The Laplace transform of the given initial value problem is s²Y(s) - 2sY(s) + 5Y(s) = 0.
Take the Laplace transform of the differential equation. Let's denote the Laplace transform of y(t) as Y(s). Using the properties of Laplace transforms and the derivatives property, we have:
L(y''(t)) - 2L(y'(t)) + 5L(y(t)) = s²Y(s) - 2sY(s) + 5Y(s) = 0.
Simplify the equation obtained from the Laplace transform. Rearrange the terms:
s²Y(s) - 2sY(s) + 5Y(s) = 0.
Solve for Y(s). Factor out Y(s) from the equation:
Y(s)(s² - 2s + 5) = 0.
Solve the quadratic equation s² - 2s + 5 = 0 to find the roots. The roots are given by:
s = (2 ± √(-16))/2 = 1 ± 2i.
Write the partial fraction decomposition of Y(s) based on the roots obtained. Since the roots are complex, we have:
Y(s) = A/(s - (1 + 2i)) + B/(s - (1 - 2i)).
Solve for A and B using algebraic manipulation. Multiply both sides of the equation by the denominators and then substitute the roots:
Y(s) = [A/(1 + 2i - 1 - 2i)]/[s - (1 + 2i)] + [B/(1 - 2i - 1 + 2i)]/[s - (1 - 2i)].
Simplify the equation:
Y(s) = A/(4i) * [1/(s - (1 + 2i))] + B/(-4i) * [1/(s - (1 - 2i))].
Apply the inverse Laplace transform to obtain the solution y(t):
y(t) = A/4i * e^((1 + 2i)t) + B/(-4i) * e^((1 - 2i)t).
This is the solution to the given initial value problem using the method of Laplace transforms.
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Round the angle measure to the nearest hundredth of a degree
The measure of the angle is 30.00 degrees
How to find sin⁻¹(0.5)?
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. It is concerned with the measurement of angles and the calculation of sines, cosines, and tangents of those angles.
These ratios are known as trigonometric functions and they are used to relate the ratios of the sides of a right triangle to the angles opposite or adjacent to the right angle.
Since sin 30° = 0.5
Thus, sin⁻¹(0.5) = 30.00 degrees
Note: Angle 30° is one of the special angles and trigonometric are very easy recall.
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The graph below shows how much money Isaac earns by washing cars. 1. Explain if the graph above displays a proportional relationship, including evidence to support your reasoning. 2. Explain what the coordinate shown on the line represents given the problem context provided.
1. The graph will represent a proportional relationship if these two following statements are correct.
The intercept is of zero.The slope of the line is constant.2. The coordinate (x,y) represents the amount of money y earned for washing x cars.
What is a proportional relationship?A proportional relationship is a linear function with an intercept of zero modeled as follows:
y = kx.
In which k is the constant of proportionality.
The input and output variables for this problem are given as follows:
Input x: number of cars washed.Output y: amount of money earned.Hence the coordinate (x,y) represents the amount of money y earned for washing x cars.
Missing InformationThe problem is incomplete, hence the answer is given in general terms using concepts of proportional relationships.
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- To meet your sales quota at work, you need to average $125 per day. So far you have sold $150,
$175, $90, and $155. How much do you need to sell on the last day to meet your quota for this
week?
Hence he need to sell $55 on the last day to meet his quota for this
week.
What is average ?
average can be defined as the ratio of the sum of observations and total number of observations.
Given,
To meet your sales quota at work, you need to average $125 per day.
So far he sold $150, $175, $90, and $155.
so ,
let last day he need to sell be x .
average = 150+175+90+155+x/ 5
125 = 570+x / 5
625 = 570 + x
x = 55
Hence he need to sell $55 on the last day to meet your quota for this
week.
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Why 5 Is it possible to construct a triangle with lengths of its sides as 4 cm 3 cm and 7 cm give reason for your answer?
With sides that are 4 cm, 3 cm, and 7 cm long, a triangle cannot be built. Because the sum of the other two sides does not exceed that of the third side.
