Complete the following proof. Show all of your work.
Prove: The segment joining the midpoints of two sides of a triangle is parallel to the third side.
1. Assign (x, y) coordinates to points A, B, and C.
2. Calculate the (x, y) values for points M and N.
3. Calculate the slope of MN.
4. Calculate the slope of AB.
5. Show that the slopes are equal. What can you conclude? B
Answers
Since m_MN = m_AB, we can conclude that the segment joining the midpoints of two sides of a triangle is parallel to the third side.
What is triangle?Triangle is a three-sided polygon with three angles. It is one of the basic shapes in geometry and is usually considered the simplest type of polygon. Triangles are distinguished by their sides, angles, and whether or not they are equilateral, isosceles, or scalene. The sides of a triangle are always connected, and the angles of a triangle always add up to 180°.
Let A(x1, y1), B(x2, y2), and C(x3, y3) be the vertices of a triangle. Then the point M, which is the midpoint of the side AB, coordinates M(x1 + x2 / 2, y1 + y2 / 2). Similarly, the point N, which is the midpoint of the side AC, has coordinates N(x1 + x3 / 2, y1 + y3 / 2).
The slope of the segment MN is given by:
m_MN = (y1 + y3 / 2 - (y1 + y2 / 2)) / (x1 + x3 / 2 - (x1 + x2 / 2))
The slope of the segment AB is given by:
m_AB = (y2 - y1) / (x2 - x1)
Substituting the coordinates for M and N into the equation for m_MN, we get:
m_MN = (y1 + y3 / 2 - (y1 + y2 / 2)) / (x1 + x3 / 2 - (x1 + x2 / 2))
= (y3 - y2) / (x3 - x2)
Substituting the coordinates for A and B into the equation for m_AB, we get:
m_AB = (y2 - y1) / (x2 - x1)
= (y3 - y2) / (x3 - x2)
Since m_MN = m_AB, we can conclude that the segment joining the midpoints of two sides of a triangle is parallel to the third side.
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Related Questions
URGENT - A triangle is shown in the imagine below. Answer the questions related to it.
Answers
Answer:
a) sin(β) = 7/6 is false for any β
b) angle β is not a real angle
Step-by-step explanation:
You want to use the Sine Law to show that a triangle with sides of lengths 6 and 10 and a smallest angle of 44.5° is impossible.
a) BetaThe Sine Law tells you side lengths are proportional to the sines of their opposite angles.
sin(β)/10 = sin(44.5°)/6
sin(β) = 10(0.7)/6
sin(β) = 7/6
The angle β does not exist. The maximum value of the sine function is 1. There are no angles for which the sine is greater than 1.
b) ImpossibleThe triangle is impossible because angle β cannot exist.
In effect, the angle 44.5° is too large, or side 10 is too long, or side 6 is too short for the figure to be a triangle.
__
Additional comment
In the realm of complex numbers, β ≈ π/2 -0.569618i radians, about (90 -32.64i)°. You cannot draw this complex angle with your real pencil.
a) For any, sin(β) = 1.17 is false.
b) Angle (β) is not a true angle.
What is trigonometry?In the branch of mathematics known as trigonometry, the relationships between the sides and angles of triangles are examined.
It involves the study of trigonometric functions like sine, cosine, tangent, cotangent, secant, and cosecant that describe the connections between the sides and angles of a right triangle.
(a) Using the Sine Law, we have:
sin(β)/10 = sin(44.5°)/6
Multiplying both sides by 10, we get:
sin(β) = (10/6)sin(44.5°)
sin(β) = 1.17
However, this is impossible since the sine of an angle can never be greater than 1. Therefore, there is no value of β that satisfies the equation, and we cannot find the value of B.
(b) The fact that the sine of β is greater than 1 means that there is no angle whose sine is equal to that value.
Therefore, there is no possible angle for β that can satisfy the given conditions of the triangle.
This means that the triangle is impossible and cannot exist with the given side lengths and angle measures.
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Write three dollars and 65 cents
as an improper fraction (and then reduce).
Answers
Answer:
300/65
Step-by-step explanation:
Answer:
3 65/100 then the simplified version is 13/20
Step-by-step explanation:
Knowing that the given sides are congruent, find the perimeter of the
triangle. *Congruent is EQUAL
9x-3
4x+7
Answers
Answer:
Step-by-step explanation:
9x-3=4x+7
+3
9x=4x+10
-4x=-4x
5x=10
divide the 5 by 10 which equals 2.
x=2
The following are equivalent radicals
3√3 = √27
-
3√6 = √54
3√7 = √63
What would be an equivalent form of 3√11 ? Explain
how you know.
