The sides of a triangle are as follows. The second side of a triangle is twice the length of the first side. The third side of the triangle is half of the second side. If the perimeter of the triangle is 36 units, what is the measure of all the sides? Write and solve an equation.

The correct options for the line are-

Coordinate are: (−2, 0) and (2, −4)

Slope m = y2 - y1 / x2 - x1

m = (-4 + 0)/(2 + 2)

m = -4/4 = -1

Slope is -1 , option A incorrect.

For the y-intercept

Use point (-2, 0) and slope m = -1

0 = -1(-2) + b

b = -2

As, y-intercept is -2.  

Option B correct.

Find equation of the line.

y = mx+ b

y = -x + (-2)

y = -x - 2

Thus, equation of the line is y = −x − 2.

Option C is also correct.

Find the x-intercept.

if y = 0

0 = -x - 2

x = -2

Thus, line intersects the x-axis at (−2, 0).

The option D is also correct.

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The complete question is-

Which statements about the line that passes through (−2, 0) and (2, −4) are true? Select all that apply.

A. The slope of the line is 1.

B. The line intersects the y-axis at (0, −2).

C. The equation of the line is y = −x − 2.

D. The line intersects the x-axis at (−2, 0).

Polynomial identities can be applied to complex numbers in the same way they are applied to real numbers, by replacing the real numbers in the polynomial with complex numbers and following the same algebraic rules and procedures.

What is a complex number?

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1.

Polynomial identities are equations that are always true for any values of the variables, regardless of whether the variables represent real numbers or complex numbers. Complex numbers can be represented in the form of a + bi, where a and b are real numbers and i is the imaginary unit.

For example, the identity[tex](x + y)^2 = x^2 + 2xy + y^2[/tex] is true for any real numbers x and y, and it also applies to complex numbers a + bi and c + di. The identity becomes [tex](a + bi + c + di)^2 = (a + c)^2 + 2(a + c)(bi + di) + (bi + di)^2[/tex]

Another example is the identity [tex](a+bi)^2 = a^2 + 2abi - b^2[/tex] which is always true for any complex number a+bi. This identity applies to any complex number and can be used to simplify polynomials involving complex numbers.

Hence, polynomial identities can be applied to complex numbers in the same way they are applied to real numbers, by replacing the real numbers in the polynomial with complex numbers and following the same algebraic rules and procedures.

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Yes, 8, 15, 17 is a Pythagorean Triple and sides of a right triangle.

What is meant by Pythagorean?

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Yes, 8, 15, and 17 are triples of Pythagoras, and the sides are right triangles.

The Pythagorean theorem, also known as the Pythagorean theorem, is the fundamental relationship between the three sides of a right triangle in Euclidean geometry.

According to this rule, the area of ​​a square with two other sides is the area of ​​the square whose side is the opposite side of the hypotenuse, or right angle.

The name of this phrase evokes the Greek philosopher Pythagoras, who lived around 570 BC. was born. This theorem has probably been proven most frequently of all mathematical theorems using various techniques.

There are numerous proofs, some dating back thousands of years, involving both geometric and algebraic proofs.

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Classifying Triangle by Sides or Angles: Equilateral triangle, Isosceles triangles, Scalene triangle, Right triangle, Obtuse triangle, Acute triangle, Equiangular triangle.

A triangle is said to be equilateral or equiangular if each of its three sides is the same length. In the well-known Euclidean geometry, an equilateral triangle is also equiangular, meaning that each of its three internal angles is 60 degrees and congruent with the others.

A triangle having three acute angles is known as an acute triangle. A triangle having one obtuse angle and two acute angles is known as an obtuse triangle. No Euclidean triangle may contain more than one obtuse angle because in Euclidean geometry, a triangle's angles must add up to 180°.

A right triangle, also known as a right-angled triangle, right-perpendicular triangle, orthogonal triangle, or previously rectangled triangle, is a triangle with one right angle, or two perpendicular sides. The foundation of trigonometry is the relationship between the sides and various angles of the right triangle.

Obtuse-angled triangles or obtuse triangles are triangles with any one of its angles being an obtuse angle or more than 90°. Only 180° is equivalent to the internal angles of an acute triangle.

A scalene triangle is a triangle with three different side lengths and three different angle measurements. The total of all internal angles, however, is always equal to 180 degrees.

A triangle with two sides that have the same length is said to be isosceles. It can be stated as having exactly two equal-length sides or at least two equal-length sides, with the latter definition containing the equilateral triangle as an exception.

