

The roots of the given quadratic equation [tex]\frac{x^{2} }{3} - \frac{5x}{3} - 6= 0[/tex]   after solving are  [tex]\alpha = \frac{5 + 47i }{2}[/tex]  and  [tex]\beta = \frac{5 - 47i }{2}[/tex] .

Given,

          Quadratic Equation is   [tex]\frac{x^{2} }{3} - \frac{5x}{3} - 6= 0[/tex]

Multiply L.H.S. and R.H.S. by 3, and we get:

            [tex]x^{2} -5x - 18 = 0[/tex]

Here,  [tex]a= 1, b= -5 ,c= -18[/tex]

Let's find out the determinant,

         [tex]D = b^{2} - 4ac[/tex]

      [tex]D= (-5)^{2} - 4(1)(-18)\\D = 25 - 72\\D = -47\\[/tex]

Therefore, the roots of the quadratic equation will be :

       [tex]\alpha = \frac{-b + \sqrt{D} }{2a}[/tex]     and    [tex]\beta= \frac{-b - \sqrt{D} }{2a}[/tex]

Now, Substitute the value of [tex]a,b,c[/tex]

      [tex]\alpha = \frac{5 + 47i }{2}[/tex]     and    [tex]\beta = \frac{5 - 47i }{2}[/tex]

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locating a number's component. Doing percentage and decimal calculations proportional relationships.So,  7/6 times taller is Sally than May.

Why is it important to use fractions?

Identifying the portion of a number is one application for fractions. figuring out percentages and decimals. Verhältnis und Proportion.

Learning more complex mathematical concepts requires a strong foundation in fractions. The greatest way to introduce algebra is through fractions because they are a student's initial exposure to mathematical abstraction.

Let's call Jenny's height H. Next, we have this:

Sally's height is equal to 7/8*H.

May's height is equal to 6/8"H.

We only need to divide Sally's height by May's height to compare their heights:

Sally's height divided by May's height equals (7/8)*H / (6/8)*H.

= (7/8) / (6/8) = (7/8) * (8/6) = 7/6

Sally therefore outranks May by 7/6.

Therefore, the answer is 7/6.

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The orthogonal trajectories for the family of curves 3x^2 − y^2 = C are circles with the equation x^2 + y^2 = C/3.

To find this, we can take the derivative of the original equation 3x^2 − y^2 = C with respect to x, which will give us 6x. Then, we can take the negative reciprocal of this derivative, which is -1/6x. This is the slope of the orthogonal trajectories.

We then can use the point-slope form of a line to get y - y1 = -1/6x (x - x1). Since the slope of the orthogonal trajectories is always -1/6x, we can just set the equation y - y1 = -1/6x (x - x1) equal to 0 and solve for x to get x1 = -y1/6.

We can then plug this value back into the point-slope equation to get y - y1 = -1/6(-y1/6) (x - (-y1/6)), which simplifies to y - y1 = y1/6 + x. Now, if we let y1 = -C/3, then we get y - (-C/3) = C/6 + x, which can be rewritten as x^2 + y^2 = C/3.

Therefore, the orthogonal trajectories for the family of curves 3x^2 − y^2 = C are circles with the equation x^2 + y^2 = C/3.

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The amount in the bank after 2.5 years will be $241.875.

Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time. Mathematically -

[tex]$A = P (1 + rt)[/tex]

{A} = final amount

{P} = initial principal balance

{r} = annual interest rate

{t} = time (in years)

Given is the amount deposited by Velvet in a bank for 2.5 years at a rate of 3%.

The amount in the bank after 2.5 years will be -

A = 225(1 + 0.03 x 2.5)

A = 225(1 + 0.075)

A = 225 x 1.075

A = 241.875

Therefore, the amount in the bank after 2.5 years will be $241.875.

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The area that the forest would occupy after 7 years is 2288 km²

An equation is an expression showing the relationship between numbers and variables.

Let y represent area of the forest after x years.

y = abˣ

a is the initial value and b is the multiplication factor.

