The distance between the two points in simplest radical form (3, 2) and (-5, -7) is [tex]\sqrt{40}[/tex]
Find the distance?We must determine the hypotenuse's length in order to answer the question because the hypotenuse's endpoints have been provided.
We must determine the hypotenuse's length in order to answer the question because the hypotenuse's endpoints have been provided.
The formula, Distance =, determines the distance between the two positions (x1, y1) (x2, y2).
The formula, Distance =, determines the distance between the two positions (x1, y1) (x2, y2).
Distance = [tex]\sqrt{(x2 - x1)(y2 - y1)}[/tex]
Distance between the two positions (3, 2) and (--5, --7) equals
Distance =root of ( -5 - 3,-7 - 2)
root of (-8 * -5) = [tex]\sqrt{40}[/tex].
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Visitors to a carnival pay an admission fee and must buy a ticket for each ride they go on.
Last year the total cost for a visitor to go on n rides was modeled by the expression 4.00 +1.50n.
This year, the carnival doubled the cost of a ride ticket but kept the admission fee the same.
Which expression models the total cost for a visitor to go on n rides at the carnival this year?
The expression models the total cost for a visitor to go on n rides (c) is 4.00+1.50n
What is Expressions?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)
An expression, often known as a mathematical expression, is a finite collection of symbols that are well-formed in accordance with context-dependent principles.
total cost for a visitor to go on n rides (c) = 4.00+1.50n
Admission fee = 4.00
Cost of each ride = 1.50
Now the cost of each ride is doubled.
New cost of each ride = 2×1.50 = 3.00
New cost model (C') for n rides = 4.00+ 3n
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in exercises 11 and 12, determine if b is a linear combination of subscript[a, 1], subscript[a, 2], and subscript[a, 3].
Yes b can be written as linear combination of 3 vectors
What is linear combination of vector?A linear combination of vectors is a sum of scaled vectors, where the scaling factors are scalars. In other words, it's a linear combination of the form a_1v_1 + a_2v_2 + ... + a_n*v_n, where v_1, v_2, ..., v_n are vectors and a_1, a_2, ..., a_n are scalars. This is a fundamental concept in linear algebra and vector spaces and is used in many areas of mathematics and physics.
if b is linear combination of [tex]a_1,a_2,a_3[/tex]
then [tex]xa_1+ya_2+za_3=b[/tex]
from here we got :
x+5z = 2
-2x+y-6z = -1
2y + 8z = 6
after solving the above equation we get the solution as
{x,y,x} = {2,3,0} is a solution to the above equation so b is a linear combination of [tex]a_1,a_2,a_3[/tex]
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Complete question:
determine if b is a linear combination of [tex]a_1,a_2,a_3[/tex]
[tex]a_1[/tex] = [tex]\left[\begin{array}{ccc}1\\-2\\0\end{array}\right][/tex] , [tex]a_2 = \left[\begin{array}{ccc}0\\1\\2\end{array}\right][/tex] , [tex]a_3 = \left[\begin{array}{ccc}5\\-6\\8\end{array}\right][/tex] , b= [tex]\left[\begin{array}{ccc}2\\-1\\6\end{array}\right][/tex]
Water is flowing from a horizontal pipe 48 feet above the ground. The falling stream of water has the shape of a parabola whose vertex (0, 48) is at the end of the pipe (see figure). The stream of water strikes the ground at the point (10 sqrt3 , 0). Find the equation of the path taken by the water
The water stream's parabola has the equation y = 48 - (x2)/(20*sqrt(3)), with the vertex at (0, 48) and the point of impact at (10sqrt(3), 0).
The equation of the parabola created by the water stream has a vertex at (0, 48), which is where the pipe ends. It is a quadratic equation. The pipe serves as the middle of the parabola's base, which is symmetrical and has a vertex at that location. The vertex form of a parabola can be used to determine the equation of the parabola, which is y = 48 - (x2)/(20*sqrt(3)). The parabola opens downward, with the highest point at (0, 48) and the lowest point at (10sqrt(3), 0), where the water stream hits the ground, according to the equation. The parabola's vertex and point of impact with the ground are described by the equation for the water's journey.
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Marissa bought 7 ounces of yogurt for $3.50.
What is the unit price?
