6.9.4 Journal: Similar circles6.9.4 Journal: Similar Circles

Journal

Geometry Sem 1

Points Possible:20

Name:


Date:



Scenario: Prove That All Circles Are Similar
Instructions
View the video found on page 1 of this Journal activity.
Using the information provided in the video, answer the questions below.
Show your work for all calculations.
The Students' Conjectures:The two students have different methods for proving that all circles are similar.
1. Complete the table to summarize each student's conjecture about how to solve the problem. (2 points: 1 point for each row of the chart)
Classmate Conjecture
John



Teresa





Evaluate the Conjectures:
2. Intuitively, does it make sense that all circles are similar? Why or why not? (1 point)








Construct the Circles:
3. Draw two circles with the same center. Label the radius of the smaller circle r1 and the radius of the larger circle r2. Use the diagram you have drawn for questions 3 – 10. (2 points)
















4. In your diagram in question 3, draw an isosceles right triangle inscribed inside the smaller circle. Label this triangle ABC. (1 point)
5. What do you know about the hypotenuse of △ABC? (2 points)




6. In your diagram in question 3, extend the hypotenuse of △ABC so that it creates the hypotenuse of a right triangle inscribed in the larger circle. Add point Y to the larger circle so it is equidistant from X and Z. Then complete isosceles triangle XYZ. (1 point)
7. What do you know about the hypotenuse of △XYZ? (2 points)




8. How does △ABC compare with △XYZ? Explain your reasoning. (2 points)








9. Use the fact that △ABC ≈ △XYZ to show that the ratio of the radii is a constant. (2 points)








Making a Decision
10. Who was right, Teresa or John? (1 point)




Further Exploration:
11. What is the circumference of the circle that circumscribes a triangle with side lengths 3, 4, and 5? (4 points)











Transcript: Similar Circles
The video begins with a young woman talking in front of a blank screen.

Audio:

I'm Teresa. My friend John and I need to prove that all circles are similar. It seems obvious, right?! Of course they're similar, they’re all circles!

[Many circles of different sizes and colors pop up onto the screen.]

Um, this is making me a little dizzy.

[The circles disappear.]

But we do need to prove that all circles are mathematically similar.

Here’s the way John looks at it: Remember what we learned about similar triangles?

[Two triangles appear on the screen. One is small and the other is large.]

We can take one triangle, and move it on top of another triangle.

[The small triangle is placed on top of the large triangle.]

Then, we dilate it to show that they are similar. Like that.

[The small triangle is dilated to the size of the large triangle.]

John says we can do the same thing with circles.

[Two circles appear on the screen. One is small and the other is large.]

Take any two circles, and move them so that they have the same center.

[The small circle is moved on top of the large circle.]

Then, you can dilate or contract the circles until they are the same size.

[The small circle is dilated to the size of the large circle.]

Taa-daa! The circles are similar. I have another way to prove it.

[Two triangles appear on the screen. The small triangle has sides of length 2, 2, and 3 and the large triangle has sides of length 6, 6, and 9.]

We also know that triangles are similar if all of their corresponding sides have the same ratio.

[The corresponding sides of the triangle are highlighted. On-screen text: 2 over 6 equals 2 over 6 which equals 3 over 9 which equals 1 over 3 Similar!]

Well, the same idea should also work with circles.

[Two circles appear on the screen. One circle is small and the other circle is large.]

If the corresponding parts of two circles have the same ratio, then the circles must be similar. And lucky for us, everything about a circle can be described with its radius!

[Beneath the small circle is written "equals r sub 1." Inside the large circle is written "equals r sub 2." On-screen text: Diameter equals 2r, Circumference equals 2pi r, and Area equals pi r squared.]

So, if the radii of these circles have a constant ratio, then the circles are similar.

[On-screen text: If r sub 1 over r sub 2 equals k, a constant, then the circles are similar.]

What's more, I think I can prove all this by using inscribed triangles. But I need your help.

[A triangle is inscribed in each of the circles using the diameter of the circles as their bases.]

Can we actually use inscribed right triangles to show that all circles are similar?

The average rate of change of the function is zero, the difference quotient is  6x + 3h - 1 and the value of (x, y) are (3/8, 29/4).