Answers
11 = 1 x 11
Since there are no perfect squares that divide evenly into 11, we cannot simplify the radical any further. Therefore, an equivalent form of 3√11 is simply 3√11.
We can confirm this by rationalizing the denominator of the radical expression:
3√11 * (√11/√11) = 3√(11*11)/√121 = 3√121/√121 = 3(11)/11 = 3
This shows that 3√11 is in simplest form and cannot be simplified any further
Hurit Imala works 40 hours a week as a clerk in a local store. She earns $8.50 an hour. Assuming she is paid for 52 weeks annually, what is her gross annual pay?
Answers
Answer:
Hurit Imala's gross annual pay is $17,680
Step-by-step explanation:
Hurit Imala works 40 hours a week, so her weekly pay would be:
40 hours/week x $8.50/hour = $340/week
To find her annual pay, we need to multiply her weekly pay by the number of weeks in a year:
$340/week x 52 weeks/year = $17,680/year
Therefore, Hurit Imala's gross annual pay is $17,680.
Mary is planning to buy a mobile phone that costs $500. She can choose any of five different payment plans with varying simple interest rates and periods:
plan 1: 6% for 10 months
plan 2: 8% for 12 months
plan 3: 10% for 14 months
plan 4: 12% for 18 months
plan 5: 14% for 12 months
Which plan offers the lowest monthly payments?
Answers
As a result, Plan 4's $38.89 monthly premium is the lowest.
How to find the lowest monthly payments ?The monthly premium you pay for health insurance. You typically have to pay additional fees for your health care in addition to your premium, such as a deductible, copayments, and coinsurance.
(P + Simple Interest) / (500 + 58.33) / 14 = $42.02 is the formula for the monthly payment.
Plan 4: P = $500
t = 18/12 years, and r = 0.12
Simple interest equals $500 * 0.12/* (18/12) = $90.
(P + Simple Interest) / t = ($500 + $90) / 18 = $38.89 is the monthly payment.
Plan 5: P = $500
t = 12/12 years, and r = 0.14.
Simple interest is equal to $500 * 0.14 * (12/12) = $70.
(P + Simple Interest) / t = ($500 + $70) / 12 = $47.50 is the amount due each month.
As a result, Plan 4's $38.89 monthly premium is the lowest.
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State the amplitude, period, midline and an equation involving the sine function for the
graph shown below using a horizontal shift between 0 and 2pi
Amplitude:
Period:
Midline: y =
f(x) =
Answers
The amplitude is 2 units.
The period is 2π units.
f(x) = 2 sin(x - π/2)
What is sine function?An oscillation that repeats itself repeatedly in many natural events is described by the sine function, which is a mathematical function. It is described as the proportion of a right triangle's hypotenuse to the side that faces an angle. The typical notation for the sine function is sin(x), where x represents the angle.
Amplitude:
The distance between the graph's midline and its maximum or minimum point is known as the amplitude. The midline in the following graph is located at y=0, midway between the greatest value of 2, and the minimum value of -2. The amplitude is therefore 2 units.
Period:
The distance among two consecutive graph points with the same value and direction is known as the period. The above graph repeats itself every 2π units of x and has one complete cycle between x=0 and x=2π. The duration is therefore 2π units.
Midline:
The midline, also referred to as the average of the highest and minimum values, traverses the graph's centre horizontally. The x-axis or the line y=0 serves as the midline in the presented graph.
It is possible to write the sine function equation for the provided graph with a horizontal shift between 0 and 2 as:
f(x) = 2 sin(x - π/2)
The graph fluctuates up and down near the midline, hence the sine function is utilised. The highest value occurs at x=π/2, which is π/2 units to the right of x=0, and the graph continues every 2π units of x, therefore the horizontal shift is π/2. The sine function multiplies the amplitude of two units.
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Read the following text from a student essay. How can the writer best improve his or
her organization?
Ponies are a type of horse, but there are significant differences between ponies and
horses. Ponies are generally shorter than horses. Ponies also tend to live longer than
horses. Horses are considered to be over fifty-eight inches tall, and ponies are less
than that. Horses usually live to be twenty-five to thirty years old. Most ponies live
to be thirty-five or even forty.
by adding missing commas
moving the first sentence to the end of the paragraph
by moving the first sentence to the second paragraph
by reordering sentences to keep the related ideas together
Answers
The writer can best improve their organization by reordering sentences to keep related ideas together.