$0.35 per ounce
$0.50 per ounce
$3.50 per ounce
$5.00 per ounce
Answer:
$0.50
50 cents per ounce.
Step-by-step explanation:
To get the unit price, you take the cost of the package divided by the parts it has. Since Marissa has a package of yogurt that costs $3.50 and there are seven ounces in the package we have this equation:
($3.50)/(7 ounces)= $0.50 = 50 cents = the second answer available.
Consider the slope field shown: (a) For the solution that satisfies y(0) 0,sketch the solution curve and estimate the following: y(I) ~ 0.75 and y(-1) 0.5 (b) For the solution that satisfies Y(O) = 4, sketch the solution curve and estimate the following: Y(0.5) -0.5 and Y(-1) 0.75 (c) For the solution that satisfies Y(O) = -1, sketch the solution curve and estimate the following: Y(I) ~ and y(-1) ~
For the solution that satisfies, sketch the solution curve and estimate the following y(0)=0: y(1)≈0.75, y(-1)≈0.5; Y(0)=4: Y(0.5)≈-0.5, Y(-1)≈0.75; Y(0)=-1: Y(1)≈-1.5, Y(-1)≈-1.5.
(a) To sketch the solution curve for the given slope field, beginning at the point (0, 0) and following the arrows, draw a curved line that gradually moves from left to right, gradually increasing in value and then slowly decreasing. The estimated value of y(1) is 0.75 and y(-1) is 0.5.
(b) To sketch the For the solution that satisfies, sketch the solution curve and estimate the following y(0)=0: y(1)≈0.75, y(-1)≈0.5; Y(0)=4: Y(0.5)≈-0.5, Y(-1)≈0.75; Y(0)=-1: Y(1)≈-1.5, Y(-1)≈-1.5. curve for the given slope field, beginning at the point (0, 4) and following the arrows, draw a curved line that gradually moves from right to left, gradually decreasing in value and then slowly increasing. The estimated value of Y(0.5) is -0.5 and Y(-1) is 0.75.
(c) To sketch the solution curve for the given slope field, beginning at the point (0, -1) and following the arrows, draw a curved line that gradually moves from left to right, gradually increasing in value and then slowly decreasing. The estimated value of Y(1) is and y(-1) is -1.5.
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Three softball players discussed their batting averages after a game.
Probability
Player 1 four sevenths
Player 2 five eighths
Player 3 three sixths
By comparing the probabilities and interpreting the likelihood, which statement is true?
Player 1 is more likely to hit the ball than Player 2 because P(Player 1) > P(Player 2)
Player 2 is more likely to hit the ball than Player 1 because P(Player 2) > P(Player 1)
Player 3 is more likely to hit the ball than Player 1 because P(Player 3) > P(Player 1)
Player 3 is more likely to hit the ball than Player 2 because P(Player 3) > P(Player 2)
Player 2 is more likely to hit the ball than Player 1 because P(Player 2) > P(Player 1). Therefore, option B is the correct answer.
What is the probability?Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.
We know that, probability of an event = Number of favourable outcomes/Total number of outcomes
Given that, three softball players discussed their batting averages after a game.
Probability:
Player 1: 4/7 =0.57
Player 2: 5/8 =0.625
Player 3: 3/6 =0.5
Therefore, option B is the correct answer.
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6 1/2 divided by 1 2/3 as a mixed number in simplest form
The value of 6 1/2 divided by 1 2/3 is 3 9/10
How to determine the quotient of the numbersFrom the question, we have the following parameters that can be used in our computation:
6 1/2 divided by 1 2/3
When represented as an expression, we have
6 1/2 / 1 2/3
Convert to improper fractions
13/2 / 5/3
Express as products
13/2 * 3/5
So, we have
39/10
Express as mixed number
3 9/10
Hence, the solution is 3 9/10
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the bus has a maximum capacity of 30 passengers write as an inequality
The inequality of the statement is x ≤ 30
How to determine the inequalityFrom the question, we have the following parameters that can be used in our computation:
The bus has a maximum capacity of 30 passengers
Represent the bus capacity with x
This means that the value of x cannot exceed 30
The term cannot exceed implies less than or equal to
So, we have
x ≤ 30
Hence, the inequality is x ≤ 30
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Draw the graph of y=f(x) and y= 1/f(x) on the same axes.
f) f(x) = -(x+4)²+1
are there any specific steps to get the answers here?