1.

The function f(x) = x² - x² + 3 is a constant function, since the two terms x² - x² cancel out. Therefore, the average rate of change of f(x) on the interval [0,2] is zero, since the function has the same value (3) at both endpoints.

2.

The difference quotient for the function f(x) = 3x² - x + 4 is:

(f(x+h) - f(x)) / h

= ((3(x+h)² - (x+h) + 4) - (3x² - x + 4)) / h

= (3x² + 6xh + 3h² - x - h + 4 - 3x² + x - 4) / h

= (6xh + 3h² - h) / h

= 6x + 3h - 1

3.

To solve the system of equations:

2x + y = 8

-4x + 2y = 13

We can use either substitution or elimination method. Here we'll use the elimination method:

Multiplying the first equation by 2, we get:

4x + 2y = 16

Adding this equation to the second equation, we eliminate y and get:

4x + 2y - 4x + 2y = 16 + 13

4y = 29

y = 29/4

Substituting y = 29/4 into the first equation, we get:

2x + 29/4 = 8

2x = 8 - 29/4

2x = 3/4

x = 3/8

Therefore, the solution to the system of equations is (x,y) = (3/8, 29/4).

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The positions are located at points A

In the given question Emily generated 10,000 random digits between 0 and 10 we have to find the approximate location of the median is in interval A/B/CV and the approximate location of the mean is in interval A/B/CV.

The mean can roughly be found in interval B, whereas the median can roughly be found in interval A.

How to find the median and mean's placement in a table

The mean

We can utilize the following inputs for our calculation as we got them from the question:

The histogram

We can observe from the histogram that it is a uniform histogram.

This indicates that the mean is in the middle.

So, we have

Mean = point A

The median

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

The histogram

We can observe from the histogram that it is a uniform histogram.

This indicates that the median is situated in the middle.

So, we have

Median = point A

Hence, the positions are located at points A

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The complete question is:

Emily generated 10,000 random digits between 0 and 10. Here are the results:

Frequency

1000

800

600

400

200

04

A B

0 1 2 3 4 5 6 7 8 9 10

Digit

The approximate location of the median is in interval A/B/C

The approximate location of the mean is in interval A/B/C

Answer:

284

Step-by-step explanation:

open up the triangle

so you get 3 sides for the length  13+10+13=36 * 5 2/3width =204

then add the 2 triangles surface

1/3 * 10base  * 12height * 2 =80

total 284

Taking the quotient between the total distance and the gallons consumed we can see that:

Mpg = 37 miles per gallon.

It is just given by the quotient between the number of miles traveled and the number of gallons used.

We know that at the start the odometer reads 69247 miles and at the end the odometer reads 69765 miles, taking the difference we can see that the total distance traveled is:

69765 mi - 69247 mi = 518 mi

And you consumed 14 gallons of gas, then the miles per gallon is:

Mpg = 518mi/14 gal = 37 miles per gallon.

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Answer:

1st term(a)=0.25

2nd term(t2)=0.5

common difference (d)=0.25

nth term(tn)=a+(n-1)d

=0.25+(n-1)0.25

=0.25+0.25n-0.25

=0.25-0.25+0.25n

=0.25n

Answer: nth term of the geometric sequence = (n^-1)/2

Step-by-step explanation:  The nth term of a GP is Tn = arn^-1

where a is the first term 0.25  

r is the common ratio 0.5/0.25 or 1/0.5 or 2/1, which is 2

Therefore the nth term = 0.25 x 2 x n^-1

= 0.5n^-1

=(n^-1)/2

The one-digit integers that satisfy the inequality "k < 0" are:

-1, -2, -3

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

k < 0

A number line representation of the inequality "k < 0" is as follows:

-3 -2 -1 0 1 2 3

---------<----------

The inequality k < 0 represent all the numbers less than 0 on the number line.

This includes all negative one-digit integers, which are -1, -2, and -3.

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On solving the provided question, we can say that the quadratic function [tex]f(x)=x^{2}+10x+24.[/tex] , so f(-2) = 8

Quantities and their variations, equations and related structures, shapes and their placements, and locations where they can be found are all included in the study of mathematics.

