Consider the series Σ (13x)" n=0 (a) Find the series' radius and interval of convergence. (b) For what values of x does the series converge absolutely? (c) For what values of x does the series converge conditionally?
Answers
(a) The series has a radius of convergence of 2/13 and an interval of convergence of -1/13 < x < 1/13.
(b) The series converges absolutely for -1/13 < x < 1/13.
(c) The series converges conditionally at x = -1/13 and x = 1/13.
(a) To find the radius and interval of convergence for the series Σ (13x)^n, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges.
Let's apply the ratio test to the given series:
lim (n→∞) |(13x)^(n+1)/(13x)^n|
= lim (n→∞) |13x|^(n+1-n)
= lim (n→∞) |13x|
For the series to converge, we need the absolute value of 13x to be less than 1:
|13x| < 1
This implies -1 < 13x < 1, which leads to -1/13 < x < 1/13.
Therefore, the series converges for the interval -1/13 < x < 1/13.
The radius of convergence is half the length of the interval of convergence, which is 1/13 - (-1/13) = 2/13.
(b) For the series to converge absolutely, we need the series |(13x)^n| to converge. This occurs when the absolute value of 13x is less than 1:
|13x| < 1
Solving this inequality, we find that the series converges absolutely for the interval -1/13 < x < 1/13.
(c) For the series to converge conditionally, we need the series (13x)^n to converge, but the series |(13x)^n| does not converge. This occurs when the absolute value of 13x is equal to 1:
|13x| = 1
Solving this equation, we find that the series converges conditionally at the endpoints of the interval of convergence, which are x = -1/13 and x = 1/13.
(a) To find the radius and interval of convergence for the series Σ (13x)^n, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges.
Let's apply the ratio test to the given series:
lim (n→∞) |(13x)^(n+1)/(13x)^n|
= lim (n→∞) |13x|^(n+1-n)
= lim (n→∞) |13x|
For the series to converge, we need the absolute value of 13x to be less than 1:
|13x| < 1
This implies -1 < 13x < 1, which leads to -1/13 < x < 1/13.
Therefore, the series converges for the interval -1/13 < x < 1/13.
The radius of convergence is half the length of the interval of convergence, which is 1/13 - (-1/13) = 2/13.
(b) For the series to converge absolutely, we need the series |(13x)^n| to converge. This occurs when the absolute value of 13x is less than 1:
|13x| < 1
Solving this inequality, we find that the series converges absolutely for the interval -1/13 < x < 1/13.
(c) For the series to converge conditionally, we need the series (13x)^n to converge, but the series |(13x)^n| does not converge. This occurs when the absolute value of 13x is equal to 1:
|13x| = 1
Solving this equation, we find that the series converges conditionally at the endpoints of the interval of convergence, which are x = -1/13 and x = 1/13.
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Related Questions
Kate is a buyer for a men’s fashion retail store. She will order a new cloth overcoat from Paris for the fall fashion season. Based on her experience, she expects to sell at least 100 coats, and at most 400, but she feels that any number of sales in between is equally likely. Therefore, she estimates that her sales are uniformly distributed between 100 and 400. The total cost to the store is $100 per coat, and the retail price is set at $180. Any coats left over at the end of season would be sold at $60 each.
part 1: a) How many coats should Kate buy if she wants to maximize profits?
part 2: b) Assume Kate buys the number of coats suggested in part a), what is the probability that the coats sell out? What is the probability that they do not sell out?
Answers
Part 1: Kate should buy 100 coats to maximize profits.Part 2: The probability that the coats sell out is 0.25 (25%), and the probability that they do not sell out is 0.75 (75%).
To maximize profits, Kate should consider the scenario where she sells all the coats without any left over at the end of the season.
Since the sales are uniformly distributed between 100 and 400, buying 100 coats ensures that she meets the minimum expected sales of 100. Purchasing more than 100 coats would increase costs without a guarantee of higher sales, potentially leading to excess inventory and lower profits.
Given that the sales are uniformly distributed between 100 and 400 coats, Kate's purchase of 100 coats covers the minimum expected sales.