A triangle is referred to as a three-sided polygon that has three vertices. The connection of the three sides ends to end at a single point forms the angle of the triangle. The sum of these angles adds up to 180 degrees.
The triangle's three sides are specified as being 4 cm, 3 cm, and 7 cm. Add the two sides together and see if the result exceeds the third side to determine whether these sides constitute a triangle.
Here, the sum of two sides 4 cm and 3 cm is calculated as,
A+B = 4+3
=7 cm
This sum is equal to the third side C = 7 cm.
Therefore, [tex]A+B\ngtr C[/tex]. So, the given sides don't form a triangle.
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The band wanted to do a fundraiser to raise money for the competition trip to Disney land they had an initial amount of money in their booster club account and each band member was expected to raise the same amount the equation 300x + 750 = 23,250 represents the total amount of money in the account after they were done fundraising if z represent the number of students in the band what does 750 likely represent
The value 750 in equation 300x + 750 = 23,250 represents the initial amount in the club account.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 3 = 9 is an equation.
We have,
Initial amount in the club account = m
Amount raised by each member = n
All members raised the same amount.
Number of members = x
The total amount raised by all the members = p
nx + m = p
This is similar to the equation 300x + 750 = 23,250
So,
n = 300
m = 750
p = 23,250
Now,
m = 750 is the initial amount in the club account.
Thus,
750 represents the initial amount in the club account.
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Which of the following is a zero of the polynomial x³ 3x² 3x?A - 1B - 2C 1D 2
A polynomial is a kind of algebraic expression in which the exponents of all variables should be a whole number.
The Power of the variables in any polynomial has to be a non-negative integer. A polynomial consists of constants and variables, but we cannot perform division operations by a variable in polynomials.
The example of polynomial is: 3x^2 + 5.
given the equation is x^3+3^2+3x.=0
(x³-x) +(3x²-3) =0
= x (x²-1) +3(x²-1)
= (x²-1) (x+3) = 0
(a² + b²) = (a+b) (a-b)
(x²-1) = (x+1) (x-1)
⇒(x+1) (x-1) (x+3) =0
x =- 3, -1, 1
So, the required zeros are -3, -1,1.
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Are complex numbers the same as real numbers?
Although every complex number is a real number, not all complex numbers are real numbers.
What connection exists between real numbers and complex numbers?On the number line, the real numbers can be plotted.The number line cannot be used to plot the complex numbers.The symbol designates it.Complex numbers make up all real numbers.Real numbers are not all complex numbers.Although every complex number is a real number, not all complex numbers are real numbers. The symbol Z C Z in mathbb C ZC denotes the collection of all complex numbers.If a number can be expressed as a+bi using the real integers a, b, it is complicated. So, certainly, a complex number can be both real and totally fictitious.To learn more about real numbers refer to:
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What are the 5 rules of factoring?
The five rules for factoring are taking the difference grouping the numbers, dividing, multiplying, taking the greatest common factor and by taking the LCM.
There are certain times when we encounters such kind of problems which includes the factoring of the components of the polynomial.
Ine Sach kind of cases there are several methods for factoring the components of the polynomial like taking the greatest common factor or finding the LCM of the components in order to make the polynomial more simplified.
Also we can take out the most common digit or the number from the whole polynomial which comes under the category of grouping.
Also we can use the method of taking the differences or like taking the division and multiplication of certain digit into the polynomials in order to do the factoring.
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You start with $575 in your bank account and deposit $60 each month. Write a linear or exponential equation
Answer:
A linear equation can be used to represent the situation where a fixed amount is added to the original balance each month. The equation would have the form:
y = mx + b
where x is the number of months, y is the balance in the bank account, m is the monthly deposit amount, and b is the initial balance. In this case, we have:
y = 60x + 575
This equation can be read as: "The balance in the bank account (y) is equal to 60 dollars (m) added to the account each month (x), plus the initial balance of 575 dollars (b)."