The relationship between a group of inputs, each of which has a related output, is referred to as a "function." A function is a relationship between inputs and outputs where each input produces a single, unique result. Every function has a domain and a codomain, often known as a scope. F is typically used to denote functions (x). An x is the input. Based on the following criteria, there are four primary categories of functions that are available: on functions, one-to-one functions, many-to-one functions, within functions, and on functions.

the quadratic function [tex]f(x)=x^{2}+10x+24.[/tex]

By substituting different values of x

For x = 0

if f(0) = 24

For x = 1

if f(1) = 35

For x = 2

if (2) = 48

For x = -2

so f(-2) = 8

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Answer:

x = 9

Step-by-step explanation:

x + 3 = 2x - 6
3 + 6 = 2x - x
9 = x
x = 9

Hope this helped have a great day!

Answer:

132

Step-by-step explanation:

The pattern is adding 18 napkins, so Friday would be 114 napkins and Saturday would be 132

The function designates an area in the X-Y plane where its value is 1/2 and an area where it is 0. By applying the formula |X| + |Y| = 1, the line separating these two regions is defined as equation.

When two expressions are joined by an equal sign, a mathematical statement is called an equation. An equation is something like 3x - 5 = 16. By solving for x, we discover that x equals 7, which is the value for the variable.

Equations can be classified as either conditional equations or identities. Any value of the variables results in an identity being true. Only certain values of the variables in a conditional equation result in truth.

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The complete question is: Let f(x)={1/2} = 0 and |X| + |Y| < 1, find the value of x.

Answer:

Supernova

Step-by-step explanation:

A supernova is a powerful and luminous explosion of a star, which is what the sun is.

The rate at which a particular animal grows is proportional to the difference between its adult height and current height. As the animal grows, the difference between the adult height and current height decreases. T

Therefore the rate of growth decreases. This is because, as the animal grows, it reaches a point where it cannot grow any taller and the rate of growth slows down. In other words, the rate of growth is proportional to the difference between the adult's height and current height. the rate of growth is also related to the age of the animal.

Younger animals tend to grow faster than older animals because their adult height is still far away, so the difference between their current height and adult height is greater.As the animal gets older and closer to its adult height, the rate of growth decreases.

furthermore, the rate of growth is also affected by the animal's diet and overall health. If the animal is malnourished or unhealthy, then its rate of growth will be slower than if it had a healthy diet and lifestyle. In conclusion, the rate of growth of  a particular animal is proportional to the difference between its adult height and current height

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Answer:

The two angles are 32 degrees and 48 degrees.

Step-by-step explanation:

If one angle of a triangle is 100 degrees, then the sum of the other two angles is 80 degrees (180 - 100). If the magnitude of the other two angles is in the ratio 2:3, then we can call the first angle 2x and the second angle 3x. The sum of these two angles is 5x, so 5x = 80, and x = 16. So, the two angles are 2x = 32 degrees and 3x = 48 degrees.

Answer:

B. 9[tex]\frac{1}{2}[/tex]

Step-by-step explanation:

Rewriting the two fractions in the expression:

=[tex]\frac{[(8)(2)]+1}{8} + \frac{[(8)(7)]+3}{8}[/tex]

=[tex]\frac{17}{8} + \frac{59}{8}[/tex]

Since the denominators of both the fractions is the same, then the denominator of the resulting simplified fraction will also be the same. In addition, everything written in the numerator level will be copied down as it is:

=[tex]\frac{17 + 59}{8}[/tex]

=[tex]\frac{76}{8}[/tex]

Divide the numerator and denominator by the Highest Common Factor '4':

= [tex]\frac{19}{2}[/tex]

Apply Long Division since:

Choose the highest quotient to be multiplied by 2 to give a number closest to 19 but still less than 19. The quotient turns out to be 9. The remainder is 1

∴ Sum of the expression = 9[tex]\frac{1}{2}[/tex]

For each of North Asia's 26 million citizens, a single ICBM missile launch represents an opportunity cost of 0.19 tons of grain.

A cost is the value of money that was used on the creation of goods or the provision of a service and is now unavailable for use in producing, investigation, retail, & accounting. While calculating a purchase cost, the sum of both the acquisition's costs is taken into account.