A certain forest covers an area of 3400 km², hence a = 3400 km². The area decreases by 5.5%, hence b = 100% - 5.5% = 94.5%. Hence:

y = 3400(0.945)ˣ

After 7 years:

y = 3400(0.945)⁷ = 2288 km²

The area of the forest after 7 years is 2288 km²

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Derivative of f(x) = 8√(10x⁴+x²-1)⁹ is (1440x³ + 72x)(10x⁴ + x² - [tex]1)^{(7/2)}[/tex].

The derivative is what calculus is all about. The instantaneous pace at which a function changes with regard to one of its variables is known as the derivative. Finding the slope of the tangent line to the function at a certain location is equal to doing this.

Calculus defines a derivative as the rate at which one quantity, y, changes in relation to another, x. The differential coefficient of y with respect to x is another name for it. Finding a function's derivative is the process of differentiation.

The typical representation of a function's derivative is d/dx (f(x)) (or) df/dx (or) Df(x) (or) f' (x). Let's examine the technical definition of a derivative. Two points on a function f(x) curve are (x, f(x)) and ((x + h), f(x + h)), respectively. If so, [f(x + h) - f(x)]/(x + h - x) = [f(x + h) - f(x) / h] will be the slope of the secant line across these locations. The second point overlaps the first point in the image below when the distance between two points is almost equal to 0 (i.e., when h approaches 0), and the secant line turns into the tangent line.

Given that;

f(x) = 8√(10x⁴+x²-1)⁹

So, First derivatives is ;

[tex]\begin{aligned}& \frac{d}{d x}[f(x)]=f^{\prime}(x) \\& \frac{d}{d x}\left[8\left(10 x^4+x^2-1\right)^{\frac{9}{2}}\right] \\& =8 \cdot \frac{d}{d x}\left[\left(10 x^4+x^2-1\right)^{\frac{9}{2}}\right] \\& \left.=8 \cdot \frac{9}{2}\left(10 x^4+x^2-1\right)^{\left(\frac{9}{2}\right.}-1\right) \cdot \frac{d}{d x}\left[10 x^4+x^2-1\right] \\\end{aligned}[/tex]

[tex]\begin{aligned} & =36\left(10 \cdot \frac{d}{d x}\left[x^4\right]+\frac{d}{d x}\left[x^2\right]+\frac{d}{d x}[-1]\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}} \\& =36\left(10 \cdot 4 x^3+2 x+0\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}} \\& \left.=36\left(40 x^3+2 x\right)\left(10 x^4\right)+x^2-1\right)^{\frac{7}{2}} \\& =\left(1440 x^3+72 x\right)\left(10 x^4+x^2-1\right)^{\frac{7}{2}}\\\end{aligned}[/tex]

So, the Derivative of f(x) = 8√(10x⁴+x²-1)⁹ is (1440x³ + 72x)(10x⁴ + x² - [tex]1)^{(7/2)}[/tex].

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The population being studied in each of the four years is:

The question that was posed to each respondent is:

The term "population" refers to all citizens who are present in a nation or who are temporarily absent from it, as well as all aliens who have made a nation their permanent home. The number of residents in a certain area is depicted by this indicator. The annual variations in population brought on by births, deaths, and net migration are measured as growth rates.

(a) Data and Calculations:

Percentage of employed individuals ages 25-29 with at least a bachelor's degree

in 2016 = 40%.

In the year 2000 = 32%,

in 1985 = 25%,

in 1964 = 16%.+

The percentage has continued to increase steadily.

(b) Numbers are not indicated by categorical data. They have a qualitative character. Instead, categories are used to organize the data. Data about ethnicity, sex, age group, and educational attainment are some examples of categorical data.