Who is called writer?
Writers are individuals who use written words to communicate ideas, stories, or information to a particular audience. They may work in various fields such as journalism, creative writing, copywriting, technical writing, or academic writing. Writers can be novelists, poets, essayists, screenwriters, playwrights, and more, and their work can take many forms, such as books, articles, blogs, scripts, and speeches. Writing is a skill that can be developed and honed through practice, and good writers are able to engage and captivate their readers with their words.
Here,
The writer can best improve their organization by reordering sentences to keep related ideas together. Here's a possible reorganization of the text:
Ponies and horses have significant differences. Horses are considered to be over fifty-eight inches tall and usually live to be twenty-five to thirty years old. In contrast, ponies are generally shorter than horses, with a height of less than fifty-eight inches, and they tend to live longer, often up to thirty-five or even forty years.
This revision keeps the sentences that relate to each other together, creating a clear and logical progression of ideas. Additionally, it removes the redundant statement of defining ponies as a type of horse, which is already well-known information.
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Given the parent function f(x)=√x and a new function g(x)=√x+6+3.
Select from the drop-down menus to correctly describe the changes on the graph from f(x) to g(x)
1. The graph will shift 6 units ___
UP , DOWN , LEFT , RIGHT
2. and 3 units ____
UP , DOWN , LEFT , RIGHT
Answers
Answer:
up down
Step-by-step explanation:
NEED HELP!! 9.1 Puzzle Time
What Do You Get When You Cross a Computer With a Lifeguard?
Answers
6. Using Pythagoras Theorem the value of x = 99. 7, The value of x is 102.56. 8. The triangle is a right triangle. 9. The triangle is acute. 10. The triangle is acute. 11. The value of x is 32.28. 12. The value of x is 40.
What is Pythagoras Theorem?The relationship between the sides of a right triangle is explained by the Pythagorean theorem, a fundamental mathematical theorem. The square of the hypotenuse's length, the longest side in a right triangle, is said to be equal to the sum of the squares of the lengths of the other two sides.
6. For a = 20, b = 99 using the Pythagoras Theorem we have:
x² = a² + b²
x² = 20² + 99²
x² = 9801
x = 99
7. For a = 60, b = 91 we have:
x² = 60² + 91²
x² = 10521
x ≈ 102.56
8. To determine whether the triangle is acute, obtuse, or right we check for using the Pythagoras Theorem.
a = 20, b = 21, and c = 29
Here,
20² + 21² = 841
29² = 841
Hence, the triangle is right.
9. For a = 15, b = 19, and c = 24. We have:
15² + 19² = 556
24² = 576
The triangle is acute.
10. For a = 11, b = 23, and c = 26. We have:
11² + 23² = 650
26² = 676
The triangle is acute.
11. Using the Pythagoras Theorem we have:
x² = 53² - 45²
x² = 1044
x ≈ 32.28
The value of x is 32.28
12. x² = 41² - 9²
x² = 1600
x = 40
The value of x is 40.
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Answer:
Step-by-step explanation:
3. The t test for two independent samples - Two-tailed example
“Bullying,” according to noted expert Dan Olweus, “poisons the educational environment and affects the learning of every child.” Bullying and victimization are evident as early as preschool, with the problem peaking in middle school. Suppose you are interested in the emotional well-being of not only the victims but also bystanders, bullies, and those who bully but who are also victims (bully-victims). You decide to measure depression in a group of victims and a group of bully-victims using a 26-item, 3-point depression scale. Assume scores on the depression scale are normally distributed and that the variances of the depression scores are the same among victims and bully-victims.
The group of 30 victims scored an average of 21.5 with a sample standard deviation of 10 on the depression scale. The group of 27 bully-victims scored an average of 25.8 with a sample standard deviation of 9 on the same scale. You do not have any presupposed assumptions about whether victims or bully-victims will be more depressed, so you formulate the null and alternative hypotheses as:
H0
: μvictims
– μbully-victims
= 0
H1
: μvictims
– μbully-victims
≠ 0
You conduct an independent-measures t test. Given your null and alternative hypotheses, this is a test. To use the Distributions tool to find the rejection region, you first need to set the degrees of freedom. The degrees of freedom is .
The critical t-scores that form the boundaries of the rejection region for α = 0.01 are ± .