One ICBM rocket launch costs $1.3 billion.

5 million metric tons of grain may be bought with this amount.

North Korea has 26 million people.

Consider a scenario in which the government uses the funds to buy metric ton of maize instead of launching an ICBM missile.

Corn purchased/population (5 million/26 million) = 0.19 tons/per person received.

Hence, 0.19 tons of grain will be distributed to every citizen in the nation.

So, for each of North Singapore's 26 million inhabitants, the potential price of one ICBM launch attempt throughout terms of maize is 0.19 tonnes of corn.

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The complete question is-

North korea’s rockest deepen food crisis:

North Korea has spent a lot money on its nuclear and rocket programs. A single launch of ICBM missile costs around $1.3 billion. Since taking power in 2011, Kim jong-un authorized over 20 rocket launches, including ICBMs, beginning in January 2017. Those launches came at a great cost to the Korean people. The cost of just one ICBM launch could have been used to purchase 5 million tons of corn, for example. That would have been if enormous benefit to the Korean people, who have suffered decades of widespread poverty and periodic starvation.

According to the news wire, what is the opportunity cost of a single icbm rocket launch in terms of corn for each of North Korea’s 26 million people ?

( ) ton(s) of corn per person

The frequency of the wave is f = 0.67 hertz.

What is the frequency?

Frequency is the number of occurrences of a repeating event per unit of time.

1) The velocity of the wave can be calculated using the formula v = lambda / T, where lambda is the wavelength and T is the time period.

From the wave function, the wavelength can be determined by finding the distance between consecutive crests or troughs of the wave,

which is given by

2 * π / (5 * π / 6) = (2/5) * (6/π) m.

The time period can be determined by finding the time it takes for one complete wave cycle to occur, which is given by 2*π / (3 * π / 7) = (2/3) * (7/π) s.

So, the velocity of the wave is v = (2/5) * (6/π) m / ((2/3) * (7/π) s) = (12/15) * (6/7) * (π/π) m/s = 4/5 m/s.

2) The time period of the wave can be calculated as T = 2*π / (3 * π / 7) = (2/3) * (7/π) s.

3) The wavelength of the wave can be calculated as lambda = 2*π / (5 * π / 6) = (2/5) * (6/π) m.

4) The frequency of the wave can be calculated using the formula f = 1 / T, where T is the time period.

f = 1 / ((2/3) * (7/π) s) = (3/2) * (π/7) Hz

f = 0.67 hertz

Hence, the frequency of the wave is f = 0.67 hertz.

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The velocity of the wave is (6 × π / 35) m/s, the time period of the wave is 7 / (3 × π) s, the wavelength of the wave is 6 / 5 m, and the frequency of the wave is 3 × π / 7 s⁻¹.

A wave function, in physics, is a mathematical description of a wave, usually used in quantum mechanics to describe the behavior of subatomic particles such as electrons and protons. The wave function provides information about the position, momentum, and other properties of a particle at a given point in time and space. It is represented by a complex mathematical function that satisfies Schrödinger's equation, which describes the time evolution of a wave function in a quantum system. The wave function is central to the interpretation of quantum mechanics and is often used to calculate the probabilities of different outcomes in quantum experiments.

Given the wave function:

y(x, t) = 0.75 m × sin[(5 × π / 6 m⁻¹) × x - (3 × π / 7 s⁻¹) × t + π / 5]

The velocity of the wave can be calculated using the wavelength (λ) and frequency (f) of the wave. The velocity (v) is given by v = λ × f. The wavelength (λ) and frequency (f) can be determined from the wave function as follows:

λ = 2 × π / (5 × π / 6 m⁻¹) = 6 / 5 m

f = 3 × π / 7 s⁻¹

So, the velocity of the wave is given by:

v = λ × f = (6 / 5 m) × (3 × π / 7 s⁻¹) = (6 × 3 × π / (5 × 7)) m/s = (6 × π / 35) m/s

The time period (T) of the wave is the time it takes for the wave to complete one full cycle. It can be determined from the frequency of the wave as follows:

T = 1 / f = 1 / (3 × π / 7 s⁻¹) = 7 / (3 × π) s

The wavelength (λ) of the wave has already been calculated in step 1 as λ = 6 / 5 m

The frequency (f) of the wave has already been calculated in step 1 as f = 3 × π / 7 s⁻¹

So, the velocity of the wave is (6 × π / 35) m/s, the time period of the wave is 7 / (3 × π) s, the wavelength of the wave is 6 / 5 m, and the frequency of the wave is 3 × π / 7 s⁻¹.