U.S. Census Bureau

a. The population being studied in each of the four years is:

O college educated individuals aged 25-29

b. The question that was posed to each respondent is:

O Have your earned a bachelor's degree (or higher)?

c. O categorical

The complete question is,

Based on data from the U.S. Census Bureau, a Pew Research study showed that the percentage of employed individuals ages 25-29 who are college educated is at an all-time high. The study showed that the percentage of employed individuals aged 25-29 with at least a bachelor's degree in 2016 was 40%. In the year 2000, this percentage was 32%, in 1985 it was 25%, and in 1964 it was only 16%.+

(a) What is the population being studied in each of the four years?

O college educated individuals

O college educated individuals aged 25-29

O individuals aged 25-29

O employed individuals aged 25-29

O employed individuals

(b) What question was posed to each respondent?

O Have your earned a bachelor's degree (or higher)?

O Are you enrolled in college?

O Are you between the ages of 25 and 29?

O In what year did you receive your bachelor's degree?

O Are you employed?

(c) Do responses to the question provide categorical or quantitative data?

O categorical

O quantitative

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The graph of the y = x³ function is given by the image given at the end of the answer.

The function for this problem is defined as follows:

y = x³.

The domain and the range of the function are given as follows:

Some points on the graph of the function are given as follows:

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The domain and the range of the composite function are given as follows:

a) Domain: {3, 5, 6, 7}.

b) Range: {0, 9}.

The composite function of f(x) and g(x) is given by the following rule:

(f ∘ g)(x) = f(g(x)).

It means that the output of the inside function serves as the input for the outside function.

The points on the composite function are given as follows:

The domain is the set of input values, hence it is given by:

{3, 5, 6, 7}.

The range is the set of output values, hence it is given by:

{0, 9}.

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The solid S is a triangular prism whose base is a triangle with vertices (0, 0), (1,0), and (0, 1). The cross-sections perpendicular to the x-axis are equilateral triangles of the same height.

The solid S is a three-dimensional shape with a triangular base. The base is a triangle with vertices (0, 0), (1,0), and (0, 1). This triangle has sides of length 1, and the angles at each vertex are equal. The equilateral triangles are the cross-sections of the solid perpendicular to the x-axis. This means that each side of each triangle is the same length, and all the angles are equal. Each cross-section has the same height, determined by the height of the solid. The shape of the solid is a triangular prism, with triangular sides that connect the three vertices. This shape has a three-dimensional volume, and the volume can be calculated by multiplying the area of the base by the height of the solid.

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Answer:

5 oranges and 10 apples were sold  

Step-by-step explanation:

Let x be the number of oranges sold and y be the number of apples sold

Since 15 fruits were sold in all we get the following equation

x + y = 15 ....................[1]

Each orange costs $1 so if x oranges were sold that would fetch x dollars in sales = x dollars in sales

Each apple costs $2 so if y apples were sold, that would result in 2y dollars in sales

Total sales in $ = x + 2y and we know this is $25 so we get the second equation as

x + 2y = 25 ..................[2]

Look at the two equations. The coefficient of the x term is the same. We can eliminate the x term by subtracting equation [1] from equation [2] as follows:

x + 2y = 25
                    -
x  +   y = 15
-----------------
        y = 10

(2y - y = y and 25-15 = 10)

Now that we know y = 10, we can use equation [1] to find x

x + 10 = 15

which makes x = 5

ANSWER

5 oranges and 10 apples were sold  

According to the given  expression There will be 910 women among the 1,560 runners.

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself.

Using mathematical operations like addition, subtraction, multiplication, and division, a combination of numbers and integers can be expressed numerically in mathematics.

I'm looking for 7/12 of 1560. I really do mean, "Divide 1560 among 12 groups and count how many there are in each category overall."

1560 divided into 12 groups.

1560/12

=130.

There are 130 people in each group. We multiply to determine how many there are in the seven groupings.

130/7

=910

Among the 1,560 runners, 910 of them will be female.

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The maximum area of rectangle is 8450 m².