In order to calculate the t statistic, you first need to calculate the standard error under the assumption that the null hypothesis is true. In order to calculate the standard error, you first need to calculate the pooled variance. The pooled variance is s2p
= . The standard error is s(M1 – M2)
= .
The t statistic is .
The t statistic in the rejection region. Therefore, the null hypothesis is . You conclude that victims have a different mean depression score than bully-victims. Thus, it can be said that these two means are different from one another.
Answers
The test statistics do not lie in the rejection so the null hypothesis does not reject.
How to solveHere we have the following information:
[tex]\bar{x}_{1}=25.3, n_{1}=25, s_{1}=9, \bar{x}_{2}=20.5, n_{2}=23, s_{2}=8[/tex]
Here degree of feedom will be
[tex]df=n_{1}+n_{2}-2=25+23-2=46[/tex]
The critical values of t are: +/- 2.687
Rejection region:
If t < -2.687 and t > 2.687, reject H0
Pooled variance is :
[tex]s^{2}_{p}=\frac{(n_{1}-1)s_{1}^{2}+\left ( n_{2}-1 \right )s_{2}^{2}}{n_{1}+n_{2}-2}=72.8696[/tex]
The standard error is
[tex]s_{M_{1}-M_{2}}=s_{p}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}=2.4664[/tex]
and t-statistics will be
[tex]t=\frac{\bar{x}_{1}-\bar{x}_{2}}{s_{p}\sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}}=1.946[/tex]
The test statistics do not lie in the rejection so the null hypothesis does not reject.
You cannot conclude...
not
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the lenghth of a picture frame is 7 inches more than th width. For what values of x is the perimeter of the picture frame greater than 154 inches
Answers
Answer:
x > 35
Step-by-step explanation:
Let's denote the width of the picture frame as "x" inches.
According to the problem, the length of the picture frame is 7 inches more than the width, so it can be expressed as "x + 7" inches.
The perimeter of the picture frame can be found by adding up the lengths of all four sides, which is given by:
Perimeter = 2*(length + width)
Substituting the values of length and width in terms of x, we get:
Perimeter = 2*(x + 7 + x)
Perimeter = 4x + 14
Now we can set up an inequality to find the values of x for which the perimeter is greater than 154 inches:
4x + 14 > 154
Subtracting 14 from both sides, we get:
4x > 140
Dividing both sides by 4, we get:
x > 35
Therefore, the values of x for which the perimeter of the picture frame is greater than 154 inches are x > 35.
A boat is heading towards a lighthouse, where Lamonte is watching from a vertical distance of 141 feet above the water. Lamonte measures an angle of depression to the boat at point � A to be 21 ∘ ∘ . At some later time, Lamonte takes another measurement and finds the angle of depression to the boat (now at point � B) to be 57 ∘ ∘ . Find the distance from point � A to point � B. Round your answer to the nearest foot if necessary.
Answers
The distance from point A to point B is 812 feet.
What is triangle?
A triangle is a form of polygon with three sides; the intersection of the two longest sides is known as the triangle's vertex. There is an angle created between two sides. One of the crucial elements of geometry is this.
Certain fundamental ideas, including the Pythagorean theorem and trigonometry, rely on the characteristics of triangles. The angles and sides of a triangle determine its kind.
Here lines AC and ED are parallel. So alternate angles are equal. Now the corresponding angles in triangle ADC and BDC.
In Δ ADC ,
=> tan 9 = opposite / adjacent = 141/AC
=> AC = 141/ tan9= 890.2389.
Now In ΔBDC,
=> tan 61 = 141/BC
=> BC = 141/tan 61 = 78.1575
Now distance AB = AC - BC = 890.2389 - 78.1575 = 812 feet.
Hence the distance from point A to point B is 812 feet.
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Debbie saves $900 each summer with her summer job. She plans to work the same job for the next four summers. She wants to save money from her summer job to pay for her first year of college. She's thinking of attending a college that costs $8,500 for one year's tuition. Her parents have said that they will pay for half of her tuition. How much more money will Debbie need by her freshman year of college to afford her first year's tuition?
Answers
Step-by-step explanation:
If Debbie saves $900 each summer for four summers, she will have saved a total of:
$900 x 4 = $3600
Her parents have agreed to pay half the tuition, which is $8,500 / 2 = $4250.
Therefore, Debbie will need to come up with the remaining amount:
$8,500 - $4,250 = $4,250
Since Debbie will have saved $3,600 from her summer job, she will need an additional:
$4,250 - $3,600 = $650
Debbie will need $650 more to afford her first year's tuition.
