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Theorems apply for equilateral triangles are theorem 1 and 2, In theorem 1 each angle of an equilateral triangle is the same and measures 60 degrees each while in theorem 2 a triangle is said to be equilateral if and only if it is equiangular.

Theorem can be defined as  a statement which has been proved true by a special kind of logical argument called a rigorous proof.

There are different types of theorem such as:

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The nature of the roots for the equation ax² + bx + c is

α + β = -b/a and αβ = c/a

where α and β are the two roots of the equation.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

A quadratic equation ax² + bx + c.

Now,

There will be two roots of the equation α and β,

Now,

The nature of the roots are

α + β  = -b/a

αβ = c/a

Thus,

The roots are α and β and α + β = -b/a and αβ = c/a.

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The nature of roots of quadratic depends on its discriminant, formula is D=√(b²-4ac).

It is the value of the unknown variable for which the quadratic equation holds true. Its nature can be real, imaginary, rational, irrational, equal and unequal as well that depends on the value of the discriminant.

For equation ax²+ bx + c =0 , expression b²- 4ac is called discriminant  and denoted by D.

Given equation is ,  ax²+ bx + c =0

then, according to quadratic formula for the roots of quadratic equation

x = {-b±√(b²-4ac)}/4ac

Let us take a real number

k > 0

√k can be defined and is a positive quantity  but √-k can not be defined.

The quantity which is under square root in the expression for roots is

b²-4ac, which is called discriminant of the quadratic equation. It is represented by D.

If D>0 and is perfect square then the equation will have two real, rational and distinct roots.

If D>0 and is not a perfect square then the equation will have two real, irrational and distinct roots.

If D=0 the equation will have two real and equal roots.

If D<0, the equation will have no real roots.

Hence, nature of roots formula for quadratic equation depends on the value of √b²-4ac.

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Answer:

Step-by-step explanation:

You want to know the pay for a paper route and dog walking if 2 days' paper route pay and 3 days' dog walking pay will total half of $110, while dog walking pays $5 less than the paper route.

Let p represent the pay for a day's pay for the paper route. Then (p-5) is the day's pay for dog walking. The total pay is ...

  2p +3(p -5) = 110/2

Simplifying the equation, we have ...

  5p -15 = 55

  p -3 = 11 . . . . . . . . divide by 5

  p = 14 . . . . . . . . add 3

  p -5 = 14 -5 = 9 . . . . . pay for dog walking

The paper route and dog walking pay 14 and 9 dollars per day, respectively.

__

Additional comment

We could write two equations, and solve by substitution. We would get the same single equation by doing that. In this solution method, it works well to let the variable represent the largest contributor (highest pay).

We could also solve the two equations by any of several other methods. The attached graph shows the solution by graphing.

the graphing system of linear inequalities of two variables are bounded by solid line or dashed line.

The linear inequalities of two variables are the mathematical expressions denoted by greater than > or less than< or combination of ≤ or ≥ between two algebraic terms.

the way we solve linear inequalities are same as the way in how we solve linear equation. The graph of linear inequalities of two variables are bounded by solid line or dashed line depending on the unequal sign presented.

hence, there is a difference between the graph of linear equation and linear inequalities.

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From the examples given in the question, we can say that Both are illustrations of social change that is taking place at various rates.

The Latin phrase "per centum," which means "by the hundred," is where the English word "percentage" comes from. Percentage segments are those with a numerator of 100. In other words, it is a connection where the whole is always deemed to be valued 100.

As per the given data,

Approximately 12,000 years ago, people began to domesticate grain in what we now call the Middle East. And it spread around the world over 6,000 to 7,000 years.

In this, it is taking a lot of time, which means that the rate is slower.

But on the other hand,

In 1984, 8 percent of u.s. households had a computer, but that number was above 70 percent in 2008.

But in this case, in around 24 years there are 70% of households have computers, which means that the rate is higher.

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The equation of the line would be y - 20 = 12(x - 1).

Option (C) is correct.

What is the slope-point form of the line?

For the line having slope "m" and the point (x1, y1) the equation of the line passing through the point (x1, y1) having slope 'm' would be

                     y - y1 = m(x - x1)

Let's suppose on the y-axis there will be a number of copies sold and on the x-axis, there will be a number of days.

