I need help with these problems 1-11

"If in two triangles, sides of one triangle are proportionate to (i.e., in the same ratio of) the sides of the other triangle, then their corresponding angles are equal and consequently the two triangles are comparable," asserts the Side-Side-Side (SSS) criteria.

Take a look at the triangles below.

- We can see that all three pairs of the sides of these triangles are congruent. - This is also known as "side-side-side" or "SSS." According to the SSS criteria for triangle congruence, two triangles are congruent if they have three pairs of congruent sides.

SSS Congruence Rule Theorem

When one triangle's three sides are identical to the corresponding three sides of another triangle, two triangles are said to be congruent.

The aforementioned theorem will now be proven.

Given: [tex]\triangle A B C[/tex] and [tex]\triangle P Q R[/tex] such that AB=PQ, BC=QR and AC=PR.

To prove: [tex]\triangle A B C \cong \triangle PQR[/tex]

Construction: Let BC be the longest side of [tex]\triangle A B C[/tex] and so QR is the longest side of [tex]\triangle P Q R.[/tex]

Draw PS so that [tex]\angle R Q S=\angle C B A[/tex] and [tex]\angle Q R S=\angle B C A[/tex].

Join SQ and SR.

In [tex]\triangle ABC[/tex] and [tex]\triangle SQR[/tex],

BC=QR Given

[tex]\angle C B A=\angle R Q S[/tex] By construction

[tex]\angle B C A=\angle Q R S[/tex] By construction

[tex]\triangle A B C \cong \triangle SQR[/tex] By ASA congruence

[tex]\angle A=\angle S[/tex] By CPCTC

AB=SQ By CPCTC

AC=SR

Now, AB=PQ and AB=SQ => PQ=SQ

Similarly, AC=PR and AC=SR => PR=SR

Hence, [tex]\triangle ABC \cong \triangle PQR,[/tex]

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The discriminant of a quadratic equation is equal to the expression (b^2 - 4ac) where a, b, and c are the coefficients of the equation. In this case, the discriminant is equal to (5^2 - 4(2)(3)) = 25 - 24 = 1.

The discriminant of a quadratic equation is a numerical value that can be used to determine the nature of its roots. The discriminant is equal to the expression (b^2 - 4ac) where a, b, and c are the coefficients of the equation. In this case, the equation is 2x² 5x 3 0. We can identify the coefficients of the equation as a = 2, b = 5, and c = 3. Using the formula, the discriminant is equal to (5^2 - 4(2)(3)) = 25 - 24 = 1. This means that the equation has two real and equal roots, as the discriminant is equal to zero. This can be seen by factoring the equation, which gives us (2x - 3)(x + 1) = 0. The roots of the equation are then x = 3/2 and x = -1.

Discriminant = b^2 - 4ac

= (5)^2 - 4(2)(3)

= 25 - 24

= 1

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From the parallelogram AECF, lines A F is similar to E C.

You should know that two or three lines are congruent if they are the same or if they look alike.

The given parameters  are

Parallelogram A E CF (given)

Parallelogram A B C D                   given

In every parallelogram, opposite side are parallel and similar

From parallelogram ABCD, line AB  = DC = FD

Also,  AB=DC=BE

In the figure also, ΔAFD=ΔBCE   (similar triangles)

Therefore, A F  similar to E C

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The total number of sunny days for the three years is 103 days.

This year there were 17 sunny days.

There were 17 sunny days this year, 18 fewer than last year. This is to mean that, last year, sunny days were 18 days more than this year, Which can be found by;

18+17 = 35 sunny days

Sunny days two years ago were three times the sunny days this year which is;

3×17 = 51 sunny days

Therefore, the total number of sunny days for the three years is calculated by adding the total number of sunny days for the three years:

17 + 35 + 51 = 103 sunny days

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Answer:

C, n= -1 or 1.