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A central angle of a circle whose radius is 14 centimeters subtends an arc whose length is 7pi/15 centimeters. What is the radian measure of the central angle?
Answers
π/30 R is the radian measure of the central angle.
What is a circle, exactly?
The collection of all points in the plane that make up a circle are all equally spaced from a certain point known as the "centre," making the form a closed two-dimensional shape.
The reflection symmetry line is created by all lines that traverse the circle. Additionally, it is symmetrical in rotation around the centre at all angles.
Total circle circumference would be π d = 28 π
this arc is a fraction of the total
7π/15 / 28π x 2π = π/30 R
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FREE
Find the Sin, Cos and Tan of the right triangle Round your answers to the nearest tenth.
2. Sin=
3. Cos=
d. Tan=
Answers
Using trigonometric functions, we can find the value of the angles as:
Sinθ = 3/5
Cosθ = 4/5
Tanθ = 3/4
Define trigonometric functions?Six basic trigonometric operations are used in trigonometry. These operations are described by trigonometric ratios. The six basic trigonometric functions are the sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function. The foundation for trigonometric identities and functions is the ratio of the sides of a right-angled triangle. The sine, cosine, tangent, secant, and cotangent values are computed for the perpendicular side, hypotenuse, and base of a right triangle using trigonometric formulas.
Here as per the question,
Opposite side of the angle = 3 units.
Adjacent side of the angle = 4 units.
Hypotenuse of the triangle = 5 units.
Sinθ = opposite side/ hypotenuse
= 3/5
Cosθ = adjacent side/ hypotenuse
= 4/5
Tanθ = opposite side/adjacent side
= 3/4
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Another method of identifying an outlier is to investigate whether there is evidence that a value might have come
from a population with a mean different from the mean of the population of the other values.
Let X and Y represent random variables. X is distributed normally with mean u, and standard deviation o.
and Y is distributed normally with mean u, and standard deviation o. Consider I randomly selected value of Y
and I randomly selected values of X.
-
(b) Consider the difference Y-X.
(i) In terms of μ, and .. what is the mean of the difference Y - X?
(ii) In terms of n and standard deviation . what is the standard deviation of the difference Y - X?
Answers
The standard deviation of the difference Y - X depends on the standard deviations of both X and Y, and is given by the formula below.
(b) (i) The mean of the difference Y - X is μY - μX, which is 0 since both X and Y have the same mean, μ.
(b) (ii) The standard deviation of the difference Y - X is √(σY^2 + σX^2), where σY and σX are the standard deviations of Y and X, respectively. Assuming that the values of X are independent of the value of Y, we can use the formula for the variance of the difference of two random variables to get:
Var(Y - X) = Var(Y) + Var(X) - 2Cov(Y, X)
where Cov(Y, X) is the covariance between Y and X. Since Y and X have the same mean, we have:
Cov(Y, X) = E[(Y - μ)(X - μ)] = E[(Y - μ)X] - μE[X] = E[(Y - μ)X] - μ^2
where E[] denotes the expected value. Since X and Y are independent, we have:
E[(Y - μ)X] = E[Y - μ]E[X] = 0
Therefore, we get:
Var(Y - X) = Var(Y) + Var(X) = σY^2 + σX^2
Taking the square root of both sides gives us the standard deviation of Y - X:
SD(Y - X) = √(σY^2 + σX^2)
Therefore, the standard deviation of the difference Y - X depends on the standard deviations of both X and Y, and is given by the formula above.
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Can you pls help. ASAP
Answers
The true statement would be: The theoretical probability of landing on 11 is 12.5% while the experimental probability of landing on 11 is 30%
What is experimental probabilityExperimental probability is a type of probability that is based on actual experimental results or data. It is the ratio of the number of times an event occurs to the total number of trials or experiments conducted.
For example, if you were to toss a coin 100 times and record the number of times it lands on heads, the experimental probability of the coin landing on heads would be the number of times it landed on heads divided by the total number of coin tosses (100 in this case). If the coin landed on heads 45 times out of 100, then the experimental probability of the coin landing on heads would be 45/100 or 0.45.
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Please help me. I need help real bad.
Answers
Answer:
θ ∈ {π/4, π/2, 3π/4, 5π/4, 3π/2, 7π/4}
Step-by-step explanation:
You want the solutions to sin(2θ)sin(θ)=cos(θ) on [0, 2π).