On day 1, he sold 20 copies so point would be (1, 20)

On day 2, he sold 32 copies so point would be (2, 32)

Slope of the line = (y2 - y1)/(x2 - x1)

                            = (32 - 20)/(2 - 1)

                         m = 12

By using the slope-point form of the line,

       y - y1 = m(x - x1)

       y - 20 = 12(x - 1)

Hence, the equation of the line would be y - 20 = 12(x - 1).

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Answer:

[tex]2^{\frac{1}{2}}[/tex]

Step-by-step explanation:

Taking the square root of a number is equivalent to putting that number to the power of 1/2.

Step-by-step explanation:

x^(a/b)

means

[tex] \sqrt[b]{ {x}^{a} } [/tex]

the numerator of the exponent is the actual "to the power of" factor.

the denominator defines the degree of the root to be taken.

as any integer number is also a fraction like

6 = 6/1

this applies to any form of rational numbers like

x^0.25 = x^(25/100) = x^(1/4) =

[tex] \sqrt[4]{ {x}^{1} } = \sqrt[4]{x} [/tex]

with irrational numbers as exponents this gets trickier, as with infinite numbers without a pattern after the decimal point we have a problem to precisely specify the degree of the root, as we also have a problem to specify the irrational number itself.

we try to find the origin of the irrational number (like a square root)

[tex] {x}^{ \sqrt{3} } [/tex]

or

[tex] \sqrt[ \sqrt{3} ]{ {x}^{2} } = {x}^{2 \div \sqrt{3} } [/tex]

so,

[tex] \sqrt{2} = \sqrt[2]{2} = {2}^{1 \div 2} [/tex]

The point on the ellipse that is farthest from the origin will be the point that is closest to one of the foci.

To find the points on the ellipse that are nearest to and farthest from the origin, we need to consider the equation of the paraboloid and the equation of the ellipse.

The equation of a paraboloid is given by:

[tex]z = ax^2 + by^2 + c[/tex]

where a, b, and c are constants.

The equation of an ellipse is given by:

[tex](\frac{x}{a})^2 + (\frac{y}{b})^2 = 1[/tex]

where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

If the plane intersects the paraboloid in an ellipse, then the equations of the paraboloid and the ellipse must be equal at the points of intersection. Therefore, we can set the equation of the paraboloid equal to the equation of the ellipse and solve for the x and y values.

[tex]z = ax^2 + by^2 + c[/tex]

[tex](\frac{x}{a})^2 + (\frac{y}{b})^2 = 1[/tex]

Substituting the equation of the paraboloid into the equation of the ellipse, we get:

[tex](\frac{x}{a})^2 + (\frac{y}{b})^2+c = 1[/tex]

We can then solve this equation for the values of x and y that correspond to the points on the ellipse that are nearest to and farthest from the origin.

The point farthest away from the origin occurs at the point on the ellipse that is farthest from the center, which is called the "foci" of the ellipse. The distance from the center of the ellipse to the foci is called the "eccentricity" of the ellipse. The eccentricity is denoted by the letter "e" and is calculated using the following formula:

[tex]e = \frac{\sqrt{(a^2 - b^2)}}{a}[/tex]

where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

Therefore, to find the point on the ellipse that is farthest from the origin, we need to calculate the eccentricity of the ellipse and then use this value to determine the location of the foci. The point on the ellipse that is farthest from the origin will be the point that is closest to one of the foci.

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The 3 ways to prove similarity are Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS).

The three theorems are as follows :

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Answer:

Given the point-slope form of a line, y = mx + b, to get the y-intercept of a line, you first have to solve for b by setting x = 0. This gives you the b-value of y = 0 + b = b. We see that b is the y-intercept.

Step-by-step explanation:

Answer:

See below

Step-by-step explanation:

The point-slope form of a line looks like
y - [tex]y_{1}[/tex]  = m( x - [tex]x_{1}[/tex])

The number that is the place of the 'm' is the slope.

For example:

If we had the equation

y - 3 = 12(x +4)

The slope would be 12.

Answer:

1/15

step by step

divide both numerator and denominator by 2

2÷2=1

30÷2=15

The product is 225.

Augmentation is an activity that addresses the fundamental thought of rehashed expansion of a similar number. The numbers that are multiplied are referred to as the factors, and the result that is produced when two or more numbers are multiplied together is referred to as the product of those numbers. Repeated addition of the same number can be made easier by using multiplication.

Given 15 x 15

the product can be done in many ways,

like add 15times 15 we get the answer,

or 15 is written as 10 + 5

so (10 + 5) x 15

10 x 15 + 5 x 15

150 + 75 = 225

Therefore, 15 x 15 is  225

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The value of function f(3) is -1

What is a Mapping Diagram?