Given that there is fence only on 3 sides and one length side have river flowing trough it

so the perimeter of the plot is given by:

l + w + w

=> 2w+l

where w is width and l is length of the plot

given , perimeter = 260

=> 2w+l = 260  ---- (i)

area of the plot is = l*w

putting the value of w from equation (i) we get

(260-2w)*w

=> 260w-2w²

so A = 260w-2w²

To maximise A we have dA/dw=0

=> d(260w-2w²)/dw

=> 260-4w

so 260 - 4w=0

=> 4w = 260

=> w = 65

l = 260 -2 w

=> 260 -2 *65

=> 260-130

=> 130

so Maximum area of the rectangle = 130*65 = 8450

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The total reimbursement for the employee will be $21.75.

It is important to note that a word problem in mathematics simply refers to a question that is written as a sentence or in some cases more than one sentence which requires an individual to use his or her mathematics knowledge to solve the real life scenario given.

In this situation, the employee is reimbursed at a rate of 29 cents per mile and he ravels at a 25 miles 3 days a week.

The total reimbursement will be:

= 0.29 × 25 × 3

= $21.75

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Complete question

An employee is reimbursed at a rate of 29 cents per mile. Travels at a 25 miles 3 days a week. Find the total reimbursement.

This figure's surface measures 2332 square inches.

The area or region that an object’s surface occupies is known as its surface area. Volume, on the other hand, refers to how much room an object has. There are numerous shapes and dimensions in geometry, including spheres, cubes, cuboids, cones, cylinders, etc. Each form has its own volume and surface area.

Given, The figure below is a net for a triangular prism. Side a = 46inches, side b = 12 inches, side c = 32inches, and altitude d = 22inches

From the figure,

The surface area of triangle = 1/2* d * a +  1/2* d * a

The surface area of the rectangle = b *a  +2 * c * b

Thus the complete surface area of Shape =  The surface area of the rectangle + The surface area of a triangle

Total surface area =d*a + b*a + 2*c*b

Total surface area =  22 * 46 + 12 * 46 + 2 * 32 * 12

Total surface area = 2332

Therefore, The surface area of this figure is 2332 square inches.

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Complete question:

To distinguish between linear growth and exponential growth, you need to look at the equation of the function and the shape of the graph.

Linear functions have a constant rate of change, meaning that the slope of the line is always the same. The equation for a linear function is y = mx + b, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line.

Exponential functions, on the other hand, have a variable rate of change, meaning that the slope of the line changes as the x-value increases. The equation for an exponential function is y = ab^x, where a and b are constants. The graph of an exponential function is a curve that increases or decreases rapidly.

If the graph is a straight line it is linear, if it is curved and increases or decreases rapidly then it is exponential, otherwise, it would be considered as other.

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Constructors, instance variables, accessors/mutators, and methods the user needs to manipulate the number of objects of the class should all be public.

Public members of a class are those that are visible and accessible to users of the class. These include constructors, instance variables, accessors/mutators, and methods that the user needs to manipulate the objects of the class. These members should be designed and implemented carefully, since they form the interface between the class and its users. Properly designed public members can ensure that the class is easy to use and understand. Making the wrong members public can lead to confusion, bugs, and difficult-to-maintain code.

Constructors, instance variables, accessors/mutators, and methods the user needs to manipulate the number of objects of the class should all be public.

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Net income is the amount of accounting profit a business has left over after covering all of its expenses. Sales revenue is multiplied by COGS, SG&A, depreciation and amortisation, interest expenditure, taxes, and any other expenses to arrive at net income.

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The length of a circle's radius where the diameter's endpoints are at (2,4) and (-3,-1) is 5/2.

Then,

[tex]$D=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}[/tex]

As a result, the diameter's end points are (2,4) and (-3,-1)

[tex]d=\sqrt{(-3-2)^2+(-1-4)^2}[/tex]

[tex]d=\sqrt{(-5)^2+(-5)^2}[/tex]

[tex]\mathrm{d}=\sqrt{25+25}=\sqrt{50}=5 \sqrt{2}[/tex]

[tex]$\text { We know Radius } & =\frac{\text { Diameter }}{2} \\[/tex]

[tex]$& =\frac{5 \sqrt{2}}{2}=\frac{5}{\sqrt{2}}[/tex]

Hence, the radius of given circle is [tex]$} \Rightarrow \frac{5}{\sqrt{2}}[/tex]

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The  height of her eye level above the sea is 45.6 km

Height talks about how high the persons eye is above te sea level.

