What is special about rigid transformations?

Answer:

x=5

Step-by-step explanation:

supplementary angles adds to 180

25x+5+10x=180

35x+5=180

35x=180-5

35x=175

divide

x=5

1/3 is the slope of the line represented by the equation y = 1/3x + 4.

In other words, the slope is the ratio of the change in height to the horizontal distance between any two locations on the line. It is calculated as the ratio of the "rise" divided by the "run" between two points on a line. The angle that a straight line forms with the positive x-axis direction is known as its inclination. The slope or gradient of a line is the tangent of the angle it creates with the positive x-axis direction. A line's slope is typically represented by the symbol m.

Given

Equation

y = 1/3x + 4

Using slope intercept form,

y = mx + b

m = slope

m = 1/3

1/3 is the slope of the line represented by the equation y = 1/3x + 4.

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We can see (either graphically or algebraically) that the lowest point, or minimum, occurs when x = 3 and y = −4. That is the location of the vertex.

The coordinate of the vertex is (3 , −4)

given equation

y = [tex]x^{2}[/tex] - 6x + 5

  = [tex](x - 3)^{2}[/tex] - 4

We can see (either graphically or algebraically) that the lowest point, or minimum, occurs when x = 3 and y = −4. That is the location of the vertex.

The coordinate of the vertex is (3 , −4)

quadratic function :

A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero.

The graph of a quadratic function is a curve called a parabola. Parabolas may open upward or downward and vary in "width" or "steepness", but they all have the same basic "U" shape. The picture below shows three graphs, and they are all parabolas.

All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola.

You know that two points determine a line. This means that if you are given any two points in the plane, then there is one and only one line that contains both points. A similar statement can be made about points and quadratic functions.

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Answer:

[tex]m\angle DAC=35^{\circ}[/tex]

Step-by-step explanation:

Arc [tex]BC=50^{\circ}[/tex] (inscribed angle theorem)

Arc [tex]AB=130^{\circ}[/tex] (semicircle)

[tex]m\angle ADB=65^{\circ}[/tex] (inscribed angle theorem)

[tex]m\angle DEA=80^{\circ}[/tex] (linear pair)

[tex]m\angle DAC=35^{\circ}[/tex] (angle sum of a triangle)

Angles can be measured using one of two standard units. Degrees are the most widely used unit of measurement.

∠ABD = 39°

Therefore, the correct option is c) 39°

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 The y-intercept is the point on the graph where it crosses the y-axis, and has coordinates of (0, b). The y-coordinate is the value of b in the slope-intercept form. The Slope is - ⅖.

Y-intercept = (0, 4), where b = 4

Given the linear equation in standard form, 2x + 5y = 20, where A = 2, B = 5, and C = 20:

Start by transforming the standard equation into its slope-intercept form, y = mx + b, where m = slope, and b = y-intercept.

Subtract 2x from both sides:

2x -2x + 5y = - 2x + 20

5y = -2x + 20

Divide both sides by 5 to isolate y:

y = - ⅖x + 4 ⇒ This is the slope-intercept form where the slope, m = -⅖, and the y-intercept, b = 4.  The y-intercept is the point on the graph where it crosses the y-axis, and has coordinates of (0, b). The y-coordinate is the value of b in the slope-intercept form. Therefore, the y-intercept is (0, 4).

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The antiderivative of f(x) = tan⁻¹(x) is x tan⁻¹(x) − ln(x²+1)/2 + C

The term antiderivative in math is referred as a function that reverses what the derivative does

Here we have given the expression f(x) = tan⁻¹(x).

Now, the antiderivative of the expression is calculated by doing the integral

=> ∫tan⁻¹(x)dx

By using the integration by parts ∫udv = uv−∫vdu

Here let us consider that u=tan⁻¹(x) and dv=dx

And the value of du is calculated as,

=> du =  ∫tan⁻¹(x)dx =  dx/(x² + 1)

And the value of v = x.