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Two equations that represent the graph shown below as sine function y = 0.909x−3 and as cosine function y = −0.416x −3.

The sine graph or sinusoidal graph is an up-down graph and repeats every 360 degrees i.e. at 2π.

The general form of a cosine function is:

y = a cosb (θ±c) ± d

Here, amplitude, a  is = 1

For b, period is = π

2π/b = π

b = 2

c = 0

vertical shift = D = -3

Putting the value into formula

y = 1 cos2 (θ + 0) - 3

y = 1 ×−0.416(θ + 0) - 3

y = −0.416θ−3  

y = −0.416x −3  (cosine function)

y = 1 sin2 (θ + 0) - 3

y = 1 ×0.909(θ + 0) - 3

y = 0.909θ−3

y = 0.909x−3 (sine function)

Thus, two equations that represent the graph shown below as sine function y = 0.909x−3 and as cosine function y = −0.416x −3.

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The probabilities for this problem are given as follows:

a) The first free throw that he makes is the fourth shot: 0.0430 = 4.30%.

b) The first is the second or the third: 0.3459 = 34.59%.

c) He misses the first three: 0.0741 = 7.41%.

The mass probability formula, giving the probability of x successes, is of:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are given by:

The probability of missing the first three is P(X = 0) when n = 3, hence:

0.42³ = 0.0741 = 7.41%.

For item a, the probability is the above probability multiplied by 0.58 (making the fourth), hence:

0.58 x 0.0741 = 0.0430 = 4.30%.

For the first being the second or the third, the probabilities are given as follows:

Hence:

0.2436 + 0.1023 = 0.3459 = 34.59%.

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The two parts of the exponential notation can be named as the "Base" and the second part is number raised to base called as Exponent .

Consider the Exponential Expression : aⁿ ;

The above exponent notation is read as the number "a" to the power of "n" ,

it means that the number "a" is multiplied "n" number of times .

The two parts of this Exponential Notations are :

the number "a" is called the base and "n" is called as the power or exponent .

Therefore , the two parts of the exponential notation are the base and the power/exponent .

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Answer:

2

Step-by-step explanation:

The equation that can be used to find the car's distance, d, at any time is  d = 30 × t.

Speed is simply referred to as distance traveled per unit time.

Mathematically, Speed = Distance ÷ time.

Given that;

First, we determine the speed of the car.

Speed = Distance ÷ time

Speed = 180 / 6

Speed = 30 m/s

Now, to get the distance traveled at any time, plug the value of the speed of the car into the speed formula and solve for distance d.

Speed = Distance ÷ time

Speed = d / t

d = Speed × t

d = 30 × t

Therefore, the equation is d = 30 × t.

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D) The example (2x^2 + 5x) + (4x^2 - 4x) = (4x^2 - 4x) + (2x^2 + 5x) illustrates the commutative property of addition for polynomials.

The commutative property of addition for polynomials states that the order in which polynomials are added does not affect the result of the sum. In other words, it states that if we have two polynomials A and B, the result of adding A and B (A+B) will be the same as adding B and A (B+A).

For example, consider the polynomials (2x^2 + 5x) and (4x^2 - 4x). If we add these two polynomials in the order given, (2x^2 + 5x) + (4x^2 - 4x) = 6x^2 + x. Now, if we add these two polynomials in the reverse order, (4x^2 - 4x) + (2x^2 + 5x) = 6x^2 + x. We can see that the order of the polynomials does not affect the result of the sum, and thus it holds the commutative property of addition.

Option (2x^2 + 5x) + (4x^2 - 4x) = (4x^2 - 4x) + (2x^2 + 5x) illustrates the commutative property of addition for polynomials, because it shows that the order of the polynomials does not matter and it gives the same answer.

It is important to note that the other options does not illustrate the commutative property.

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Therefore , the solution of the given problem of ratio comes out to be Jessie votes to Rachel votes is 6:5.

A ratio is the simple expression "a / b," where "b" must not be greater than zero, for an ordered pair of numbers "a" and "b." Two ratios are combined to form a ratio. The ratio might be represented as 1:3 if there was just one man and three girls. 3/4 are girls, while 1/4 are boys. A part is a statistic, a percentage of something's size, or a distinction between two or more objects.

Here,

Given: Rachel received 50 votes,

Mario got 40 votes, and Jessie got 60.

To calculate the proportion of Jessie votes to Rachal votes:

Thus,

=> 60 : 50

=>6:5

Therefore , the solution of the given problem of ratio comes out to be Jessie votes to Rachel votes is 6:5.