Answer: To graph y = f(x) = -(x+4)²+1 and y = 1/f(x) on the same axes, you can follow these steps:
Plot the y-intercept of y = f(x) which is (0,-1)
For y = f(x), the parabola opens downward since the coefficient of x² is -1, thus the vertex is at (-4,1)
Plot the x-intercepts of y = f(x) which are x = 4 and x = -8
The parabola of y = f(x) is symmetric about the y-axis, so the graph will be reflected about the y-axis.
For y = 1/f(x), the x-intercepts will be the same as y = f(x) and the y-intercept is (0,1)
To graph y = 1/f(x), we can start by plotting the x-intercepts and the y-intercept, and then connecting the points.
Label the x-intercepts, y-intercepts, and vertex for both graphs
Shade the regions above the parabola of y = f(x) and below the graph of y = 1/f(x)
Note: It is important to remember that the graph of y = 1/f(x) is a reflection of the graph of y = f(x) across the x-axis, so the y-intercept of y = 1/f(x) is (0,1)
Step-by-step explanation:
GIVING 50 POINTS FOR THE RIGHT ANSWER!!!!
What is 34 + 4/9 x 1/2
Answer:
308/9
Step-by-step explanation:
Answer: 34 + 4/9 x 1/2 = 34 + 2/9 = 36/9.
Decimal Form: 34 + 4/9 x 1/2 in decimal form is 34.22222222.
Step-by-step explanation:
If a plane can travel 490 miles per hour with the wind and 430 miles per hour against the wind, find the speed of the plane in still air.
Answer:
460
Step-by-step explanation:
Let b = base speed and w = wind power.
Here's what we know:
b + w = 490
b - w = 430
From here, we can use the elimination method. If we treat each variable and constant as columns, we do the following:
b + b = 2b
w - w = 0
490 + 430 = 920
Leaving us with:
2b = 920
Divide by 2:
2b / 2 = b
920 / 2 = 460
So b = 460 mph
You read online that a 15 ft by 20 ft brick patio would cost about $2,275 to have professionally installed. Estimate the cost of having a 22 by 29 ft brick patio installed.
Answer:
1,547
Step-by-step explanation:
15' x 20' = 300 square feet
2,275 / 300 = $7.583 per square foot
12 x 17 = 204 square feet
204 x 7.583 = 1,547
If there are 24 apples and 12 bananas for a fruit basket, fill out all of the possible ratios of apples to bananas that could be made
Answer:
Step-by-step explanation:
for every 24 apples there is 12 bananas
so if you dived it by 2 it will be 12 apples 6 bananas
so it will always be half amount of bananas
Using trial and improvement, find the numbers that should replace A and B in the
rule for this sequence.
Rule:
Start at 6
Multiply by A then subtract B each time
6-8->
-36->
Using trial and improvement the numbers that should replace A and B is 20 and 34.
What is the trial and improvement process?If you are unable to solve a problem using standard algebraic techniques, try using the trial and improvement approach. If you want to get a C in math, you must comprehend this question because it typically is worth 3 marks.
According to the topic
6 = A * 2 - B = 2A - B
SZ = A - S - B = SA - B {Find the relationship}
:: 3A = 36 - 8 = 28
/ A = 28/3 = 9
; B = 2A - 8 = 18 - 8 = 10
the rule expression. 14n-10
: n = 3 , 10 . 3 - 10 = 20
n = 4 11 . 4 - 10 = 34
Using trial and improvement the numbers that should replace A and B is 20 and 34.
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Which equation has a solution of x= 3/4
The equation has a solution of x= 3/4 will be B. 8x = 6
How to calculate the equation?An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.
In this case, the value of x in 3x = 6 will be:
x = 6 / 3
x = 2
The value of x in 8x = 6 will be:
x = 6 / 8
x = 3 / 4
The value of x in 4x = 2 will be:
x = 2/4
x = 1/2
The value of x in 3x = 15 will be:
x = 15 / 3
x = 5
The correct option is B.
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Complete question
Which equation has a solution of x= 3/4?