The probability of selling out can be calculated by finding the proportion of the range covered by the desired sales (100 out of 300). Therefore, the probability of selling out is 100/300 = 0.25 or 25%. The probability of not selling out is the complement, which is 1 - 0.25 = 0.75 or 75%.
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37 of the 998 digital video recorders (dvrs) in an inventory are known to be defective. what is the probability that a randomly selected item is defective?
Answers
The probability that a randomly selection of 998 digital video recorders in a inventory one of them is defective is 3.71%
To solve this exercise the formula and the procedure that we have to use for probability is:
% probability= (achieving success / possible outcomes) * 100
Information about the problem:
Achieving success = 37Possible outcomes = 998Applying the formula to calculate the probability we get:
% probability= (achieving success / possible outcomes) * 100
% probability= (37/ 998) * 100
% probability= (0.0371) * 100
% probability= 3.71%
What is probability?Probability is the function that measures the chances that an outcome of a random event will be as expected. In this way you can measure how likely it is that an event will happen.
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ANSWER ASAP PLEASE. TAKE YOUR TIME TO DETERMINE IT. IT CAN HAVE MORE THAN 1 ANSWER.
1.) -7
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
2.) -√81
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
3.) 8.444....
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
4.) 9/10
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
5.) √42
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
Answers
Answer:
1.) -7
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
2.) -√81
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
3.) 8.444....
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
4.) 9/10
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
5.) √42
A. Rational
B. Irrational
C. Natural
D. Whole
E. Integer
what is the highest common factor of 21 and 6 thanks :)))))
Answers
Answer:
The highest common factor of 21 and 6 is 3
Step-by-step explanation:
(a) Place a right triangle with leg lengths of 7 units and 9 units in the coordinate plane. (b) Find the length of the hypotenuse. Round to the nearest tenth.
Answers
The hypotenuse is therefore around 11.4 units long.
DESCRIBE THE RIGHT ANGLE TRIANGLE?If one of a triangle's angles is 90 degrees or more, the triangle is said to be right-angled. The other two angles add up to 90 degrees. The sides that make up the right angle are perpendicular and the triangle's base.
where the hypotenuse length is c and the leg lengths are a and b, respectively.
The Pythagorean theorem can be used to determine the length of the hypotenuse of a right triangle with 7 and 9 unit legs:
a² + b² = c²
If we substitute 7 for letter a and 9 for letter b, we get:
7²+ 9² = c²
49 + 81 = c²
130 = c²
The result of squaring the two sides is: c = 130) 11.4
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What 3 events caused ww1?
Answers
The real causes of World War I included politics, secret alliances, imperialism, and nationalistic pride.
The nations of Europe were continually vying for dominance and forming alliances in the years preceding the conflict. In 1881, Germany formed an alliance with Italy and Austria-Hungary. All of these nations committed to defending one another in the event that France attacked them. But after that, Italy went and formed a covert alliance with France, declaring they would not support Germany.
In 1892, France and Russia formed an alliance in response to Germany's alliances. Britain and France signed a contract in 1904. France, Britain, and Russia established the Triple Entente in 1907. Germany believed that the strong alliance that surrounded them posed a serious danger to their existence and regional dominance.
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shirabi spent $208 on a sewing machine to make purses. she spends a total of $10 on thread, fabric, and accessories for each purse and plans to charge $36 for each purse. the equation represents her break-even point, when x represents the number of purses sold. 208 10x
Answers
Shirabi needs to sell a minimum of 20 purses in order to reach her break-even point, where her total revenue equals her total costs.
To find the break-even point, we need to determine the number of purses that Shirabi needs to sell in order to cover her costs and not incur any loss. The equation representing her break-even point is given as:
208 + 10x
Here, 208 represents the cost of the sewing machine, and 10x represents the total cost of thread, fabric, and accessories for each purse, multiplied by the number of purses sold.
To find the break-even point, we need to set the equation equal to zero:
208 + 10x = 0
Now, let's solve for x:
10x = -208
x = -208/10
x = -20.8
Since the number of purses sold cannot be negative, we can disregard the negative solution. Therefore, Shirabi's break-even point is when she sells 20 purses.