The equation so formed is

3.6√h = 13.5 Kilometers

Making √h the subject of the relation we have

√h  = 3.75 Kilometers

Making h the subject by squaring both sides of the equation we have

h = √3.75 Kilometers above sea level

h = 1.9 Kilometers above sea level rounded

Taking these various steps, the height of the eye is 1.9 kilometers above sea level.

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Correct question:

If a person's eye level is h meters above sea level and she can see d kilometers to the horizon, then d = 3.6√h . Suppose the person can see 13..5 kilometers to the horizon

The smallest number of first-place votes a candidate could win with under the Plurality method is 105.

The division is one of the basic arithmetic operations in math in which a larger number is broken down into smaller groups having the same number of items.

Given that, an election with 521 votes.

considering an election with 521 votes and 5 candidates up for the election

Dividing the votes amongst 5 candidates

= 521/5 = 104.2  

So, the least number of first-place votes needed by a candidate using the plurality method  would be = 105 votes

104.2+ 104.2 + 104.2 + 104.2 + 104.2 = 521 (Dividing the votes equally)

104+104+104+104+104= 520

Thus, the remaining vote = 521 - 520 = 1

Least first-place vote = 104+1= 105 votes

Therefore, the smallest number of first-place votes a candidate could win with under the Plurality method is 105.

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After two hours, Cesar and Camden are both 48 kilometers from their homes as per the coordinates.

A pair of numbers called coordinates are used to locate a point or a form in a two-dimensional plane. The x-coordinate and the y-coordinate are two numbers that define a point's location on a 2D plane.

given, Cesar and Camden are brothers and live in the same house. Cesar is 80 kilometers away from home and bikes toward home at a rate of 16 kilometers per hour. Camden is 20 kilometers away from home and keeps walking away from home at a rate of 14 kilometers per hour.

Linear Equation for Cesar:

y = 80- 16x

Linear equation for Camden:

Y = 14x + 20

The coordinates (2, 48) is showing the position of both brothers after 2 hours.

(2, 48) this graph represents that both brothers will be met at 48 kilometers if they are walking the same road.

therefore, As per the given coordinates Cesar and Camden are both 48 kilometers away from home after 2 hours.

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simplest polynomial function with zeroes -2 and 3 is expanded form is f(x) = [tex]x^2 - x - 6[/tex].

To write a polynomial function with given zeros, we can use the factorization method. The zeros of a polynomial function are the solutions of the equation f(x) = 0. So, if we know the zeros of a polynomial function, we can use them to write the function in factored form.

For example, if the zeros of the polynomial function are -2 and 3, we can write the function as:

f(x) = (x + 2)(x - 3)

This function can be expanded into:

f(x) = [tex]x^2 - x - 6[/tex]

It's important to notice that the degree of the polynomial function is determined by the highest exponent of the variable in the equation, in this case 2.

It's also important to notice that the zeroes of the function are -2 and 3, as when you substitute them in the equation, the result is zero.

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The complete question is:

What is the simplest polynomial function with zeroes -2 and 3? Write the equation in the expanded form.

The error in writing the polynomial 3m²-5m⁵+m⁴ in standard form

as -5m³+3m²+m⁴ is, Power of  -5m is 5 not 3 and powers should be in descending order

The polynomials are the algebraic expressions, which have multiple powers. Polynomials are always written in a form of descending powers.

And based on powers of polynomials, we can categorize polynomials into following categories-

1. Linear Equation

2. Quadratic Equation

3. Cubic Equation

Given that,

The polynomial-        3m²-5m⁵+m⁴

The standard form-    -5m³+3m²+m⁴

the errors are in standard form of given polynomial,

 3m²-5m⁵+m⁴

For converting it into standard form,

So, it can be arranged in descending powers

-5m⁵+m⁴+ 3m²

By comparing it with given standard form

It can be seen that there are two errors in given form

1.   Power of -5m is 5 not

2.  Variables should be arranged in descending powers

Hence, the correction is power -5 and descending order of powers

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The next number in the list would be -11.