A. 3x = 6
B. 8x = 6
C. 4x = 2
D. 3x = 15
The land area of Statesville, NC, is 20.7 square miles. If the entire US population (currently about 322.9 million) were put inside the city limits of Statesville, determine the number of square feet that each person would have. Round your answer to two decimal places. [Hint: Remember that a mile is 5,280 feet, so a square mile is (5,280 x 5,280) square feet.]
In the land area of Statesville, NC there would be 133.54 persons per square foot.
What is a unitary method?A unitary method is a mathematical way of obtaining the value of a single unit and then deriving any no. of given units by multiplying it with the single unit.
Given, One mile is 5280 feet so a square mile is
(5280 × 5280) = 27878400 square feet.
Therefore, The number of persons in each square foot should be,
= 322900000/27878400 persons per square foot.
= 3229000/278784 persons per square foot.
= 133.54 persons per square foot.
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on Saturday there was a -4 change in inventory. How many phones were In inventory on Sunday?
Answer:
14
Step-by-step explanation:
First we can find Wednesdays inventory by doing 16-5, giving us 11
We can do the same thing for Saturday. Doing -2+20 because the change was positive 20, not negative. That brings Saturdays inventory to 18
They said Saturdays change was -4
So taking the 18 and subtracting 4 gets you Sundays inventory of 14
Si el costo de 1 Kg de azucar es $8?cual es el precio de 3 3/4 kg?
Answer:
$30
Step-by-step explanation:
1kg= $8
3kg *$8= $24
3/4= 0.75
$8 *0.75= $6
$24+$6= $30
3.75 kg * $8= 30
* = multiply
Is this sequence arithmetic or geometric?
List at least 5 terms of the sequence
Write a recursive definition for the sequence
This formula states that each term in the sequence is equal to the previous term plus 3.
Is this sequence arithmetic or geometric?This sequence is arithmetic, because the difference between consecutive terms is consistent.The first five terms of the sequence are 1, 3, 5, 7, and 9.The recursive definition of the sequence is a(n) = a(n-1) + 2, where a(1) = 1.This sequence appears to be arithmetic, as the difference between consecutive terms is constant.This can be seen by looking at the graph, where the y-values of each point are regularly spaced apart.The first five terms of the sequence are 2, 5, 8, 11, and 14.A recursive definition of the sequence is a formula that can be used to generate subsequent terms of the sequence.In this case, the recursive definition of the sequence is a(n) = a(n-1) + 3, with a(1) = 2.For example, the fourth term in the sequence (11) is equal to the third term (8) plus 3, and the fifth term (14) is equal to the fourth term (11) plus 3.This sequence appears to be arithmetic, as each consecutive term increases by the same amount. The difference between each term is constant, so this is a defining characteristic of arithmetic sequences.The first five terms of this sequence are 0, 2, 4, 6, 8.\A recursive definition of this sequence is a formula that describes the relationship between each term and the preceding term. The recursive definition of this sequence is an_n = a_(n-1) + 2, where a_n is the nth term of the sequence and a_1 = 0. This definition states that each term is two more than the previous term.To learn more about the sequence refer to:
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Rework problem 27 from section 1.2 of your textbook, about the three subsets X1, X2, and X3 that partition a set X, except assume that the number of elements in X, is 4 times the number of elements in X2, the number of elements in X3 is 6 times the number of elements in X2, and n(X) = 99. (1) n(X1) = (2) n(X2) = (3) n(X3) =
The value of three subsets is [tex]n(x_{1})=36, n(x_{2})=9, & n\left(x_3\right)=54[/tex].
A set is a collection of objects or elements, grouped in the curly braces, such as {a, b, c, d}. If a set A is a collection of even number and set B consists of {2,4,6}, then B is said to be a subset of A, denoted by B⊆A and A is the superset of B.
The elements of sets could be anything such as a group of real numbers, variables, constants, whole numbers, etc. It consists of a null set as well.
If set A has {X, Y} and set B has {X, Y, Z}, then A is the subset of B because elements of A are also present in set B.