Shirabi needs to sell a minimum of 20 purses in order to reach her break-even point, where her total revenue equals her total costs. Selling fewer than 20 purses would result in a loss, while selling more than 20 purses would generate a profit.
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A storekeeper of an electronics company may have to deal with many types of materials that may kept in the store. Explain with suitable examples, FIVE (5) classes of materials that a storekeeper may be involved. (25 marks, 400 words)
Answers
Storekeepers in electronics companies deal with various types of materials. Five classes of materials include electronic components, raw materials, finished products, packaging materials, and maintenance supplies.
Electronic Components: Storekeepers are responsible for managing a wide range of electronic components such as resistors, capacitors, integrated circuits, connectors, and other discrete components. These components are essential for assembling electronic devices and are typically stored in organized bins or cabinets for easy access.
Raw Materials: Electronics companies require various raw materials for manufacturing processes. Storekeepers handle materials like metals, plastics, circuit boards, cables, and other materials needed for production. These materials are usually stored in designated areas or warehouses and are monitored for inventory levels.
Finished Products: Storekeepers are also responsible for storing and managing finished products. This includes fully assembled electronic devices such as smartphones, computers, televisions, and other consumer electronics. They ensure proper storage, tracking, and distribution of these products to customers or other departments within the company.
Packaging Materials: Packaging plays a crucial role in protecting and shipping electronic products. Storekeepers handle packaging materials such as boxes, bubble wrap, foam inserts, tapes, and labels. They ensure an adequate supply of packaging materials and manage inventory to meet packaging requirements.
Maintenance Supplies: Electronics companies often require maintenance and repair supplies for their equipment and facilities. Storekeepers handle items like tools, lubricants, cleaning agents, safety equipment, and spare parts. These supplies are necessary to support ongoing maintenance activities and ensure the smooth operation of machinery and infrastructure.
Overall, storekeepers in electronics companies deal with a diverse range of materials, including electronic components, raw materials, finished products, packaging materials, and maintenance supplies. Effective management of these materials is crucial to ensure smooth operations, timely production, and customer satisfaction.
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What property is illustrated in the following equation? 7 + 0= 7 Associative Property of Multiplication Commutative Property of Addition Distributive Property Identity Property of Addition
Answers
Answer:
Property of Addition
Step-by-step explanation:
What would an equation be for a line with a slope of 3 and a y-intercept of 7?
Answers
Answer:
y = 3x + 7
Step-by-step explanation:
Using the formula y = mx + b we can create this equation.
m = slope
b = y -intercept
based on the information provided I know that
Slope = 3
and
y-intercept = 7
Substitute:
y = mx + b
y = 3x + 7
y = 3x + 7 is the equation when the slope is 3 and the y intercept is 7.
Let us use the slope-intercept form for this, with m as the slope and b as a constant.
y = mx + b
7 = 3(0) + b [since the y-int is 7, the x value will be zero]
b = 7
y = 3x + 7
what is the cost of installing a fence around a rectangular shaped lot if the cost of the fence is $3.25 per linear foot and the lot is 80 ft. wide and 120 ft. deep?
Answers
The cost of installing a fence around an 80 ft. wide and 120 ft. deep rectangular lot, with the fence priced at $3.25 per linear foot, will be $1,300.
To determine the cost of installing a fence around a rectangular lot, you need to calculate the total length of the fence required and then multiply that by the cost per linear foot. The given dimensions of the lot are 80 feet wide and 120 feet deep.
First, calculate the perimeter of the rectangular lot. The perimeter of a rectangle is given by the formula P = 2L + 2W, where L is the length (or depth) and W is the width. In this case, the perimeter is P = 2(120) + 2(80) = 240 + 160 = 400 feet.
Next, multiply the total length of the fence by the cost per linear foot, which is $3.25. So, the cost of installing the fence is 400 feet × $3.25 per linear foot = $1,300.
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four bags of candy cost 9 dollars what's the cost per bag
Answers
Answer:
2.25
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...