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The rectangular form of z1 is -4 + j3 and the polar form is 5∠153°. The rectangular form of z2 is 1 - j and the polar form is √2∠-45°. The other expressions have similar rectangular and polar forms.

The rectangular form of complex numbers is represented by two coordinates, one real, and one imaginary. The polar form of complex numbers is represented by the magnitude and angle of the number.

In the expression (a), z1 = -4 + j3; the rectangular form is -4 + j3, and the polar form is 5∠153°. In the expression (b), z2 = 1 - j; the rectangular form is 1 - j, and the polar form is √2∠-45°.

In the expression (c), z1 + z2 = -3 + 2j; the rectangular form is -3 + 2j, and the polar form is 2.82∠29.14°. In the expression (d), jz2 = j(1 - j); the rectangular form is j(1 - j) and the polar form is 1∠-135°.

In the expression (e), z1(-1) = 1/z1; the rectangular form is 1/z1, and the polar form is 0.25∠-33.7°. In the expression (f), z1/z2 = (-4+j3)/(1-j); the rectangular form is (-4+j3)/(1-j), and the polar form is 2.24∠-118.4°.

In the expression (g), ez2 = e(1 - j); the rectangular form is e(1 - j), and the polar form is 2.72∠-44.99°. In the expression (h), z1z1 = (-4+j3)(-4+j3); the rectangular form is (-4+j3)(-4+j3), and the polar form is 25∠306°.

In the expression (i), z1z2 = (-4+j3)(1-j); the rectangular form is (-4+j3)(1-j), and the polar form is 7∠-96.6°.

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Answer:

-3

Step-by-step explanation:

The slope of the line that passes through the points (-1, 4) and (1, -2) can be found using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points. Substituting the values, we get:

m = (-2 - 4) / (1 - (-1))

m = -6 / 2

m = -3

So, the slope of the line is -3.

Answer:

Step-by-step explanation:

9/12 + 10/12

Answer: Final Balance is $11,591.87 and the total compound interest is $1,731.87.

Step-by-step explanation:

For a distribution that is skewed right, the true statement is (c) mean> median

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

a distribution that is skewed right,

For a distribution that is skewed right, it means that there are more values on the right tail of the distribution and the tail extends further to the right than the left.

In this case, the mean is typically greater than the median.

Hence, the correct answer is "mean > median."

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The appropriate distribution to answer questions about the New York Lotto is the hypergeometric distribution, which is used to calculate the probability of a certain number of successes in a sample of items drawn without replacement from a population.

The New York Lotto is a lottery game where participants select six numbers from a set of 51 numbers, 1 to 51. Then, six balls are randomly drawn from a barrel without replacement to determine the six winning numbers. To answer questions about the probability of various outcomes in this game, the appropriate probability distribution is the hypergeometric distribution. This distribution is used to calculate the probability of a certain number of successes in a sample of items drawn without replacement from a population. In the case of the New York Lotto, the population is the set of 51 numbers from 1 to 51, and the sample is the six numbers selected by the participant. The hypergeometric distribution can be used to calculate the probability of matching exactly two of the winning numbers.

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An expression for the given phrase is 5(r+s).

An expression is a combination of terms that are combined by using mathematical operations such as subtraction, addition, multiplication, and division.

The given phrase is 5 times as much as the sum of r and s.

Here, the sum of r and s is r+s

Now, 5 times as much as the sum of r and s is

5(r+s)

Therefore, an expression is 5(r+s).

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Answer:

60

Step-by-step explanation:

Because we are told the figure is a regular polygon, then it must be an equilateral triangle. This means all angles are equal (60 degrees each)

Therefore, the values of x that make the equation distance between (11,6) and (x,-9) equal to 17 are x = 19 and x = 3.