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Answer: -1/16

Step-by-step explanation:

3/8  --->  6/16

5/16 - 6/16 = (-1/16)

Statement about the temperature for Tuesday compared to Monday at 11:00 p.m is temperature on Monday is 50⁰ F lesser than Tuesday

A number system is defined as a system of writing to express numbers.

In the given number line each unit is 10⁰ F

On Monday at 11:00 p.m. the temperature is 40⁰ F

Tuesday at 11:00 p.m. the temperature is -10⁰ F

The difference between two temperatures is 40⁰ F-( -10⁰)

40⁰ F+10⁰F

50⁰ F

The temperature on Monday is 50⁰ F lesser than Tuesday

Hence, statement about the temperature for Tuesday compared to Monday at 11:00 p.m is temperature on Monday is 50⁰ F lesser than Tuesday

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The total area of the patio will be 20 * 40 = 800 square feet.

The proportionate relationship between a linear dimension of a feature on a model, map, or drawing of an object and the same feature on the genuine thing. A scale model is a physical representation of an object that is geometrically equivalent (known as the prototype).

Since the scale drawing is 3 inches : 10 feet, this means that every 3 inches on the drawing represents 10 feet in real life. Hence, 1 inch on the drawing represents 10/3 = 3.33 feet in real life.

So the patio measures 6 inches by 12 inches on the drawing, which corresponds to 6 * 3.33 feet by 12 * 3.33 feet in real life, or 20 feet by 40 feet.

Therefore, the total area of the patio will be 20 * 40 = 800 square feet.

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Construct a dilate, with scale factor 2, of a triangle using only a ruler and compasses.

The center of dilation—the location from where the picture is shrunk or expanded—is necessary to build a dilation. A scale factor, or the quantity that controls how big or tiny the new picture will be, is also necessary. If the scale factor is more than 1, the new picture will be bigger than the original image, creating an enlargement. We see a decrease if the scale factor is less than 1, as the new picture will be smaller than the original image.

Step-1

Let's dilate triangle PQR by a scale factor of 2 with a center of dilation at point C.

Step-2

First, we will use a straightedge to connect the center of dilation C to each vertex. We are going to extend those lines across the whole page.

Step-3

Next, we'll place the point of the compass on the center of dilation C and the pencil on vertex Q and measure that distance.

Step-4

Now, without changing the size of the compass, we  will move the point of the compass to vertex Q, and make a mark on the line that is extended through Q.

Step-5

Notice that since we measured the distance from C to Q and then used that same distance to mark the line, we have now created a new point that is twice the distance from C that Q was.

We now repeat the process for each of the other vertices of the triangle.

Step-6

These marks represent the vertices of our dilated image. We often use the same letters with an apostrophe to show that it is the corresponding vertex.

Step-7

Finally, we use the straightedge to connect all of the new vertices.

Now,See how triangle P ′ Q ′ R ′  is tw ce the size of the original triangle.

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The value of K that will make the expressions equivalent is 10.

Considering the two expressions:

7 + (1/2)x - 3 + (3/4)x and 1 whole number (1/4)x + k - 6

we shall write the expression 7 + (1/2)x - 3 + (3/4)x in the form of the expression having k and then compare to know the value of k as follows;

(1/2)x + (3/4)x = (2x + 3x)/4

(1/2)x + (3/4)x = 5x/4

7 - 3 = 10 - 3 - 3

7 - 3 = 10 - 6

so;

7 + (1/2)x - 3 + (3/4)x = 5x/4 + 10 - 6

by comparison;

5x/4 is equal to the mixed fraction 1 whole number (1/4)x

k = 10 as obviously -6 is equal to -6.

Therefore, 10 is the value of k that will make expressions equivalent.

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The x coordinate of one of the endpoints of the dilated segment is; x = 8

In the case when a coordinate point (x, y) should be dilated by a scale factor k, it means that the coordinate would be (kx, ky).

Now, since the coordinate point of dilation is given as (4 ,1), it means that if the coordinate is dilated by a factor of 2, then the new coordinate points should be at (2(4), 2(1))

= (8, 2)

Therefore we can conclude that that will be a coordinate of one of the endpoints of the dilated segment and the x-coordinate here is 8

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The perimeter of the egg comprising of arcs AB, AC, BD, and CD is π·(6-√2)/2 centimeters

The perimeter of a figure is the length of the boundary line of a plane occupied by the figure.