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I’m the figure below F is between E and G and G is between F and H if FG=5 and EG=11 and FH=8 find EH
Answers
Answer:ur mom
Step-by-step explanation:
How many pint are in 28 cups
Answers
Answer:
14 pints are in 28 cups
Step-by-step explanation:
Answer:
14 pints in 28 cup
Step-by-step explanation:
Cameron is on page 21 of his book. He reads 112 pages per minute and wants to read to at least page 84 tonight. The inequality 21+112t≥84 models this situation, where t is time in minutes. Which represents the time it will take Cameron to reach his goal?
Answers
Answer:
t ≥ 0.5625
Step-by-step explanation:
Given:
21 + 112t ≥ 84
112t ≥ 84 - 21
112t ≥ 63
t ≥ 63/112
t ≥ 0.5625
t = time in minutes
Please help me!
"Explain why this graph shows a function."
Answers
Answer:
This graph shows a function because no vertical line passes through more than one point on the graph.
Step-by-step explanation:
3. Suppose g(t) = [0.5sinc²(0.5 t) cos(2 t)], where the sinc function is defined as (3.17) on p. 100 of the textbook. (a) Apply Parseval's Theorem to determine the 95% energy bandwidth (B) of this signal, where we define the 95% energy bandwidth as:
(b) Gf²df = 0.95Eg. What is the 95% energy bandwidth of g(2t) in terms of the value of B determined in Part a. Please provide full justification for your answer.
Answers
To determine the 95% energy bandwidth (B) of the signal g(t) = [0.5sinc²(0.5 t) cos(2 t)], we can apply Parseval's Theorem. Parseval's Theorem states that the total energy of a signal in the time domain is equal to the total energy of the signal in the frequency domain. Mathematically, it can be expressed as:
∫ |g(t)|² dt = ∫ |G(f)|² df
In this case, we want to find the frequency range within which 95% of the energy of the signal is concentrated. So we can rewrite the equation as: 0.95 * ∫ |g(t)|² dt = ∫ |G(f)|² df
Now, we need to evaluate the integral on both sides of the equation. Since the given signal is in the form of a product of two functions, we can separate the terms and evaluate them individually. By applying the Fourier transform properties and integrating, we can find the value of B.
For part (b), when we consider g(2t), the time domain signal is compressed by a factor of 2. This compression results in a corresponding expansion in the frequency domain. Therefore, the 95% energy bandwidth of g(2t) will be twice the value of B determined in part (a). This can be justified by considering the relationship between time and frequency domains in Fourier analysis, where time compression corresponds to frequency expansion and vice versa.
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Write the given system of differential equations using matrices and solve. x= x + 2y - 2 y = 1+2 z' = 4x - 4y +52
Answers
The given system of differential equations can be written using matrices as follows:
X' = AX + B,
where X = [x, y, z] is the vector of variables, X' represents the derivative of X with respect to some independent variable, A is the coefficient matrix, and B is the constant matrix.
In this case, the coefficient matrix A is [[1, 2, 0], [0, 0, 2], [4, -4, 0]], and the constant matrix B is [-2, 1, 52].
To solve the system, we can find the eigenvalues and eigenvectors of the coefficient matrix A.
These eigenvalues and eigenvectors help in diagonalizing the coefficient matrix, allowing us to solve the system using the diagonalized form.
Once we have the diagonalized form, we can solve each equation individually to obtain the solutions for x, y, and z. Finally, we combine these solutions using linear combinations to form the general solution for the system.
However, without specific eigenvalues, eigenvectors, or initial conditions, it is not possible to provide the numerical solution.
If you have the eigenvalues, eigenvectors, or initial conditions, please provide them, and I can assist you in solving the system using the given matrices.
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⚠️‼️ HELP PLEASE GIVING POINTS !!
Answers
Answer:
m∠QPR = 25°.
Step-by-step explanation:
If m∠QPR = 9x + 16°, substitute in 1 for 'x':. 9(1) + 16 = 9 + 16 = 25°
two verticals of a triangle are 8 in negative 5 And 8 and 7 if the area of the rectangles Is 72 square units named the possible location of the other 2 vertical
Answers
First let's find the length of the segment created by the two given points, using the formula:
\(\begin{gathered} d=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2} \\ d=\sqrt[]{(7-(-5))^2+(8-8)^2} \\ d=\sqrt[]{12^2} \\ d=12 \end{gathered}\)Since the given points have the same x-coordinate, we have a vertical segment, so the adjacents sides to this side will be horizontal segments.
