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In a proof of the Pythagorean theorem using similarity, what allows you to state that the triangles are similar in order to write the true proportions c/a=a/f and c/b=b/e ?

a) the geometric mean (altitude) theorem
b) the geometric mean (leg) theorem
c) the right triangle altitude theorem
d) the SSS theorem

Answer:

[tex]f(n) = 1.9 +0.6n[/tex]

Step-by-step explanation:

The nth term of the arithmetic sequence is given by:

[tex]a_n = a_1+(n-1)d[/tex]            ....[1]

where

[tex]a_1[/tex] is the first term

d is the common difference and n is the number of terms.

Here, f(n) represent diameter in centimeters and n the term number in the sequence.

Given the sequence represents the diameters of circles used to create an art project:

2.5 cm, 3.1 cm, 3.7 cm , 4.3 cm

This sequence is an arithmetic sequence with

[tex]a_1[/tex] = 2.5 and d = 0.6

Since,

3.1-2.5 = 0.6,

3.7-3.1 = 0.6

4.3-3.7 = 0.6

Substitute the given values in [1] we have;

[tex]f(n) =2.5+(n-1)(0.6)[/tex]

Using distributive property, [tex]a\cdot (b+c) = a \cdot b+ a\cdot c[/tex]we have;

[tex]f(n) = 2.5+0.6n-0.6[/tex]

Simplify:

[tex]f(n) = 1.9 +0.6n[/tex]

Therefore, the equation represents the sequence of diameters is, [tex]f(n) = 1.9 +0.6n[/tex]

The equation is A ( A + 2 ) = 674 + ( A + ( A + 2 ) )

The first integer is A = 26

The second integer is B = 28

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the two numbers be A and B

where , the first integer is = A

The second integer is = B

B is the consecutive integer of A , so

The value of B = A + 2

Now , the equation will be

The product of two consecutive even positive integers is 674 more than the sum of the two integers

So , substituting the values in the equation , we get

A ( A + 2 ) = 674 + ( A + ( A + 2 ) )

On simplifying the equation , we get

A² + 2A = 674 + 2A + 2

Subtracting 2A on both sides of the equation , we get

A² = 674 + 2

A² = 676

Taking square roots on both sides of the equation , we get

A = 26

The value of B = A + 2

B = 28

Therefore , the value of A = 26

The value of B = 28

Hence ,

The first integer is A = 26

The second integer is B = 28

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The number of gallons of paint to cover the entire wall will be 3 gallons of paint.

Algebra is the study of algebraic expressions, while logic is the manipulation of those concepts.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This rule is used to answer the problem correctly and precisely.

If 1/2 gallon of paint covers 1/6 of a wall. Then the division of the numbers 1/2 and 1/6 will be given by putting a division sign between them. Then the number of gallons of paint is calculated as,

⇒ (1/2) / (1/6)

⇒ (1/2) x 6

⇒ 3

The number of gallons of paint to cover the whole wall will be 3 gallons of paint.

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The function for the area of the circle if x is the radius of the circle and y is the area of the circle is y = πx².

An object's area is how much space it takes up in two dimensions. It is the measurement of the number of unit squares that completely cover the surface of a closed figure.

Given:

The area of a circle is a function equal to the product of pi (pi) and the square of the radius (r),

The area of the circle can be written as shown below,

Area of circle = π × R²

Here π is a constant, R is the radius of the circle,

When we put the value of the radius, we will get the value of the area,

Hence, the radius is an independent function and the area is dependent on the value of R (dependent function),

Replace y with area and radius with x as shown below,

y = πx²

Thus, y = πx² is the function for the area of the circle.

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Answer: The  average rate of change of f(t) from t = 2 to t = 6 is 1.5 thousand owners per year.

Step-by-step explanation:

Given: The function below shows the number of car owners f(t), in thousands, in a city in different years t:

[tex]f(t) = 0.25^2 - 0.5t + 3.5[/tex]

Now,

[tex]f(2) = 0.25(2)^2 - 0.5(2) + 3.5=3.5\\\\= f(6) = 0.25(6)^2-0.5(6)+ =9.5[/tex]

Now, the  average rate of change of f(t) from t = 2 to t = 6 is given by :-

[tex]k=\dfrac{f(6)-f(2)}{6-2}\\\\=\dfrac{9.5-3.5}{4}=1.5[/tex]

Hence, the  average rate of change of f(t) from t = 2 to t = 6 is 1.5 thousand owners per year.

The equation is

4(q-10)=76

The amount of questions is

29 questions

133=x + (x+5) + (x+14)

133 = 3x+19

3x=114

x= 114/3 =38

5+38 =43

38+14 =52

check: 38 +43 +52 = 133

3 sides are : 38, 43 & 52

To simplify a fraction by dividing both top and bottom by the same factor.

So,  pm/ pn equals to m/n .

If any variable or factor is present in both numerator and denominator. Then, these variables or factor are cancelled out each other.

In expression pm/pn , we observed that variable p is present in both numerator and denominator. So, variable p is cancelled out to each other.

Therefore,  pm/pn = m/n

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D. All counting numbers are whole numbers.

All integers are counting numbers. This is not true. Counting numbers (also known as natural numbers) are positive whole numbers starting from 1, 2, 3, and so on. Integers include all positive and negative whole numbers, as well as zero. Thus, not all integers are counting numbers.

All rational numbers are integers. This is not true. Rational numbers include all fractions where the numerator and denominator are integers (denominator not zero). For example, 1/2 is a rational number but not an integer.

All integers are whole numbers. This is not true. Whole numbers include all non-negative integers (0, 1, 2, 3, ...), but integers also include negative numbers, which are not whole numbers.

Counting numbers (1, 2, 3, ...) are included in the set of whole numbers, which consists of all non-negative integers (0, 1, 2, 3, ...).

To fit data to a normal distribution and estimate population percentages, calculate the mean and standard deviation for the data set, use these values to construct a normal distribution, and apply the empirical rule and confidence intervals to estimate population percentages.

To use the mean and standard deviation of a data set to fit it to a normal distribution and estimate population percentages, you will need to understand key statistical principles such as standard deviation, the mean, and the normal distribution curve.

The mean, which is an average of all the values in a data set, provides a central value. The standard deviation represents the spread of the data around this mean. A lower standard deviation indicates that the values are closer to the mean, while a higher standard deviation indicates a wider spread.

A normal distribution, also known as a bell curve, shows how values in a data set are dispersed around the mean. The properties of the normal distribution (68% of data within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations) are known as the empirical rule or the 68-95-99.7 rule.

To estimate population percentages using the mean and standard deviation, you construct a confidence interval. This range provides an estimate of where you expect the true population parameter to lie with a certain level of confidence. For example, a 95% confidence interval means that you can be 95% confident that the true population mean lies within this interval.

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