FactorsWe can use the identity for sin(2θ) to write the equation in terms of sine and cosine.
(2sin(θ)cos(θ))·sin(θ) = cos(θ) . . . . replace sin(2θ)
2sin(θ)²cos(θ) - cos(θ) = 0 . . . . . . . subtract cos(θ)
cos(θ)(2 sin(θ)² -1) = 0
Zero product ruleThe zero product rule tells you a product will be zero only when one or more factors is zero. The zeros of the factors are ...
cos(θ) = 0 ⇒ θ = π/2, 3π/2
2sin(θ)² -1 = 0 ⇒ sin(θ) = ±√(1/2) ⇒ θ = π/4, 3π/4, 5π/4, 7π/4
The solutions are θ ∈ {π/4, π/2, 3π/4, 5π/4, 3π/2, 7π/4}.
__
Additional comment
The solutions are the x-intercepts of the graph of sin(2θ)sin(θ) -cos(θ), the values of x where this expression is zero. A graphing calculator finds them nicely.
At West Point School District, enrollment increased from 320 students in 2019 to 225 students in 2020. What is the percent of change in enrollment? Round to the nearest percent.
Answers
Answer:95
Step-by-step explanation: cause 320 - 225 = 95
Task 1abcd: Suppose we have n = 60 puppies (across several litters from these two dogs). For the sample, we have the (previously observed) sample proportion
= 0.3833.
Suppose we wanted to use this sample to determine if there is evidence that p = the population proportion of puppies who are spotted is greater than 0.25.
a. State the hypotheses about the population proportion p.
H0:
H1:
Answers
H0 is the null hypothesis stating with that the population proportion of
puppies who are spotted is less than or equal to 0.25,
H0: p <= 0.25
H1: p > 0.25
Why do you use the term "hypothesis"?In a scientific setting, a hypothesis (plural: hypotheses) is a tested
claim regarding the causal connection between two or more variables
or a suggested explanation for a phenomenon that has been
observed.
where H0 is the null hypothesis stating that the population proportion
of puppies who are spotted is less than or equal to 0.25, and H1 is the
alternative hypothesis stating that the population proportion of puppies
who are spotted is greater than 0.25.
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You made $75 an hour with your commission pay. This month I want to increase that by 20% so that I make
Answers
11. A local grocery store has 7 different kinds of hot sauce that you would love try.
However, you only have enough money to purchase 5 of the different flavors. How
many of the flavors of hot sauce are possible for you to purchase?
Answers
we are selecting 5 flavors out of 7 without regard to order. The number of combinations of 7 things taken 5 at a time is given by: C(7,5) = 21
what is combinations ?
Combinations, on the other hand, refer to the selection of objects from a set without considering their order. For example, the combinations of 2 letters from the set "ABC" are "AB", "AC", and "BC". The order of the letters in each combination does not matter, so "AB" and "BA" are considered the same combination.
In the given question,
This is a combination problem, since we are selecting 5 flavors out of 7 without regard to order. The number of combinations of 7 things taken 5 at a time is given by:
C(7,5) = 7! / (5! * 2!) = (76543) / (54321) = 21
Therefore, you can purchase 21 different combinations of 5 flavors of hot sauce
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If a 4 digit number 4ab5 is divisible by 55 then determine the value of b-a
Answers
Answer:
Step-by-step explanation:
Solution:
A four-digit number 4ab5 is divisible by 55. Then the value of b - a is 1.
find the surface area and volume of this retangular prism 7cm by 6cm by 5cm
Answers
The volume of the rectangular prism is 210cm³.
Define rectangular prismA rectangular prism is a three-dimensional solid shape that has six rectangular faces. Each face of the prism is a rectangle, and the opposite faces are congruent and parallel. The rectangular prism is also known as a rectangular cuboid or a rectangular parallelepiped. The rectangular prism has eight vertices (corners), twelve edges, and six faces. It is a special case of a parallelepiped, which is a three-dimensional solid with six parallelogram faces.
The surface area of the rectangular prism is:
Surface Area = 2(lw + lh + wh)
Surface Area = 2(7cm x 6cm) + 2(7cm x 5cm) + 2(6cm x 5cm)
Surface Area = 84cm² + 70cm² + 60cm²
Surface Area = 214cm²
Hence, the rectangular prism has a 214 cm² surface area.
We must multiply the rectangular prism's length, width, and height in order to get its volume.