An exponential equation can also be used to represent the situation where the balance in the bank account increases by a fixed percent each month, it would be:
y = a(1+r)^x
Where a is the initial amount, r is the interest rate and x is the number of months.
Step-by-step explanation:
What is equation easy?
A mathematical equation can be defined in a variety of ways.
Equation that is linear.A quadratic equation.Cubic Formula.Define the term equation?A mathematical equation can be defined in a variety of ways. The definition of an equation in algebra is a mathematical statement that proves two mathematical expressions are equal.
Equation that is linear.
There could be multiple variables in a linear equation. If a variable's highest power is always 1, an equation is said to be linear. A one-degree equation is another name for it.A quadratic equation.
A second-order equation is this one. At least one of the variables in quadratic equations needs to be raised to exponent 2.Cubic Formula.
Third-order equation.There should be at least one variable raised to exponent 3 in cubic equations.To know more about the equation, here
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A triangle has angle measures 23° and 35°. What is the measure of the third angle?
Answer:
122°
Step-by-step explanation:
All triangles have a total of 180°.
Angle 1 will be X
Angle 2 will be Y
Angle 3 will be z
180 - (x + y) = z
1. Find the present value of $5000 to be received in 2 years if the money can be invested at 12%
interest rate compounded continuously.(1 Point)
$6272
O $6365.75
$6356.25
The present value of $5000 to be received in 2 years at a 12% interest rate compounded continuously is $6356.25. This can be calculated by using the formula PV = FV/(1 + r)^t, where PV is the present value, FV is the future value, r is the interest rate, and t is the time period. Therefore, PV = 5000/(1 + 0.12)^2 = $6356.25.
Answer:
C) $6356.25------------------------------
Use the continuous compound equation:
A = Pe^(rt), where A - future value, P - present value, r - annual interest rate, t - number of yearsSubstitute and calculate:
A = 5000*e^(0.12*2) = 6356.25 (rounded)The matching choice is C.
Mario and Russell plot a triangle on a coordinate grid and then perform a
rotation and a translation. Russell thinks the original triangle and final triangle
are congruent but Mario disagrees. Which student is correct and why?
Russell is correct. Since neither rotations no translations preserve distance, the
triangles are congruent.
O Mario is correct. Since neither rotations not translations preserve distance, the
triangles are not congruent.
Mario is correct. Since both rotations and translations preserve distance, the triangles
are congruent.
►
Russell is correct. Since both rotations and translations preserve distance, the
triangles are congruent.
The right answer is SSS since two triangles are said to be congruency when each of their three sides and Mario is correct.
How can you show a triangle is congruent?Triangles are said regarded as being congruent if their size and shape actually match exactly. The word "congruent" contains the sign. When a triangle's three sides and three angles match those of a another triangle's three sides and three angles, two triangle are said to be congruent.
Here,
begin by writing the names of the triangles' vertices, beginning with triangles DAB and BCD, as shown below.
AB = CD
DA = BC
In both triangles, side BD is the same, therefore
BD = BD
We can infer that the triangles DAB & BCD are congruent that use the SSS feature of a congruence triangle because they have the same number of sides on each side.
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What is called longitude Class 6?
Vertical lines known as longitudes are used to determine the west or east of the meridian in Greenwich, England. Latitude and longitude work together to help cartographers, geographers, and others find specific locations on the globe.
How is longitude measured?Distance from the primary meridian is measured in terms of longitude. Longitude lines, often known as meridians, are fictitious lines that divide the Earth. They sprint from pole to pole from north to south, but they gauge the distance in either an east or west direction. The units of longitude are degree, minute, and second.
The North and South Poles serve as the meeting points for imaginary lines that circle the globe vertically (up and down) and are used to calculate longitude. Meridians are the name for these lines. One arc degree of longitude equals one meridian. There are 360 degrees in an orbit around the Earth.
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!!!!50 POINTS!!!!
It will take you t seconds to carry the water, which
you will do y times a day. What is the total amount of
time you will be carrying water?
How many hours would you spend carrying water in a
year?