Step-by-step explanation: The bars around 3x+6 are absolute value bars (| |), meaning that whatever lies inside of them will turn positive. An example: |-9| ➡️ 9. So, since we do have the absolute value bars, multiplying 3 by -1 would not effect the equation. So therefor we could use a regular 1 or a negative one to make the equation equal to 9.

Answer:

Octagon

Step-by-step explanation:

An octagon has 4 sets of parallel lines and 8 sides.

When the population standard deviation is not known, the sample size needed for a confidence interval can be determined using the t-distribution.

The t-distribution is used to calculate the margin of error, which is the amount of variation that is allowed in the sample results. The sample size needed for a confidence interval is determined by calculating the critical value of the t-distribution for a given level of confidence and degree of freedom. The degree of freedom is the number of independent observations in the sample minus one.

The confidence level determines the critical value of the t-distribution, which is used to calculate the margin of error. The sample size is then calculated by dividing the margin of error by the desired confidence level. This method allows us to determine the sample size needed for a confidence interval even when the population standard deviation is unknown.

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The base in term of the height of the triangle is written as ;b = 2A/h.

A closed triangle has three sides. It is among the fundamental geometric shapes because it has three vertices and three sides.

Thus, the formula for the are area of the triangle is -

A = 1/2 bh

In  which,

b = base of the triangle.

h = height of the triangle.

In term of the height.

b = 2A/h

Thus, the base in term of the height of the triangle is written as;b = 2A/h.

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The explicit form that would be used to find the nth term in the sequence is aₙ = 2 + 3n

What is arithmetic sequence?

The difference between each consecutive term in an arithmetic sequence is constant.

An explicit formula for an arithmetic sequence is a = d(n - 1) + c, where d is the common difference between consecutive terms.

According to the given question:

The first term in an arithmetic sequence is 5 that is a = 5

Now, the third term in the sequences is -1

a₃ = a + 2d

By substituting the value of a and a₃ we can find d for the arithmetic sequence.

-1 = 5 + 2d

2d = -6

d = -3

So, the explicit form that would be used to find the nth term in the sequence is

aₙ = 5 - (n - 1)(-3)

Simplifying further to get

aₙ = 2 + 3n

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Therefore, the solution to this problem is  the option 3) Gross =$204.1   million is the correct option.

Regression analysis, which is used in statistical modeling, is a collection of statistical procedures for determining the associations between a dependent variable and one or more independent variables.

Here,

Given: Multiple regression model to predict domestic gross earning.

=>  Gross = -27 + (1.1*Budget)-(0.43*runtime)+(2.6*return) + error

We have,

budget = $75 million

runtime = 120 minute

rotten tomatoes =77

=>Gross = -27 + (1.1*75)-(0.43*120)+(2.6*77)

=> Gross =$204.1   million

Therefore, the option 3) Gross =$204.1   million is the correct option.

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x (x - 1) (x + 1) = [tex]x^{3}[/tex]- x

Distribute

1. (x + 1) [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3}[/tex] - x

2. Distribute

   (x + 1) [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3}[/tex] - x

    [tex]x^{3 }[/tex] + 1 [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

3. Multiply by 1

   [tex]x^{3 }[/tex] + 1 [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

4. Distribute

  [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - x (x + 1) = [tex]x^{3 }[/tex] - x

  [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - ([tex]x^{2}[/tex] + x) = [tex]x^{3 }[/tex] - x

5. Distribute

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - ([tex]x^{2}[/tex] + x) = [tex]x^{3 }[/tex] - x

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - [tex]x^{2}[/tex] - x = [tex]x^{3 }[/tex] - x

6. Combine like terms

   [tex]x^{3 }[/tex] + [tex]x^{2}[/tex] - [tex]x^{2}[/tex] - x =  [tex]x^{3 }[/tex]- x

   [tex]x^{3 }[/tex] - x = [tex]x^{3 }[/tex] - x

7. Move terms to the left side

   [tex]x^{3 }[/tex] - x = [tex]x^{3 }[/tex] - x

  [tex]x^{3 }[/tex] - x - ([tex]x^{3 }[/tex] - x) = 0

8. Distribute

 [tex]x^{3 }[/tex] - x - ([tex]x^{3 }[/tex] - x) = 0

 [tex]x^{3 }[/tex] - x - [tex]x^{3 }[/tex] + x = 0

9. Combine like terms

  [tex]x^{3 }[/tex] - x - [tex]x^{3 }[/tex] + x = 0

   - x + x = 0

10. Combine like terms

    - x + x = 0

     0 = 0

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The two types of functions are linear and non linear functions.