What is Inductive reasoning?

Inductive reasoning is a form of reasoning that derives a general principle from a set of observations. It entails drawing broad conclusions from specific observations. Deductive reasoning differs from inductive reasoning.

Based on the pattern in the list, the next number in the sequence using inductive reasoning would likely be -11.

The pattern appears to be that each number is the previous number minus 3.

Hence, By using inductive reasoning the next number in the list would be -11.

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The number in the 17th position in the list is 7425

We start by eliminating some of the options. The smallest possibility for the larger number is if the larger number were twice the smaller number. [tex]$2457\times2=4914$[/tex], since [tex]$2457$[/tex] the least value in the group. In order for the largest possibility to be less than or equal to 7542, the largest number in the set, when multiplied by 2, it must be twice as large as the largest number in the set.This happens to be [tex]$2754\times2=5508$[/tex]. As a result, the number would need to be even and between 4914 and 5508. 5472 and 5274 are the only even numbers in the set and in this range. Both of these numbers are not twice a number in the set, according to a cursory check. The number cannot be four times or higher than any other number in the set because [tex]$2457\times4=9828$[/tex], well past the range of the set. Therefore, the number must be triple another number in the set. The least possibility is [tex]$2457\times3=7371$[/tex]and the greatest is [tex]$2475\times3=7425$[/tex], since any higher number in the set multiplied by 3 would be out of the range of the set. Reviewing, we find that the upper bound does in fact work, so the multiple is [tex]$2475\times3=7425[/tex].

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The expected value of a random variable is the sum of the probabilities of all possible outcomes which is 10.083

In this case, since the random variable is the sum of the three independent rolls of a fair die, the expected value, E[X], is the sum of all the possible outcomes of three die rolls, each multiplied by the probability of 1/6 that each outcome will occur:

E[X] = (1 + 2 + 3 + 4 + 5 + 6) × (1/6)³

= 3.5

The variance of a random variable is the expected value of the squared deviation from the mean. In this case, the variance for X is:

Var(X) = E[X^2] - (E[X])^2

= (1² + 2² + 3² + 4²+ 5² + 6²) × (1/6)³ - (3.5)²

= 10.0833

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The measure of angle C is 149°.

In mathematics, an angle is a figure made up of two rays that share a terminal and are referred to as the sides of the angle and the vertices of the angle, respectively.

Angles, which are frequently expressed in terms of degrees or radians, are used to quantify the size of turns.

The law of cosines, the law of sines, and the Pythagorean theorem are only a few examples of mathematical equations that use angles, which are a key component of geometry.

When understanding shapes, angles are equally crucial.

The angles within a triangle, a quadrilateral, and a polygon can provide information on the number of sides a shape has in addition to whether or not it is regular.

Angles are also used to calculate the arcs and circles of an object.

From the given figure,

It is a pentagon, sum of sides of a pentagon is 540°

It is clear that,

90 + 90 + 90 + 6x + 1 + (8x - 11) = 540°

⇒ 260 + 14x = 540°

⇒ 14x = 540 - 260

⇒ 14x = 280

⇒ x = 20

∠ C = 8 × 20 - 11 = 160 - 11 = 149°

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The label of the triangles based on the center are;

A) Incenter

B) Orthocenter

C) Circumcenter

D) Centroid

1) The centroid of a triangle is defined as the point in which the three medians of the triangle intersect.

2) The orthocenter of a triangle is defined as the point where all the three altitudes of the triangle cut or intersect each other.

3) The incenter of a triangle is defined as the intersection point of all the three interior angle bisectors of the triangle.

4) The circumcenter of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect.

Looking at the given triangles, we can conclude that triangle A shows the incenter, triangle B shows the orthocenter, triangle C shows the circumcenter while triangle D shows the centroid,

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