In order to find the other vertices, first let's find the length of the horizontal sides, using the area formula:
\(\begin{gathered} A=l\cdot w \\ 72=12\cdot w \\ w=\frac{72}{12}=6 \end{gathered}\)We can find the vertices by adding 6 to the x-coordinate of the given points, because then we will have the two horizontal sides of the rectangle:
\(\begin{gathered} (8,-5)\to(8+6,-5)=(14,-5) \\ (8,7)\to(8+6,7)=(14,7) \end{gathered}\)So the possible locations for the other 2 vertices are (14, -5) and (14, 7).
Solve this inequality for x. 81-1'1/5x<55
Answers
The solution for the inequality is x < 11/130.
The problem we dealing with is related to inequality which is referred to as An inequality in mathematics could be a relationship that produces a non-equal comparison between two integers or other numerical expressions.
Since we are provided with inequality which is 81 - 11/5x < 55
Now multiplying both by 5x we get
81 × 5x - (11/5x) × 5x < 55 × 5x
=> 405x - 11 < 275x
=> 405x - 275x < 11
=> 130x < 11
=> x < 11/130
so it is the solution for the inequality 81 - 11/5x < 55
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The prism shown in the diagram has a volume of 39 units^3. What is the volume of the pyramid? Explain your reasoning. I WILL GIVE BRAINLIEST ASAP FOR A QUICK ANSWER
Answers
The volume of the pyramid is 13 units³.
What is the volume of the pyramid?
A pyramid is a 3-dimensional shape that is made up of a rectangular base and 4 triangular faces. The volume of a pyramid is one-third that of a prism.
Volume of a pyramid = 1/3 x prism
1/3 x 39 = 13
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Annelise buys a car that is one year old for $13600.
The value of this car has reduced by 15% of the value when it was new.
Calculate the value of the car when it was new.
Answers
Answer:
$16000Step-by-step explanation:
Initial value is x, current value is $13600, price reduction by 15%.
x - 15% = 13600x - 0.15x = 136000.85x = 13600x = 13600/0.85x = 16000URGENT EASY MATH PLEASE
Answers
Answer:
Step-by-step explanation:
Let's solve your inequality step-by-step.
8x−7≤−6−(1−8x)
Step 1: Simplify both sides of the inequality.
8x−7≤8x−7
Step 2: Subtract 8x from both sides.
8x−7−8x≤8x−7−8x
−7≤−7
Step 3: Add 7 to both sides.
−7+7≤−7+7
0≤0
Answer:
All real numbers are solutions.
Answer:
All real numbers are answers
Step-by-step explanation:
PLZ mark brainliest!
I need help Asap File is below.
Answers
Answer: Shade 23 squares on the bigger grid and 203 on the smaller one. 0.23 > 0.203
Step-by-step explanation:
Help, please! I'm rush rn and I want to make sure that my answers are correct so please answer thisss I hope there is an user that understands my situation rn
4. Translate the mathematical sentence showing the perimeter of the squash field into English sentence.
5. Write a mathematical sentence that shows the area (A) of the beans field.
6. Mr. Guzman would like his bean plants to grow to a height of (x + 3). Write the mathematical phrase in solving for the volume of the bean plants if they reach a height of (x + 3).
Answers
\(\huge \bf༆ Answer ༄\)
Question 4 : Perimeter of squash field ~
2 [ (x + 1) + 4y ]English sentence :
Length of the field is 4 times y and Width of the field is (x + 1), and hence it's perimeter is 2 times the sum of Length and Width.