I need an expression, evaluation, and explanation! ASAP
Answer:
1. t*y
2. 365*t*y
Step-by-step explanation:
Identify variables
T (time) seconds
Y amount of times per day
T * Y = amount of time you would spend carrying water in a day, 24 hrs
365 * t * y = 365 (amount of days per year, multiplied by the amount of time you carry water per day.)
Which are two solutions for the given equation 3x 4y 7?
The solutions of the given equation are given below:
(i) 3x+4y =7
(ii) x=6y
(iii) x+πy =4
(iv) x-y =4
All these equations are having infinitely many solution.
Infinitely Many solution:
An equation can have an infinite number of solutions if certain conditions are met. If the lines are coincident and have the same y-intercept, the system of equations has infinitely many solutions. If two lines have the same y-intercept and slope, they are really on the same line.
(i) Given :
3x + 4y = 7 --------------------------- (1)
On putting x = 0 in equation 1
3x + 4y = 7
⇒ 3 × 0 + 4y = 7
⇒ 0 + 4y = 7
⇒ 4y = 7
⇒ y = 7/4
Therefore, ( 0, 7/4) is a solution of given equation.
On putting y = 0 in equation 1
3x + 4 × 0 = 7
⇒ 3x + 0 = 7
⇒ 3x = 7
⇒ x = 7/3
Thus, (7/3, 0) is a solution of given equation.
Hence, 2 out of the infinitely many solutions of the given equation are(0,7/4), (7/3,0).
(ii) x = 6y
Let x = 0
⇒ x = 6y
⇒ 0 = 6y
⇒ y = 0/6
⇒ y = 0
(0,0) is the solution.
Let x = 1
⇒ x = 6y
⇒ 1 = 6y
⇒ y = 1/6
(1,1/6) is the solution.
Hence, 2 out of the infinitely many solutions of the given equation are (0,0), (1,1/6).
(iii) x + πy = 4
let x = 0
⇒ x + πy = 4
⇒ 0 + πy = 4
⇒ πy = 4
⇒ y = 4/π
(0, 4/π) is the solution.
Let y = 0
⇒ x + πy = 4
⇒ x + 0(y) = 4
⇒ x + 0 = 4
⇒ x = 4
(4, 0) is the solution.
Hence, 4 out of the infinitely many solutions of the given equation are (0, 4/π), (4, 0).
(iv) x - y = 4
Let x = 0
⇒ x - y = 4
⇒ 0 - y = 4
⇒ -y = 4
⇒ y = - 4
(0,- 4) is the solution.
Let x = 1
⇒ x - y = 4
⇒ 1 - y = 4
⇒ -y = 4 - 1
⇒ -y = 3
⇒ y = - 3
(1,- 3) is the solution.
Hence, 4 out of the infinitely many solutions of the given equation are(0,- 4), (1,- 3).
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Factorise the following.
2x^2+7x+3
Step-by-step explanation:
either x=-1/2
or x=-3
compare with ax²+bx+c=0
Solve the following problem
Note that the area of the above triangle is given as: 42.
This is solved using the SOH CAH TOA Rule and the Formula of the Right Angled Triangle
The hypotenuse is the longest side of a right triangle, a "opposite" side is the one across from a given angle, and a "adjacent" side is the one close to a given angle. Right triangle sides are described using specific terminology. A right triangle's hypotenuse is always the side opposite the right angle.
To find the area of the triangle we must obtain, the Base and the Height, that is FD and A F.
Note that we have
∠F = 90°
∠D = 30°
Since the sum of angles in a Δ = 180°,
∠A = 180 - (90 + 30)
∠A = 60°
To find the area of the triangle we must obtain, the Base and the Height, that is F D and A F.