Then the expression is written as,

=>  ∫tan⁻¹(x)dx = x tan⁻¹(x) - ∫x/(x² + 1)dx

As per the concept of the constant of integration, it can be simplified as,

=> x tan⁻¹(x) − ln(x²+1)/2 + C

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If there are no duplicate digits, then and only then, is a number considered to be unique. For instance, unique numbers include 20, 56, 9863, 145, etc., but non-unique numbers include 33, 121, 900, 1010, etc.

Any two items that are the only ones of their kind are equally unique. Unique is defined as the only one like it. There is not a singular most. Each number is distinct because there is only one other one like it. For instance, if you subtract a three-digit number, like 345, from its reverse, 543, you get a difference of 198.

If the number An'obtained is subtracted from a number An made up of n consecutive digits in ascending order by reversing the digits of An, then the difference is always a constant. This constant is termed as the Unique number Un.

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The expression f (0) in a function is used to represents the y-intercept on the graph of the function f (x).

The point where the graph cuts the y - axis is known as the y - intercept whereas the point where it cuts the x - axis is known as x - axis. The graph cuts y- axis when the value of x coordinate is 0.

The values which satisfies the equation is known as the zeros of the function. The graph of a function is the set of all points in the plane of the form (x f(x)).

The graph of f could similarly be defined as the graph of the equation y = f(x). As a result, the graph of a function is a subset of the graph of an equation.

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The amount that Sarah need to invest for the value of the account to reach $1030 will be equal to $792.

Interest that is added to a loan and deposit sum is known as compound interest. In our everyday lives, it is the notion that is employed the most frequently.

Compound interest is calculated as a sum using the interest and principal accrued over time. Compound interest versus simple interest differ primarily in this way.

As per the details given in the question,

Rate, r = 5.3%

Time, t = 5 years

Amount = $1030

As per the formula of compound interest:

x × (1 + 5.3%/4)⁽⁵⁾⁽⁴⁾ = 1030

x × (1 + 53/4000)⁽⁵⁾⁽⁴⁾ = 1030

Now, calculate the product,

x × (1 + 53/4000)²⁰ = 1030

x × [(4000 + 53)/400]²⁰ = 1030

x = 1030/(4053/4000)²⁰

x = (1030 × 4000²⁰)/4053²⁰

x = 792.

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There is a 4/5 probability of catching a T-shirt that is not a size small.

Simply put, the probability is the likelihood that something will occur.

When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes.

Statistics is the study of events that follow a probability distribution.

So, there were 50 shirts thrown into the throng in total.

There were 10 little shirts thrown.

15 medium-sized shirts were hurled.

The quantity of extra-large and big shirts.

= Total shirts - ( 10 + 15)

= 50 - 25 = 25

So, a total of 25 shirts are scattered throughout the large/extra large shirts.

A number of shirts that are NOT SMALL size are now available.

= Total shirts - Shirts with small size = 50 - 10 = 40

Probability of getting a not small t-shirt = The total not small t-shirt/The total number of t-shirts

= 40/50

= 4/5

Therefore, there is a 4/5 probability of catching a T-shirt that is not a size small.

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Answer:

c. 49

Step-by-step explanation:

(g o f) (3)

f(x) = 3x - 2

g(x) = x²

(g o f) (x) = (3x - 2)²

          = (3 × 3 - 2)²

          = (9 - 2)²

          =7²

          =49

Consider a cube of side a units, with sides along x, y and z axis and one vertex at origin, as in the figure.

Here AB and CD are two diagonals of the cube, because A and B are diagonally opposite corners, also C and D.

In the figure, coordinates of A is (0, 0, 0) and that of B is (a, a, a). Then vector AB is given by,

[tex]\vec{\rm{AB}}=\rm{\left < a,\ a,\ a\right > }[/tex]

Coordinates of C is (a, 0, 0) and that of D is (0, a, a). Then vector CD is given by,

[tex]\vec{\rm{CD}}=\rm{\left < -a,\ a,\ a\right > }[/tex]

The diagonals each have a magnitude of a√3.