The perimeter of the egg can be found by finding the sum of the lengths of the arcs that together form a composite figure of the perimeter of the egg

The length of arc AC centered at point B is found as follows;

The angle at the center B is the base angle of the isosceles right triangle

The acute angle of an isosceles right triangle is 45°, therefore, the angle at B = 45°

The radius of the arc AC, AB = AO + OB = BC

AO and OB are the radii of the circle with center O

AO = OB = 1 cm

AB = 1 + 1 = 2 = BC

The length of AC = (45/360)× 2×π×2 = π/2

AC = π/2

The radii and angle at the center of arc BD are also 2 cm and 45°

The length of arc BD is therefore, (45/360) × 2 × π × 2 = π/2

BD = π/2

The angle at the center of arc CD = ∠CED

∠CED ≅ ∠AEB (Vertical angle postulate)

∠AEB = ∠AEO + ∠BEO = 45° + 45° = 90°

∠CED = ∠AEB (Definition of congruency)

∠CED = ∠AEB = 90°

The radius of the arc CD, CE = ED = BC - EB

EB = √(1² + 1²) = √2

CE = 2 - √2

Length of arc CD = (90°/360°) × 2 × π × (2 - √2)) = π × (2 - √2))/2

CD = π·(2 - √2)/2

The radius of the arc AB = 1 cm

Angle at the center of arc AB = 180°

Length of arc AB = π × 1 = π

AB = π

The perimeter of the egg = AC + CD + BC + AB

Therefore;

The perimeter of the egg = π/2 + π × (2 - √2))/2 + π/2 + π = π·(6 - √2)/2

The perimeter of the egg is π·(6 - √2)/2 cm

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The given quadrilateral ABCD is a rectangle, and the value of x is -2.

A closed quadrilateral has four sides, four vertices, and four angles. It is a form of polygon. By connecting four non-collinear locations, it is created. Quadrilaterals always have a total internal angle of 360 degrees.

A quadrilateral with parallel sides that are equal to one another, and four equal vertices is known as a rectangle. It is also known as an equiangular quadrilateral for this reason.

Given that ABCD is a rectangle.

In rectangle all the angles are equal, hence equate the two angles as follows:

4x + 8 = 6x + 12

6x - 4x = 8 - 12

2x = -4

x = -2

Hence the value of x is -2.

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A factor of a polynomial function is a polynomial that divides the function evenly when using polynomial division.

To check if x - 8 is a factor of the function f(x) =[tex]2x^3 - 17x^2 + 64[/tex]  , we can use polynomial division.

When dividing f(x) by x - 8, the remainder should be zero.

x - 8 is a factor of the function when the remainder is zero.

In this case, the quotient is [tex]2x^2 + x + 8[/tex] , and the remainder is zero, so x - 8 is a factor of the function f(x) = [tex]2x^3 - 17x^2 + 64.[/tex].

So we can say [tex]f(x) = (x-8)(2x^2 + x + 8)[/tex]

Alternatively, we can also use synthetic division to check if x - 8 is a factor of the function.

Synthetic division is a shorthand method of dividing a polynomial by a linear factor such as x - 8.

If the remainder is zero, then x - 8 is a factor of the function.

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As the triangle cannot be equilateral since the sides are not equal. So the obtuse triangle has side lengths 5, 7 and 11.

Assume that a triangle has sides that measure 5, 7, and 11.

Finding the sum of the squares on the triangle's two shorter sides and comparing it to the square on the longest side will reveal the type of triangle.

A triangle must be sharp in order to:

Let a = 5, b = 7 and c = 11

a^2 + b^2 =  (5)^2 + (7)^2 = 25 + 49 = 74

c^2 = (11)^2 = 121

(1) If the triangle is

a^2 + b^2 > c^2

Therefore, the triangle is not acute.

(2) Eqviangular means that each internal angle must be 60 degrees in length.

As a result, the triangle is not eqviangular.

(3) The triangle cannot be equilateral since the sides are not equal.

(4) We note that a^2 + b^2 = c^2.

The triangle is hence obtuse.

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A quadratic regression equation for the stream of water is,

⇒ y = -0.139x² + 1.667x

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Here, The vertex = (6, 5)

Axis of symmetry is line, x = 6

And, A third point in the parabola is, (12, 0)

Hence, The quadratic regression equation for the stream of water is,

⇒ y = - a(x - 6)² + 5

      = -a(x² - 12x + 36) + 5

      = -ax² + 12ax - 36a + 5

Since the parabola passes through point (0, 0),

i.e. -36a + 5 = 0

⇒ a = 5/36

⇒ a = 0.139

Therefore, The equation is,

⇒ y = -0.139x² + 12(0.139)x

⇒ y = -0.139x² + 1.667x

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Answer:

Its A

Step-by-step explanation:

A dilation of a line is a transformation that changes the size of the line but preserves its shape.