To obtain F D, we must use the Sine rule which states:
Sine θ = O/H; that is
Sin ∠A = F D/AD
Sin 60° = F D/14
0.86602540378 = FD/14
F D = 14 * 0.86602540378
F D = 12.1243556529
F D [tex]\approx[/tex] 12. 12
Using the above Cos Rule (CAH) we can find the Adjacent Side (that is height). Thus
Cos A = A F/AD
Cos 60° = A F/14
0.5 = A F/14; multiply both sides by 14
A F = 0.5 * 14
A F = 7
Now we have Adjacent Side (Height) = 7; and the Opposite Side (Base) = 12.12
The area of the Triangle according to the Pythagorean Rule is:
1/2 B * H
= 1/2 * 12.12 * 7
= 42.42
Thus the Area of the above triangle [tex]\approx[/tex] 42
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Angela ran for 45 mins, her pace was 9 mph. How far did she run (in miles)?
Use a trigonometric ratio to solve for y. Round to two decimal places as necessary!!
HELP, and can someone explain how to get the answer?
Using Trigonometric identities, the value of y = 11.49.
Trigonometric identities are equality conditions in trigonometry that hold for all values of the variables that appear and are defined on both sides of the equivalence.
These are identities that, geometrically speaking, involve certain functions of one or more angles.
They are not to be confused with triangle identities, which are identities that may involve angles but may also involve side lengths or other lengths of a triangle.
Using Trigonometric Identities,
Cos θ = Base / Hypotenuse
Cos 29 = 10 / y
0.87 = 10 / y
y = 10 / 0.87
y = 11.49
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help so i can get an A?
Simplified form of -5⁷÷ 5² is -5⁵.
The correct option is C.
What is expression?Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between. The mathematical operators can be of addition, subtraction, multiplication, or division.
Given expression,
-5⁷÷ 5²
Above expression can be written as
-5⁷/5²
= -1(5⁷)/5²
= -1(5⁵.5²)/5²
= -1(5⁵)
= -5⁵
Hence, -5⁵ is the simplified form of expression -5⁷÷ 5².
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Please helpppp
solve the inequality
Answer:
A) [tex]x>-2[/tex]
Step-by-step explanation:
[tex]x+4>-x \\ \\ 2x+4>0 \\ \\ 2x>-4 \\ \\ x>-2[/tex]
is 3(5a + 9) + 5a and 4(5a+6) equivalent
No, 3(5a + 9) + 5a and 4(5a+6) are not equivalent
What are equivalent expressions ?Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value(s) for the variable(s).
To know if 3(5a + 9) + 5a and 4(5a+6) are equivalent, we simplify both expressions and see if they will have result
simplifying 3(5a + 9) + 5a;
by opening the parentheses
15a+27+5a
collecting like terms
20a+27
simplifying 4(5a+6);
by opening the parentheses,
20a + 24
Therefore 20a + 27 is not thesame as 20a+24. This means 3(5a + 9) + 5a is not equal to 4(5a+6).
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Write an equation
in slope-intercept form of the line that passes through the points
The equation in slope-intercept form of the line that passes through the points is f(x) = 2x + 2
How to determine the linear function f with the given values.From the question, we have the following parameters that can be used in our computation:
f(0) = 2 , f(-2) = -2
A linear equation can be represented as
f(x) = mx + c
Substitute the known values in the above equation, so, we have the following representation
m * 0 + c = 2
m * -2 + c = -2
Solving 0 * x + c = 2, we have
c = 2
So, we have
m * -2 + c = -2
This gives
-2 * m + 2 = -2
Evaluate the like terms in the above equation
So, we have the following representation
-2m = -4
Divide both sides of the equation by -2
So, we have
m = 2
Substitute the known values in the equation f(x) = mx + c, so, we have the following representation
f(x) = 2x + 2
Hence, the function is f(x) = 2x + 2
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Complete question
Write an equation in slope-intercept form of the line that passes through the points: f(0) = 2 and f(-2) = -2
Baseball And Calculus
Answer:
Calculus in baseball. In baseball, calculus can be used to optimize the pitcher’s throw to achieve maximum efficiency. Also, calculus can be used to calculate the projectile motion of baseball’s trajectory and to predict if runners can make it to the next base on time given their running speed and the speed of a hit ball.