A mapping diagram is an image of two circles which shows two values corresponding to each other, such that one is the input value to the function and the other is the output value.

Given ordered pairs :

(- 1, 0), (0, 3), (3, - 1), (9, 9)

Domain and range from given ordered pairs.

Domain - means x - axis

Range - means y-axis

Now, take out domain and range like that,

Domain of the function : (-1, 0, 3, 9)             [ input values]

Range of the function : (0, 3, -1, 9)               [ output values]

Now, we need to determine the value of function f(3).

It means 3 is input value in f(3) so the output value at 3 is,

f(3) = -1

Hence, the value of function f(3) is -1.

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If the coordinates of point F are (x,y), then the coordinates of point F' after the dilation will be (3x, 3y).

To find the location of point F' after the triangle is dilated by a scale factor of 3, you need to multiply the coordinates of point F by the scale factor.

For example, if the coordinates of point F are (x,y), then the coordinates of point F' after the dilation will be (3x, 3y).

It's important to note that the dilation will change the size of the triangle, but not its shape. The angles of the triangle will remain the same, and the ratios of the side lengths will also remain the same.

If you have a specific triangle and the coordinates of point F, I can provide more detailed guidance on finding the coordinates of point F' after the dilation.

∴ If the coordinates of point F are (x,y), then the coordinates of point F' after the dilation will be (3x, 3y).

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The required total number of Lucy's coins,when we have 15% of quarter coins are 80.

percentage, a general worth demonstrating 100th pieces of any amount. One percent (represented 1%) is a hundredth part; in this way, 100% addresses the total and 200 percent determines two times the given amount. rate.

Quarters coins = 12

Let x be total number of coins Lucy has,then

15% of x = 12

15x/100 = 12

x = 1200/15

x = 80 coins

Thus, there are 80 coins in total.

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The value of x is 11 and ∠LMN = 190°

What is Linear Equation in One Variable?

A linear equation is a one-variable equation of a straight line. The variable's only power is 1. Linear equations in one variable of the form   ax + b = 0 are solved using simple algebraic techniques.

Solution:

Since, MO bisects the angle

we can say that, ∠LMO = ∠OMN

6x + 19 = 9x - 14

3x = 33

x = 11

so, ∠LMO = 66 + 19 = 85°

Therefor the ∠LMN = 190°

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The gradient of 3x+ 1 is conjecture states that this function can be iterated over and over until it yields the number 1, starting from any positive integer n.

A conjecture turns into a theorem when it is rigorously proven. A conjecture is not merely a tool for professional mathematicians; it is an essential step in problem-solving. It is extremely uncommon for a problem's solution to be immediately obvious while tackling problems on a daily basis. Instead, establishing a conjecture about the answer and then proving it through evidence are steps in the problem-solving process that include reviewing cases and the problem's structure. When one observes a pattern that holds true in numerous instances, hypotheses start to form. A pattern may be true in many instances, but that does not guarantee that it will be true in all instances. For a mathematical observation to be fully accepted, conjectures must be proven.

Given

The 3x + 1

Conjecture states that this function can be iterated over and over until it yields the number 1, starting from any positive integer n.

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Answer:

4700 workers

Step-by-step explanation:

We know that 60% of the worker are men

1880 people work for the company who aren't men

100% - 60% = 40%

40% of workers =  1880

40% = 0.4

If 0.4 = 1880

100% of the workers x

100% = 1

We have equations

0.4 = 1880

1 = x

Cross multiply

0.4x = 1880

Divided both sides by 0.4

x = 4700 workers total

To find the number of men workers, we take

4700 - 1880 = 2820 men workers

So, there are a total of 4700 workers!

1880 women workers & 2820 men workers!