[tex]|\vec{\rm{AB}}|=|\vec{\rm{CD}}|=\rm{a\sqrt3}[/tex]

In the figure θ is the angle between the diagonals.

We take the dot product of the vectors AB and CD to get angle between them.

[tex]\longrightarrow\vec{\rm{AB}}\cdot\vec{\rm{CD}}=|\vec{\rm{AB}}|\cdot|\vec{\rm{CD}}|\cdot\cos\theta[/tex]

[tex]\longrightarrow\rm{\left < a,\ a,\ a\right > \cdot\left < -a,\ a,\ a\right > =a\sqrt3\cdot a\sqrt3\cdot \cos\theta}[/tex]

[tex]\longrightarrow\rm{-a^2+a^2+a^2=3a^2\cos\theta}[/tex]

[tex]\longrightarrow\rm{a^2=3a^2\cos\theta}[/tex]

[tex]\longrightarrow\rm{\cos\theta=\dfrac{a^2}{3a^2}}[/tex]

[tex]\longrightarrow\rm{\underline{\underline{\cos\theta=\dfrac{1}{3}}}}[/tex]

The angle between the diagonals satisfy this equation.

Hence Proved!

a. Yes Juan is correct.

b. The given quadratic equation can be solve by using quadratic formula.

What is a quadratic equation?

A quadratic equation is a polynomial equation with a maximum degree of two. Where a ≠ 0, the equation is given by ax² + bx + c = 0.

Given equation is x² + 7x + 3.

The given equation can't factorized by using middle term or splitting method.

Factorization of quadratic equations can be done in different methods. They are:

Splitting the middle term

Using formula

Using Quadratic formula

Using algebraic identities

Assume that x² + 7x + 3 = 0

x = [-7 ± √(7² - 4 × 1 × 3)]/ 2 × 1

x = [-7 ± √37]/ 2

The factors of quadratic polynomial is ( x - [-7 ± √37]/ 2).

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The rule of the reflection of the triangle ABC to A'B'C' across the y-axis will be (x, y) to (-x, y).

It is the image of the point which is located in the opposite direction of a given point.

The rule of the refection will be given as,

Firstly, the rule for reflecting a point about the y-axis is;

While reflecting on the y-axis, we get the reflected points (x, y) by the coordinates (-x, y).

The standard of the impression of the triangle ABC to A'B'C' will be (x, y) to (- x, y).

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When constructing an angle bisector, the compass should be used to make three arcs, but these arcs do not necessarily need to have the same radius.

To construct an angle bisector, you can follow these steps:

In step 1, the radius of the first arc can be any length, as long as it intersects both sides of the angle. The radius of the second and third arcs should be the same as the first arc because they are drawn using the same width setting on the compass. However, the radius of the second and third arcs does not need to be the same as the radius of the first arc. The important thing is that all three arcs are drawn using the same width setting on the compass so that they are all equidistant from the vertex of the angle.

The complete question is:-

When constructing an angle bisector, the compass must be used to make three arcs. Do all three arcs need to have the same radius? Explain.

- Yes, because then the marks are equidistant from each other.

- Yes, because you need the points of intersection of the angle and arcs to form a parallelogram.

- No; the second two arcs must have the same radius but it can be different from the first.

- No; any three radii will work.

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According to the SAS rule, the two triangles are congruent.

In geometry  SAS rules states that if two triangle are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.

Here we have to prove that the two triangle given here under are congruent using SAS rule.

While we looking into the following diagram, we have identified that the following are given,

In the given triangle, we have know that that the triangle △GHJ and △RTS, has the side values HJ =TS

Here we also know that the angles ∠GHJ=∠STR are equal.

Then the sides of the triangle HG = TR

Therefore, the △GHJ Congruent to △RTS

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In a list or collection of numbers, the range is the difference between the lowest and highest integers. Place all the numbers in order before attempting to determine the range.