Step-by-step explanation:
Which of the following is a solution of the equation x+ 2y 7?a) x=3 y=5b) x=3 y=-5c) x=3,y=2d) x=0.y=7
(3,2) is the required solution of an equation and This is a straight line and has infinite number of solutions.
the solution is explained as :
y=2
x=7−2(2) =3
Hence, (5,1) is one of the solution of the given equation.
A straight line is an infinite length of an line that does not have any curves on it. A straight line in which it can be formed between two given points and also but both the ends extend to infinity. A straight line is formed when two points A (x1, y1) and B (x2, y2) are connected with the shortest distance between them and the line that ends are extended to infinity.
In the given representation shown below, a straight line between two points A and B is shown. A straight line AB is represented by:
←→
A B
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Why is 21 a prime number?
Answer:
We do not consider 21 as a prime number, because it can be written as a product of two smaller natural numbers
Step-by-step explanation:
Can someone tell me what number on the number line is greater than -9 for an inequality?
Answer: Any number that is to the right of it, hence greater/bigger.
-8,-7,-6,-5.....
Step-by-step explanation:
Example: If you have -8 vs -9 vs -10 dollars, You have more with -8 vs -9 or -10.
Hence anything to the right of a give number on the number line is greater.
Also, think of the opposite with negatives.
For example with positive numbers, the larger the number the greater, but with negative numbers its the other way around, the smaller the number the greater.
Which measure of center should Raul use to calculate his overall performance if he hits a line drive to the outfield on each pitch?
Since the distribution of the 10 distances is not uniform, Raul should use the median to determine his total performance.
What is median?The median is the value that divides a data sample, a population, or a probability distribution's upper and lower halves in statistics and probability theory.
It could be referred to as "the middle" value for a data set.
The fundamental difference between the mean and the median when describing data is that the median is more representative of the "normal" value because it is not skewed by a tiny fraction of exceptionally big or small values.
Due to the fact that income distribution can be very skewed, the median income, for instance, might be a better indicator of what is considered a "normal" income.
Given that it is the most resilient statistic, the median is crucial to strong statistics.
According to our question-
The home run is an outlier in the data set even if nine lengths are grouped together.
The median is the most accurate indicator of the middle of a data collection including outliers.
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Create a rectangle ABCD on a coordinate plane that meets the following conditions:
1. all four points are in different quadrants
2. point A is in Quadrant Il with coordinates (-a, b)
3. the distance from point A to point B is 3a
4. the distance from point A to point D is 4b
5. neither axis is a line of symmetry in the rectangle
Therefore , the solution of the given problem of expression comes out to be 794.0481.
Describe expression.combinations of numbers, symbols, and operations (such + and) that show how much something is worth. Examples: • 2 + 3 is the formula.
Here,
3 x/2 variable is another formula.There is no equals sign in the expression. A mathematical expression is a sentence made up of variables, numbers, or both. There is never an equal symbol in an expression. These expressions are examples. An equation is a mathematical statement proving the equality of two expressions.
Given : 5 +
=> 5 + (5.3)*(5.3)*(5.3)*(5.3)
=> 789.0481 + 5
=> 794.0481
Therefore , the solution of the given problem of expression comes out to be 794.0481.
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On solving the provided question, we can say that a rectangle ABCD on a coordinate plane, point A is in Quadrant Il with coordinates (-a, b), the distance from point A to point B is 3a
What is rectangle?A rectangle in Euclidean plane geometry is a quadrilateral with four right angles. You might also describe it as follows: a quadrilateral that is equiangular, which indicates that all of its angles are equal. The parallelogram might also have a straight angle. Squares are rectangles with four equally sized sides. A quadrilateral of the shape of a rectangle has four 90-degree vertices and equal parallel sides. As a result, it is sometimes referred to as an equirectangular rectangle. Because its opposite sides are equal and parallel, a rectangle is also known as a parallelogram.
a rectangle ABCD on a coordinate plane,
point A is in Quadrant Il with coordinates (-a, b).
the distance from point A to point B is 3a
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