It is a type of similarity transformation. A dilation can either be a reduction, in which the line is made smaller, or an enlargement, in which the line is made larger. In both cases, the line's shape is preserved, but the size is changed.

The center of dilation, also known as the scale factor, is a fixed point around which a dilation takes place. The scale factor is a value that represents the amount of change in size, it can be greater than 1 for an enlargement or less than 1 for a reduction.

In geometry, a dilation is represented by a multiplication of the coordinates of a point by a scale factor. For example, a dilation of a line with a scale factor of 2 will result in the line being twice as long as the original line.

In conclusion, a dilation of a line is a transformation that changes the size of the line but preserves its shape. The center of dilation or scale factor is a fixed point around which a dilation takes place and is represented by a multiplication of the coordinates of a point by a scale factor.

It can be either an enlargement or a reduction depending on the scale factor value.

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The Slope Intercept Form Of 2x + 3y = 8 Is y = -2x/3 + 8/3

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

2x + 3y = 8

Make 3y the subject of the formula

So, we have the following representation

3y = -2x + 8

Divide through by 3

This gives

y = -2x/3 + 8/3

Hence, the equation is y = -2x/3 + 8/3

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Complete question

contents

The Slope Intercept Form Of 2x + 3y = 8 Is

The coordinate (2, 3) lies in the first quadrant of a Cartesian plane, which is the upper-right quadrant when the origin (0, 0) is the center.

The coordinate (2, 3) lies in a Cartesian plane, which is a two-dimensional plane with an origin (0, 0) at the center. There are four quadrants on the Cartesian plane. To find which quadrant the coordinate (2, 3) lies in, we must look at the signs of the x- and y-coordinates. The x-coordinate is 2, which is a positive number, and the y-coordinate is 3, which is also a positive number. This means that the coordinate (2, 3) lies in the first quadrant, which is the upper-right quadrant when the origin is the center. The other quadrants are numbered in a clockwise direction, starting with the first quadrant, which is the upper right. The second quadrant is the upper left, the third quadrant is the lower left, and the fourth quadrant is the lower right.

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Using the formula of Simple Interest, the value for the principal amount is obtained to be P = $1333.33.

What is Simple Interest?

Simple interest is a way to figure out how much interest will be charged on a sum of money at a specific rate and for a specific duration of time. Contrary to compound interest, where we add the interest of one year's principal to the next year's principal to compute interest, the principal amount under simple interest remains constant.

The value for Simple Interest is given as = $180

The value for rate is given as = 4.5%

The value for time is given as = 3 years

To find the value for Principal amount, follow the formula for Simple Interest -

SI = P R T/100

Where P is the principal amount, R is the rate and T is the time.

Substitute the values in the equation -

180 = (P × 4.5 × 3)/100

180 = 13.5P / 100

P = (180 × 100) / 13.5

P = 18000 / 13.5

P = 1333.33

Therefore, the value for principal amount is $1333.33.

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3x+y=-1 is the equation for slope as -3 and point as (-1,2).

Algebraic expressions are used to express mathematical expressions, and algebraic expressions are used to express mathematical expressions. For example, 3x + 5 = 14 is an equation containing two expressions separated by a 'equal' sign. A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation. Linear equations are classified into three types: point-slope, standard, and slope-intercept.

Here,

slope=-3

point=(-1,2)

(y-y1)=m(x-x1)

y-2=-3(x+1)

y-2=-3x-3

3x+y=-1

The equation for slope as -3 and point as (-1,2) is 3x+y=-1.

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Constant functions have graph that are horizontal lines in the plane and are linear functions. Graph 3 is the correct answer.

One of the simplest forms of real-valued functions is a constant function, which is used to express a quantity that remains constant across time. Constant functions have graphs that are horizontal lines in the plane and are linear functions. The x-value serves as the function's domain, which is depicted on the x-axis, and the y-axis serves as its range, which is indicated by the symbol f(x).

We know that the constant functions remain constant, that is they do not change their value. Constant functions have graphs that are horizontal lines in the plane and are linear functions.

Hence, graph 3 is the correct answer.

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