A linear function is a function that can be written in the form f(x) = mx + b, where m and b are constants. This type of function has a constant rate of change, meaning the output will always increase or decrease the same amount for a given change in the input. For example, if the input increases by 3, the output will increase by 6. An example of a linear function is f(x) = 3x + 5. This function will always increase by 3 for each increase in the input, and will always have a y-intercept of 5.

A non-linear function is a function that cannot be written in the form f(x) = mx + b. This type of function has a variable rate of change, meaning the output will increase and decrease at different rates for different changes in the input. An example of a non-linear function is f(x) = x2 - 4x + 5. This function will increase and decrease at different rates for different changes in the input, and will always have a y-intercept of 5.

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The slope of the line y =3x(1 ,3) is 3.

The ratio of how much y grows as x grows by a certain amount is known as a line's slope. The slope of a line indicates how steep it is, or how much y rises as x rises. Anywhere along the line, the slope remains constant (the same). In math, we frequently want to quantify the steepness. Slopes B and C are clearly higher than slope A. While slope A is just four units high, they are both seven units high. Therefore, it would seem that height and steepness are related.

Given

Equation

y = 3x (1,3)

Using slope intercept form,

y = mx + b

m = slope

m = 3

The slope of the line y =3x(1 ,3) is 3.

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According to square numbers , 1 is a square number while 2 is not .

What does Squaring mean ?

In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 32, which is the number 9.

When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.

Thus , 1 is a square number while 2 is not .

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Answer:

1536

Step-by-step explanation:

Answer:

3,6,12,24,48,96,192,384,768,1536

I don't know which term you need so i mentioned all terms

The sum of roots of x2 + 5x + 6 = 0 is -5.

x2 + 5x + 6 = 0

To find the sum of roots, we need to factor the equation :

x2 + 5x + 6 = 0

x2 + 3x + 2x + 6

= 0

x(x + 3) + 2(x + 3)

= 0

(x + 3)(x + 2) = 0

x = -3, -2

Sum of Roots = -3 + (-2) = -5

x2 + 5x + 6 = 0

To find the sum of roots, we need to factorize the given equation. To do that, we can divide the equation into two parts, x2 + 3x and 2x + 6. We can factorize the first part as x(x + 3) and the second part as 2(x + 3). Then, we can multiply both parts to get the factorized equation, (x + 3)(x + 2) = 0. We can solve this equation to get the roots, x = -3 and -2. Finally, we can add both roots to get the sum of roots, -3 + (-2) = -5.

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The value of b when the system has an infinite number of solutions is 12.

The Given equations are

y=6x-b ---(1)

6=3x+3y ----(2)

If the system has an infinite number of solutions, then equation 1 is equal to equation 2

If a pair of linear equations have unique or infinite solutions, then the system of equation is said to be a consistent pair of linear equations

Isolate variable y in equation 2

2=x+y

Multiply by 6 on both sides

y=6x-12 -----(3)

To find the value of b, equate equation 1 and equation 3

6x-b=6x-12

b=12

Thus the value of b=12.

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Answer:

Option D is correct answer

Step-by-step explanation:

-4b + 8c + 12 - 8b - 2c + 6

-4b - 8b + 8c - 2c + 12 + 6

-12b + 6c + 18

Take 6 as common

6(-2 + c + 3)

Thus, The answer is 6(-2 + c + 3)

The two triangles are congruent if two angles and a non-included side of one triangle are congruent with two angles and the corresponding non-included side of a second triangle. So it was proved.