The statistics' range is the difference between the highest and lowest values for a specific data collection. The range could also be expressed as the difference between the highest and lowest observation. The sample interval is calculated as the greatest value less the smallest value. The sample range is a crucial measure of variability for continuous variables.

here

The x values for a function like f are contained in this collection (x).

The collection of numbers that the function assumes is known as its range.

Following the insertion of an x value, the function outputs this sequence of values.

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The rational zero theorem states that each rational zero(s) of a polynomial with integer coefficients f(x) = [tex]a_{n} x^{n} +a_{n-1} x^{n-1} +...........+a_{1} x^{} +a_{0}[/tex]

is of the form p/q, where:

->p is a factor of the constant

->q is a factor of the leading coefficient

->p and q are relatively prime (i.e., the fraction p/q is in its simplest form)

The rational root theorem, also known as the rational root test, is an algebraic theorem that states that for a polynomial equation in one variable with integer coefficients to have a solution (root) that is a rational number, both the constant term (the term without a variable) and the leading coefficient (the coefficient of the highest power) must be divisible by the fraction's denominator.

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The triangle has sides of 12, 36, and 37 and is not a right triangle.

What are Pythagoras's triples of the right triangle?

Pythagorean triples are defined as a² + b² = c², where a, b, and c are positive integers. These triples are denoted as (a,b,c). The perpendicular is a the base is b, and the hypotenuse is c of a right-angled triangle.

We have,

a triangle has sides of 12, 36, 37

A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

(12)² + (36)² = (37)²

1440 = 1369

S0, LHS ≠ RHS

Hence, the triangle has sides of 12, 36, and 37 is not a right triangle.

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several sixteens. Numerous numbers that are multiples of 16 include 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.

In mathematics, a multiple is any number that results from multiplying a given number by an integer. As an illustration, multiples of 5 include: 10, 15, 20, 25, 30, etc. 14, 21, 28, 35, 42, 49, and so on are all multiples of 7.

A number is a multiple of a number if another natural number and the given number are multiplied together.

Divide the digits into pairs starting at the end, then multiply the final group by 1, the group before it by 2, the group before that by 4, then 8, 16, and so on. Include every component. If the outcome is a multiple of 7, then so was the original number. Since 91 (137) is a multiple of 7, so is 21553.

16-digit multiples. There are 16 multiples, including 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, and 176.

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Red to tiny will have a probability of 12/20.

Describe probability.

In other terms, probability is a number that indicates the likelihood of an event occurring. Probability is defined as the ratio of favorable outcomes to all outcomes.

Probability is calculated as follows: N = N samples / N favorable outcomes

There are 20 blue numbers, of which the large ones are 17 and the small ones 3. Likewise, there are 20 red digits; the large ones are 8 and the small ones are 12.

The likelihood of receiving a little red number will be determined as follows:

Probability is calculated as follows: N = N samples / N favorable outcomes

12 good outcomes total.

There are 20 samples total.

Fill out the formula above with the value.

12/20 is the probability.

As a result, 12/20 will be the likelihood of red to little.

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The fraction Lee give the dog Gilligan is 7/10

From the question, we have the following parameters that can be used in our computation:

Initial amount of water = 5/7 gallons

Amount of water given to the dog = 1/2 gallons

The fraction given to the dog is then calculated as

Fraction given to the dog = Amount of water given to the dog/Initial amount of water

Substitute the known values in the above equation, so, we have the following representation

Fraction given to the dog = (1/2)/(5/7)

Express as products

Fraction given to the dog = 1/2* 7/5

Evaluate the products

Fraction given to the dog = 7/10

Hence, the fraction is 7/10

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The point that represents a solution to the linear equation 5x + 6y = -15 is (-1, -5 / 3)

Linear equation can be represented in different form such as standard form, general form, slope intercept form and point slope form.

Therefore, let's find the solution to the linear equation.

5x + 6y = -15

Let's put the equation in slope intercept form.