The formula y=mx+b is used to draw a simple regression graph. The inclination is B, and the y-intercept is m. It is frequently referred to simply as a "simple method combining many variables," despite the fact that the section before it is divided into distinct sections. Bivariate linear equations have two variables. For problems involving linear equations in applications, there are no proven solutions. Y=mx+b, also known as mx+b, provides a number of straightforward formulae.

Here,

Using the distance formula:

sqrt((x - 11)^2 + (-9 - 6)^2) = 17

Simplifying:

sqrt((x - 11)^2 + 225) = 17

Squaring both sides:

(x - 11)^2 + 225 = 289

(x - 11)^2 = 64

Taking the square root:

x - 11 = ±8

x = 11 + 8 or x = 11 - 8

x = 19 or x = 3

Therefore, the values of x that make the equation distance between (11,6) and (x,-9) equal to 17 are x = 19 and x = 3.

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The statement that is true about histograms graphs is: "Data is organized in equal intervals."

Academics are using graphs to logically chart or graphically depict events or quantities. A graph point is a common way to depict how different things interact. The non-linear storage structure known as a graph is made up of nodes, also referred to as vertices, and lines. Glue ought to be used to connect the edges, also known as nodes , respectively.  (4.5).

Here,

The statement "Data appears as bars that are all about the same height" is false.

A histogram is a graph that represents the distribution of numerical data by organizing it into "bins" or "intervals" and displaying the count or frequency of data points that fall within each interval. The height of each bar represents the frequency or count of data points in that interval. The bars may or may not touch depending on the data and the chosen intervals.

The statement that is true about histograms is: "Data is organized in equal intervals."

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Therefore, the probability of selecting a defective battery followed by another defective battery is 1/145.

Calculating the likelihood that a statement is true or that a particular event will take place is the centre of probability theory, a branch of mathematics. Any number between 0 and 1, where 1 is usually used to range confidence as well as 0 is routinely used it to convey possibility, can be used to represent chance. A probability diagram shows the likelihood that a particular occurrence will occur.

Here,

The probability of selecting a defective battery on the first draw is 3/30, or 1/10, since there are 3 defective batteries out of 30 total.

Since one battery has already been drawn, there are now 29 batteries left in the box, and 2 defective batteries remaining.

The probability of selecting another defective battery on the second draw, without replacement, is 2/29.

To find the probability of both events happening (selecting a defective battery followed by another defective battery), we multiply the probabilities of each event:

P(defective battery and then another defective battery) = P(defective battery on first draw) × P(defective battery on second draw)

= (1/10) × (2/29)

= 2/290

= 1/145

Therefore, the probability of selecting a defective battery followed by another defective battery is 1/145.

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Answer: The intersection of two submodules of an R-module is itself a submodule of the R-module. To show that the intersection of any nonempty collection of submodules of an R-module is a submodule, we can use induction.

Suppose we have a collection of submodules M1, M2, ..., Mn of an R-module R. We will prove that the intersection of this collection is a submodule of R.

Base case: n = 2.

In this case, M1 and M2 are submodules of R, and their intersection is also a submodule of R.

Inductive step: n = k + 1 (k > 1).

Suppose the intersection of any collection of k submodules of R is a submodule of R. Let M1, M2, ..., Mk, Mk+1 be a collection of k + 1 submodules of R.

Then, the intersection of M1, M2, ..., Mk is a submodule of R, and the intersection of this submodule and Mk+1 is also a submodule of R.

Therefore, by induction, the intersection of any nonempty collection of submodules of an R-module is a submodule.

Step-by-step explanation:

Answer:

d ≈ 5.1

Step-by-step explanation:

calculate the distance d using the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = L (3, 7 ) and (x₂, y₂ ) = M (- 2, 8 )

d = [tex]\sqrt{(-2-3)^2+(8-7)^2[/tex]

   = [tex]\sqrt{(-5)^2+1^2}[/tex]

   = [tex]\sqrt{25+1}[/tex]

    = [tex]\sqrt{26}[/tex]

    ≈ 5.1 ( to the nearest tenth )