Question 5 : Area of Beans field ~ [ Length × Width ]
(xy + 5) × (xy + 1)or, after further simplification :
x²y² + 6xy + 5Question 6 : Volume of the cuboid is [ L × W × H ]
So, The volume will be Length (xy + 1) times Width (xy + 5) times height (x + 3)
that is ~
(xy + 1) × (xy + 5) × (x + 3) (x²y² + 6xy + 5) (x + 3)x³y² + 6x²y + 5x + 3x²y² + 18xy + 15The solution to all the questions about rectangles are;
4) Perimeter of the Squash field is 2 times the sum of it's Length and Width.
5) A = x²y² + 6xy + 5
6) V = x³y² + 6x²y + 5x + 3x²y² + 18xy + 15
Area and Perimeter of Rectangles4) We can see the squash field is rectangular and the dimensions are;
Length; L = x + 1
Width; w = 4y
Formula for perimeter of a rectangle is;
P = 2(L + w)
P = 2((x + 1) + 4y)
In English sentence, we can say that;
Perimeter of the Squash field is 2 times the sum of it's Length and Width.
5) The beans field is rectangular and its' dimensions are;
Length; L = xy + 5
Width; w = xy + 1
Area of rectangular Beans field;
A = Length × Width
A = (xy + 5) * (xy + 1)
A = x²y² + 6xy + 5
6) Formula for volume of the cube formed by the height will be;
V = L * w * h
since h = (x + 3)
Thus, volume is;
V = (xy + 5) * (xy + 1) * (x + 3)
V = x³y² + 6x²y + 5x + 3x²y² + 18xy + 15
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Which expression is equivalent tofor all values of m , p , and v where the expression is defined?
m^6p^(-3)v^10.m^2p^5v^2
a. m^12p^(-15)v^20
b. m^3p^12v^7
c. m^-(18)p^20v^10
d. m^8p^2v^12
Answers
The given expression \(m^6p^{-3}v^{10} .\ m^2p^5v^2\) for all values of m, p, and v is equivalent to \(m^{8}p^{2}v^{12}\). Therefore, option D is the right choice for this question.
Monomials are algebraic expressions with single terms. They can be said to be specialized cases of polynomials.
We are given the algebraic expression - \(m^6p^{-3}v^{10}\) . \(m^2p^5v^2\)
To simplify it we will use the rules of the indices as follows -
\(a^{m}.\ a^{n} = a^{m+n}\)
Now,
\(m^6p^{-3}v^{10}\) . \(m^2p^5v^2\)
Segregating the like variables, we get,
= \((m^6.\ m^2) .\ (p^{-3}.\ p^{5}) .\ (v^{10}.\ v^{2})\)
by using the rules of indices, we will get,
= \((m^{6+2}) .\ (p^{-3+5}) .\ (v^{10+2})\)
= \((m^{8}) .\ (p^{2}) .\ (v^{12})\)
= \(m^{8}p^{2}v^{12}\)
Hence, the given expression \(m^6p^{-3}v^{10} .\ m^2p^5v^2\) is equivalent to \(m^{8}p^{2}v^{12}\).
Therefore, option D is the right choice for this question.
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Similar triangles ABE and BCD are shown on the coordinate plane. Line t passed through points A, B, and C,Select from the list to correctly complete the sentence.
Answers
Answer
The slope of segment AB is equal to the slope of the segment BC.
Explanation
Similar triangles have the same set of interior angles.
And it is one these angles, that determines the slope of the line segment AB or BC.
For the two triangles, that angle is the same because they are similar triangles.
So,
The slope of segment AB is equal to the slope of the segment BC.
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A triangle has one side that measures 4x and two sides that measure x + 5 each. Write an
expression for the perimeter of this triangle.
Answers
That’s the expression
P = perimeter
if p || , m<7 = 131°, and m<16 = 88°, find the measure of the missing angle m<8= ?
Answers
The angle 7 is given as, m<7 = 131°.
The sum of angle 7 and angle 8 is 180°.
\(m\angle7+m\angle8=180^{\circ}\)Substitute the m<7 = 131°.
\(171^{\circ}+m\angle8=180^{\circ}\)\(m\angle8=180^{\circ}-171^{\circ}\)\(m\angle8=49^{\circ}\)Thus the value of angle 8 is,
\(m\angle8=49^{\circ}\)