The volume of the rectangular prism is:
Volume = lwh
Volume = 7cm x 6cm x 5cm
Volume = 210cm³
Hence, the rectangular prism has a 210 cm³ volume.
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Simplify each complex fractions. Assume no denominators are 0
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The complex fractions are simplified to;
60. 3/2062. xy.64. (x² + x + 1)/[2x³(x-1)].How to calculate complex fractions?(1/4 - 1/5)/(1/3)
To simplify this complex fraction, first simplify the numerator by finding a common denominator:
(1/4 - 1/5) = (5/20 - 4/20) = 1/20
Then, we can divide by the denominator by multiplying by its reciprocal:
1/20 ÷ (1/3) = 1/20 × (3/1) = 3/20
Therefore, the simplified complex fraction is 3/20.
(x-y)/(x⁻¹ - y⁻¹)
To simplify this complex fraction, use the formula for the difference of squares on the denominator:
x⁻¹ - y⁻¹ = (x - y)/(xy)
Then, we can rewrite the complex fraction as a division of two fractions:
(x-y)/(x⁻¹ - y⁻¹) = (x-y) × (xy)/(x - y) = xy
Therefore, the simplified complex fraction is xy.
((2x + 1)/(x- (1/x)) - (1/x))/(2x + (2x)/(1-1/x))
To simplify this complex fraction, first simplify each of the two fractions in the numerator and denominator separately:
(2x + 1)/(x- (1/x)) = [(2x² + x)/x]/[(x² - 1)/x] = (2x² + x)/(x² - 1)
(1/x) = (x⁻¹)
(2x + (2x)/(1-1/x)) = (2x + (2x)/(x-1))
= [(2x(x-1) + 2x)/ (x-1)] / [(x-1)/(x-1)]
= (2x² - 2x + 2x)/(x-1)
= (2x²)/(x-1)
Substituting these simplified fractions back into the original complex fraction:
[(2x² + x)/(x² - 1) - x⁻¹] ÷ [(2x²)/(x-1)]
= [(2x² + x)/(x² - 1) - (1/x)] × [(x-1)/(2x²)]
= [(2x² + x) - (x² - 1)]/[x(x² - 1)] × [(x-1)/(2x²)]
= (x² + x + 1)/[2x³(x-1)]
Therefore, the simplified complex fraction is (x² + x + 1)/[2x³(x-1)].
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Image transcribed:
60. (1/4 -1/5)/(1/3)
62. (x-y)/(x⁻¹ - y⁻¹)
64. ((2x 1)/(x- (1/x)) - (1/x))/(2x + (2x)/(1-1/x)
Answer:
60. 3/20
62. -xy
64. (2x³ -x² +1)/(4x⁴ +2x³ -2x²)
Step-by-step explanation:
You want the simplified form of various complex fractions.
SimplificationFractions are added and divided in the usual ways:
a +b/c = (ac +b)/c
a/b +c/d = (ad +bc)/(bd)
(a/b)/(c/d) = (ad)/(bc)
60.(1/4 -1/5)/(1/3) = (5 -4)/(5·4)·(3/1) = 3/20
62.(x -y)/(1/x -1/y) = (x -y)/((y -x)/(xy)) = xy(x -y)/(y -x) = -xy
64.[tex]\dfrac{\dfrac{2x}{x-\dfrac{1}{x}}-\dfrac{1}{x}}{2x+\dfrac{2x}{1-\dfrac{1}{x}}}=\dfrac{\dfrac{2x^2}{x^2-1}-\dfrac{1}{x}}{2x+\dfrac{2x^2}{x-1}}=\dfrac{(2x^3-x^2+1)(x-1)}{x(x^2-1)(2x^2-2x+2x^2)}\\\\=\dfrac{2x^3-x^2+1}{2x^2(x+1)(2x-1)}=\boxed{\dfrac{2x^3-x^2+1}{4x^4+2x^3-2x^2}}[/tex]
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Additional comment
The factored form in 64 is useful for many purposes. "Simplified" usually means that factors are multiplied out. Note that we used (x²-1) = (x-1)(x+1).
If you were to describe the domain of the expression in 64, it would exclude x=1 as well as those values of x that make the remaining denominator factors zero.
The vertices of a rectangle are plotted.
A graph with both the x and y axes starting at negative 8, with tick marks every one unit up to 8. The points negative 5 comma 2, 4 comma 2, negative 5 comma negative 4, and 4 comma negative 4 are each labeled.