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Answer:

60% are men. 40% women

x - amount of men, 1 880 amount of women

60:x = 40:1 880

40x = 60(1 880)

40x = 112 800

x = 2 820

Percentage = Rate / Base

= 1 880 / (1 - 0.60)    (1-0.60 to find out the percent of not men)

= 1 880 / 0.40

= 4 700

Women: 4 700 * 0.4 = 1 880    (this statement is true)

Men: 4 700 * 0.6 = 2 820       (therefore, it is too)

Step-by-step explanation:

The mean for the discrete probability distribution is 3.73

Generally, To calculate the mean of a discrete probability distribution, you can use the formula:

Mean = ∑x * P(x)

Using this formula, the mean of the given distribution is:

Mean = (2 * 0.2) + (6 * 0.3) + (10 * 0.3) + (14 * 0.2) = 4

To calculate the standard deviation of a discrete probability distribution, you can use the formula:

Standard Deviation = √∑(x - mean)^2 * P(x)

Using this formula and the mean that we calculated above, the standard deviation of the given distribution is:

Standard Deviation = √( (2 - 4)^2 * 0.2 + (6 - 4)^2 * 0.3 + (10 - 4)^2 * 0.3 + (14 - 4)^2 * 0.2 )

= √( (-2)^2 * 0.2 + (2)^2 * 0.3 + (6)^2 * 0.3 + (10)^2 * 0.2 )

= √( 4 * 0.2 + 4 * 0.3 + 36 * 0.3 + 100 * 0.2 )

= √( 0.8 + 1.2 + 10.8 + 20 ) = √( 13.8 )

= approximately 3.73

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If Francisco wins 650.16 per month, he will earn a total of 7,801.92 per annum. Note that this is a multiplication exercise.

It is to be noted that Multiplication is a mathematical operation that involves multiplying two or more numbers together. It is often represented using the symbol "x" or "*".

Since there are 12 months in a year,  and Francisco earns 650.16 per annum,

Hence,

12 x 650.16 = 7,801.92

Thus, it is correct to state that where Francisco gets 650.16 per month, by the year's end, he will have earned a total of 7,801.92.

Note that Per Year and Per Annum are synonyms and can be used interchangeably.

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A positive y-intercept means that the line crosses the y-axis above the origin, and a negative y-intercept means that the line crosses below the origin. Arbitrary lines can be defined simply by changing the values ​​of m and b.

Slope Intercept:

The slope intercept of the straight line is used to find the equation of the straight line. The slope intercept formula requires knowing the slope of the line and the intersection of the line with the y-axis. Consider a line with slope 'm' and y-intercept 'b'. The slope-intercept shape formula for a straight line with slopes 'm' and 'b' as y-intercept is given as y = mx + b.

Positive Intercept:

A positive slope refers to a line that is slanted and slopes upward when viewed from left to right. The positive slope of a straight line can be calculated using the formula m = (y2 - y1)/(x2 - x1) = Tanθ = f'(x) = dy/dx. A positive slope means that the two quantities represented along the two axes of the coordinate system increase or decrease simultaneously. A line with a positive slope is a positive value for the ratio of climb to run. A straight line with a positive slope makes an acute angle with the positive x-axis.

                    Positive slope (m) = +slope/run = +Δy/Δx

Negative Intercept:

Negative slope means that the coordinates of the two given points are negatively related. A line with a negative slope has m < 0, and the angle θ it makes with the positive x-axis is obtuse θ, such that 90° < θ < 180°. The slope-to-run ratio of the negative slope line is negative. where slope is the change in y value, denoted by Δy, and run is the change in x value, denoted by Δx.

                          Negative slope = -rise/run = -Δy/Δx

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The nature of roots of quadratic equation 5x²- 4x +5= 0

are unreal.

An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax2 + bx + c = 0, where x is the variable, a and b are the coefficients, and c is the constant term.

Nature of Roots of the Quadratic Equation

The symbols alpha () and beta () are typically used to denote a quadratic equation's roots. Here, we'll learn more about how to identify a quadratic equation's nature without locating the equation's roots. Additionally, look at the formulas for calculating the product and sum of the equation's roots.

Without actually locating the roots (, ) of the equation, the nature of the roots of a quadratic equation can be determined. By using the discriminant value, which is a component of the quadratic equation solution formula, this is made achievable. The discriminant of a quadratic equation is denoted as "D" and has the value b2 - 4ac. Using the discriminatory value as a basis, the nature of the roots of the quadratic equation can be predicted.

Discriminant: D = b2 - 4ac

D > 0, the roots are real and distinct

D = 0, the roots are real and equal.

D < 0, the roots do not exist or the roots are imaginary.

From the given quadratic equation, we can observe that:

a = 5

b = -4

c = 5

On finding discriminant, we get:

D = b² - 4ac

D = (-4)² - (4 × 5 × 5)

D = 16 - 100

D = -84

We can notice that, D(-84) < 0, which means the roots are unreal. Therefore, no roots exist!