5x + 6y = -15

6y = -5x - 15

divide through by 6

y = - 5 / 6 x - 15 / 6

y = - 5 / 6 x - 5 / 2

Therefore, the slope of the equation is -5 / 6 and the y-intercept is - 5 / 2.

Using the equation we can find the solution of the equation.

We will substitute the value of x to get y value.

Hence,

y = - 5 / 6 x - 5 / 2

using (-1, -5 / 3)

y = - 5 / 6 (-1) - 5  /2

y = 5 / 6 - 5 / 2

y = - 5 / 3

Hence, the solution is (-1, -5 / 3)

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Answer:

Step-by-step explanation: 62 is the limit

Answer:

  a) 81/500

  b) 7/90

  c) 3/11

Step-by-step explanation:

You want to write the fractions that correspond with the decimals ...

  a) 0.162

  b) 0.0777...

  c) 0.272727...

The fraction corresponding to a terminating decimal can be written based on how you pronounce the fraction.

  0.162 = "one hundred sixty-two thousandths" = 162/1000

This can be reduced by removing a common factor of 2.

  = 81/500

The fraction equivalent to a repeating decimal can be found by a multiply and subtract process. In general, this makes the denominator be a number that has as many 9s as the decimal has repeating digits.

This has one repeating digit, 7. This means we want to convert to a fraction by multiplying by 10 to the power of 1:

  x = 0.0777...

  10x = 0.7777...

Now, we can subtract the original number:

  10x -x = 0.7777... -0.0777... = 0.7000...

Notice that the repeating digits cancel. Now we have ...

  9x = 0.7 = 7/10

Dividing by 9 gives ...

  x = 7/90

Using the above method with 10 to the power of 2 (with a 2-digit repeat), we get ...

  100x = 27.272727...

  100x -x = 27.272727... -0.272727... = 27

  99x = 27

  x = 27/99 = 3/11

__

Additional comment

In general, the repeating part of a repeating decimal can be expressed over an equivalent number of 9s. For example, the 6-digit repeat 0.142857142857... is equivalent to 142857/999999. As it happens, this one reduces to 1/7.

If there are intervening digits, as in 0.4333... for example, then they can be treated as a terminating decimal, and the repeating part can be multiplied by an appropriate power of 10 to account for the shift from the decimal point:

  0.4333... = 0.4 + 0.1 × 0.333...

  = 4/10 + 1/10 × 3/9 = 4/10 + 1/30

  = 12/30 +1/30 = 13/30

Answer: 24

Step-by-step explanation:

1*4=4

2*4=8

3*4=12

6*4=24

9*4=36

The Square root function is written in the form of f(x) = √x .

The Square Root function is defined as the function in which the parent function is written in the form of f(x) = √x .

and the inverse of the Square root function is square function , f(x) = x² .

The Domain and range of the root function is [0 , ∞) .

which means the range is the set of all positive real numbers and

hence in the square root function, result is always positive.

Some Examples of Root Functions are , √2 , √3 , √8 etc.   .

Therefore , Square Root function is explained .

The given question is incomplete , the complete question is

What is the Square Root Function ?

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No, it does not make sense that the probability of cats falling from the 5th floor would be greater than those of the 4th. The force of gravity is the same regardless of the floor, so the number of cats falling should be the same no matter what floor they start from.

The force of gravity is a constant force that is the same on all floors. This means that the same amount of force is acting on the cats, no matter which floor they start from. Therefore, the number of cats falling from the 5th floor should be the same as the number falling from the 4th floor. The number of cats falling from a higher floor may be greater if there are extra factors, such as wind, that could affect their fall, but the force of gravity does not change with the floor. Therefore, it does not make sense for the number of cats falling from the 5th floor to be greater than those of the 4th

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V = 1/3πr²h

•Mutiply both sides by 3 to cancel out the fraction

3V = πr²h

•Divide both sides by πh 3V/πh = r²

•Find the square root of both sides

√3V/πh = r

•Switch the sides to make it easier to read

r = √3V/πh

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