The two triangles are congruent if two angles and a non-included side of one triangle are congruent with two angles and the corresponding non-included side of a second triangle.

If angle ad, side ac-df, and angle cf are present, then ABCDEF is true.

To support: By the angle sum property, ABC DEF A+B+C=180° ______ (1)

In DEF D+E+F=180°___________ (2) (By angle sum property)

From (1) and (2), A+B+C= D+E+F A+E+F= D+E+F

(Given B=E and C=F) A=D ____________________ (3)

Currently, in ABC and DEF A=D (from (3))

AC = DF (Given)

C=F (Give (AAS congruency)

So it was proved.

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Answer:

52 degrees

Step-by-step explanation:

The 3 angles of a triangle add up to 180 degrees

To find angle C, use 180 - B - A = C

180 - 85 - 43 = 52

Using Pythagoras theorem the compound inequality for the possible length is 7.6 ≤ x ≤ 9

Pythagoras theorem states that “In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.

To find the possible value for the third side, we have to use Pythagoras formula which is given as;

x² = y² + z²

In the first scenario, we would assume the third side is the hypothenuse

x² = 8.3² + 3.4²

x = 9

In the second scenario, we would assume the third side is one of the legs of the triangle and the formula will change to

8.3² = y² + 3.4²

y² = 8.3² - 3.4²

y = 7.6

Writing a compound inequality to show the possible length

7.6 ≤ x ≤ 9

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Sheila measured the water level in the swimming pool every week for 4 weeks and found that the water level decreased by 8 millimeters in the first week, 5 millimeters in the second week, 12 millimeters in the third week and 5 millimeters in the fourth week. The net result was a loss of 10 millimeters of water over the 4 week period.

Sheila wanted to track the changes in the water level of a swimming pool over time. She measured the level every week for 4 weeks. In the first week, the water level dropped by 8 millimeters. In the second week, the level dropped by 5 millimeters. In the third week, the level dropped by 12 millimeters. Finally, in the fourth week, the level dropped by 5 millimeters. The net result of these measurements was a loss of 10 millimeters of water over the course of the 4 week period. This indicates that the hot and dry spell had a negative effect on the water level in the pool.

Total Water Loss = 8 millimeters + 5 millimeters + 12 millimeters + 5 millimeters = 30 millimeters

Total Water Gain = 0 millimeters

Net Loss = Total Water Loss - Total Water Gain = 30 millimeters - 0 millimeters = -10 millimeters

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The space occupied by the flat surface of a cylinder's base, also known as the base area, is quantifiable.

Given,

Base area of cylinder;

The amount of space that a flat surface at the base of a cylinder occupies is known as the base area. Since the cylinder's base is a circle, the base area of the cylinder is actually an area of a circle.

Formula for base area;

The cylinder's base area is calculated by multiplying the square of its radius by π.

Then,

"πr²" is the formula for the cylinder's base area with radius r.

That is,

Base Area of a Cylinder = (π × radius²) square units.

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Complete question is here;

What is the base area B of cylinder?

The value of k is -3, if x-1 is a factor of 4x³+3x² −4x + k

Quadratic Equation:

A quadratic equation is a second-order algebraic equation in x. The quadratic equation in standard form is ax2 + bx + c = 0. where a and b are the coefficients, x is the variable, and c is the constant term. The first condition of the quadratic equation is that the coefficients of x² are non-zero terms (a ≠ 0). To write a quadratic equation in standard form, first write the x² term, then the x term, and finally the constant term. The numbers for a, b, and c are generally written as whole numbers, not as fractions or decimals.

According to the Question:

Let P(x) = 4x³+3x² −4x + k

Given,  (x−1) is a factor of P(x)

x−1 = 0

x= 1

∴ p(1) = 4×(1)³+3×(1)² −4×(1)+k = 0

⇒ 4+3−4+k = 0

⇒ k+3=0

⇒ k=−3

Hence, the value of k is −3.

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