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The nature of roots of quadratic equation 5x²- 4x +5= 0

are unreal.

An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax2 + bx + c = 0, where x is the variable, a and b are the coefficients, and c is the constant term.

Nature of Roots of the Quadratic Equation

Discriminant: D = b2 - 4ac

D > 0, the roots are real and distinct

D = 0, the roots are real and equal.

D < 0, the roots do not exist or the roots are imaginary.

From the given quadratic equation, we can observe that:

a = 5

b = -4

c = 5

D = b² - 4ac

D = (-4)² - (4 × 5 × 5)

D = 16 - 100

D = -84

We can notice that, D(-84) < 0, which means the roots are unreal. Therefore, no roots exist!

Using the vertical line test, there are only two graphs that represent functions (graph function): graph, 1 and graph 6.

A graph function is a graph that represents a function on a coordinate plane, which means that every point on the graph fulfills the function equation. In a graph function, an x corresponds to only one y.

If we want to know whether a graph represents a function or not, we can perform a vertical line test. In this test, we can drag a vertical line from left to right on the graph. If the vertical line intersects with the graph at one point at a time, then it's a function.

Otherwise, if the vertical line intersects with the graph at two or more points at a time, then it's not a function. It could happen when an x value gives two or more y values (1 input results in multiple outputs), which means the graph is not a function.

Now we take a look at the given picture.

Graph 1: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at a time. So yes, the graph represents a function.

Graph 2: when we drag a vertical line from left to right on the graph, it intersects with the graph at two points. So no, the graph doesn't represent a function.

Graph 3: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at first. But later on, it crosses two points in the graph. So no, the graph doesn't represent a function.

Graph 4: when we drag a vertical line from left to right on the graph, it intersects with the graph at two points. So no, the graph doesn't represent a function.

Graph 5: when we drag a vertical line from left to right on the graph, it also intersects with the graph at two points. So no, the graph doesn't represent a function.

Graph 6: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at a time. So yes, the graph represents a function.

So there are only two graphs that represent functions: graph, 1 and graph 6 since the other graphs fail the vertical line test.

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78 ft is the total length of the 3 cables.

Given data :

[tex]24^{2}[/tex]+[tex]10^{2}[/tex] =  [tex]d^{2}[/tex]

576  +100 = [tex]d^{2}[/tex]

676 = [tex]d^{2}[/tex]

d = 26

d = 26 is the length of the cable.

3 x 26 = 78 ft is the total length of the 3 cables.

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The required lengths of side of triangle are 15 in, 16 in and 17 in.

Given :

The sum of the angles of a triangle is always 180°

Given that :

Sides of a triangle are consecutive integers.

Also given that,

Perimeter of triangle is 48.

Let the sides of triangle are x, x + 1 and x + 2.

x + x + 1 + x + 2 = 48

3x = 48 - 3

3x = 45

divide by 3 on both sides

x = 15

The length of the triangles  

x + 1 = 15 + 1 = 16

x + 2 = 15 + 4 = 17

The side lengths of the triangle are 15, 16 and 17.

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Using the SAS(Side-Angle-Side) Theorem, ASA(Angle-Side-Angle) Theorem, SSS(Side-Side-Side) Theorem and AA(Angle-Angle) Theorem we can easily prove that two triangles are similar.

When the sides and angles of two triangles match, the triangles are said to be congruent. When the angles of two triangles match, the triangles are said to be similar. Thus, two triangles can be superimposed side to side and angle to angle.

Hence, when the sides and angles of two triangles match that triangle is congruent and When the angles of two triangles match that triangle is similar.

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S angle is equal to 118°,Professor Delta was shooting hoops in the gym.

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No, the range and the domain are very different concepts because the domain is just a collection of natural numbers, whereas the output values are referred to as the range.

A function's authorized range of values is known as its domain. The x values in a function like f make up this set (x). A function's range is the collection of values that it can take as input. When we enter an x value, the function outputs this set of values.

here

The set of all feasible function inputs is known as a function's domain. The function f(x) = 2x can be represented by this box.

The domain is the set of all natural numbers when x = 11, 12, 13, 14, etc.,  the values that are returned are referred to as the range.

No, the range and the domain are very different concepts .

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Answer: 2 metric tons per hour

Step-by-step explanation:

We know that to get from 1/4 hour to 1 hour we have time 4

So we take 1/2 times 4 = 4/2 = 2 metric tons per hour!