What is the area of the rectangle?
15 square units
30 square units
45 square units
54 square units
Answers
The area of the rectangle is d. 54 square units.
What is rectangle?A rectangle is a closed two-dimensional geometry with four sides, four corners, and four right angles (90°). A rectangle's opposite sides are equal and parallel. Because a rectangle is a 2-D form, it has two dimensions: length and width. The length of the rectangle is the longer side, while the width is the shorter side.
The rectangle has two pairs of parallel sides, so its area can be calculated by multiplying the length of one pair of parallel sides by the length of the other pair of parallel sides.
From the given points, we can see that the length of one pair of parallel sides of the rectangle is:
4 - (-5) = 9 units
And the length of the other pair of parallel sides of the rectangle is:
2 - (-4) = 6 units
Therefore, the area of the rectangle is:
9 units × 6 units = 54 square units.
So the area of the rectangle is d. 54 square units.
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find the values of X and Y
Answers
The values of x and y for the right triangle are 4 and 7.21
How to evaluate for the values of x and y for the triangleThe perpendicular height of the right triangle divides the triangle in two triangles with the same proportions as the original triangle.
by Pythagoras, the hypotenuse for the bigger of the two triangles is calculated as;
√(6² + 9²) = 10.82
the angle opposite the side length 6 for triangle with hypotenuse 10.82 and the angle opposite the side length x for triangle with hypotenuse y are the same by proportion, so:
x/6 = 6/9 {opposite/adjacent}
x = 36/9 {cross multiplication}
x = 4
also;
4/y = 6/10.52 {opposite/hypotenuse}
y = 42.08/6
y = 7.21
Therefore, the values of x and y for the right triangle are 4 and 7.21
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find the products AB and BA to determine whether B is the multiplicative inverse of A.
Answers
B is not the multiplicative inverse of A.
Multiplicative inverse matrix:The multiplicative inverse matrix, also known as the inverse matrix or simply the inverse, is a matrix that when multiplied by a given matrix, results in the identity matrix. The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.
Here we have
[tex]A = \left[\begin{array}{cc}3&-2&\\0&-3&\end{array}\right][/tex] [tex]B = \left[\begin{array}{cc}-1&2&\\0&-4&\end{array}\right][/tex]
Product of A and B
[tex]AB = \left[\begin{array}{cc}3&-2&\\0&-3&\end{array}\right] \times \left[\begin{array}{cc}-1&2&\\0&-4&\end{array}\right][/tex]
[tex]AB = \left[\begin{array}{cc}3(-1) + (-2)(0) &3(2) + (-2)(-4)&\\0 + 0 &0 +(-3)(-4)&\end{array}\right][/tex]
[tex]AB = \left[\begin{array}{cc}-3 &14&\\0 &12&\end{array}\right][/tex]
Product of B and A
[tex]BA = \left[\begin{array}{cc}-1&2&\\0&-4&\end{array}\right] \times \left[\begin{array}{cc}3&-2&\\0&-3&\end{array}\right][/tex]
[tex]BA = \left[\begin{array}{cc}1(3) + 2(0)&-1(-2)+2(-3)\\0(3)+0 &0(-2)+(-4)(-3) \end{array}\right][/tex]
[tex]BA = \left[\begin{array}{cc}-3&-4\\0&12\end{array}\right][/tex]
Here AB ≠ BA ≠ I
Therefore,
B is not the multiplicative inverse of A.
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Complete Question given in the picture
5x1 + 6x1/10 + 7x1/100 + 2x1/1,000
Answers
The simplified expression is 84/125.
What is expression?An expression is a combination of numbers, variables, and mathematical symbols that represent a mathematical relationship or formula. Expressions can be simple or complex and can include operations such as addition, subtraction, multiplication, and division, as well as exponents, radicals, and functions.
In the given question,
To simplify the expression 5x1 + 6x1/10 + 7x1/100 + 2x1/1,000, we can start by converting all the fractions to have a common denominator of 1,000. This gives:
5x1 + 6x100/1,000 + 7x10/1,000 + 2x1/1,000
Next, we can add the terms that have the same denominator:
(6x100 + 7x10 + 2x1)/1,000
Simplifying the numerator, we get:
(600 + 70 + 2)/1,000
Adding the terms in the numerator, we get:
672/1,000
Simplifying the fraction, we can divide both the numerator and denominator by the greatest common factor, which is 8:
84/125
Therefore, the simplified expression is 84/125.
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