Let the number of elements in [tex]$x_2$[/tex] be [tex]$x$[/tex], then
[tex]$$\begin{aligned}& n\left(x_1\right)=4 x \\& n\left(x_2\right)=x \\& n\left(x_3\right)=6 x\end{aligned}$$[/tex]
Given that number of elements in [tex]$x=99$[/tex]
[tex]$$\begin{aligned}\Rightarrow & n\left(x_1\right)+n\left(x_2\right)+n\left(x_3\right)=99 \\\Rightarrow & 4 x+x+6 x=99 \\\Rightarrow & 11 x=99 \\\Rightarrow & x=9 \\\therefore & n\left(x_1\right)=4 x=4 \times 9=36 \\& n\left(x_2\right)=x=9 \\& n\left(x_3\right)=6 x=6 \times 9=54\end{aligned}$$[/tex]
Therefore, the value of [tex]& n\left(x_3\right)[/tex][tex]=54[/tex].
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at a small coffee shop, the distribution of the number of seconds it takes for a cashier to process an order is approximately normal with mean 276 seconds and standard deviation 38 seconds. which of the following is closest to the proportion of orders that are processed in less than 240 seconds? following is closest to the proportion of orders that are processed in less than 240 seconds?
А
0.17
B
0.25
0.36
D
083
E
0.95
As per the interpolation method, the proportion of orders that are processed in less than 240 seconds is 0.17
The interpolation method is defined as a method of deriving a simple function from the given discrete data set such that the function passes through the provided data points.
Here we have given that at a small coffee shop, the distribution of the number of seconds it takes for a cashier to process an order is approximately normal with mean 276 seconds and standard deviation 38 seconds.
And the proportion of orders that are processed in less than 240 seconds is calculated as,
Here we know that 1 standard deviation below mean = 16% = M - d
Then we apply the value on it, then we get,
=> 276 s - 38 s = 238 s
Here we know that the mean of distribution is,
=> 50% which is equal to 276 s
And the proportion less than 240 s is consider that x%, then by using the interpolation method it can be calculated as,
=> (240 - 238) / (276 - 238) = (x - 16) / (50 - 16)
When we simplify this one then we get,
=> 1/19 = x - 16 / 34
Which can be rewritten as,
=> 19(x - 16) = 34
=> 19x - 304 = 34
=> 19x = 338
=> x = 17.78%.
Therefore the closest to the proportion of orders that are processed in less than 240 seconds is 17% that is equal to the value of 0.17.
Hence the correct option is (A).
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Which graph shows the new image of the rectangle after a translation of two units to the left?
The new image of the rectangle after a translation of two units to the left is of option D
How to draw a shifted figure if the shifts are provided?Suppose the graph is drawn on the coordinate plane.
Let the shifting be (x,y) → (x+a, y+b)
Then, add 'a' to all x coordinates of the graph's points. Add 'b' to all y coordinates of the graph's points.
The resultant set of new points will be plotted.
Given;
A rectangle area on graph with coordinates (1,0), (6,0),(6,2), (1,2)
Now after the shift of 2 unit left, the x coordinates will be affected;
=(1-2,0), (6-2,0),(6-2,2), (1-2,2)
=(-1,0), (4,0),(4,2), (-1,2)
Therefore, the coordinates of shifted graph will be (-1,0), (4,0),(4,2), (-1,2)
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Question 23 of 41
Which of the following is equivalent to the product below?
√9.√5√2
OA. 18√5
OB. 10√3
OC. 3√10
OD. 5√18
On the coordinate plane, the segment from D(-6, 3) to E(2, 3) forms one side of ADEF. The
triangle has an area of 28 square units. Select all of the points where F could be.
(5,-4)
(1, 10)
(7,4)
(-6, 10)
On the coordinate plane, the segment from D(-6, 3) to E(2, 3) forms one side of ADEF. The possible point F are (5,-4), (1, 10), (7,4), (-6, 10).
How many points does a F earn?Since we are aware that the triangle's area is 28 square units and that the base is the segment DE, which has a length of 8 units, we can construct the equation:
Area equals (1/2) × base × height = (1/2) × 8 × height=28
We can calculate the height and discover that it is 7 units.
Point F must be situated 7 units above or below the line DE since the triangle's height is perpendicular to the base DE. Therefore, any position that is 7 units above or below the line DE, which is determined by the equation y = 3, is considered to be at point F.
So the possible point F are (5,-4), (1, 10), (7,4), (-6, 10)
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Complete question -
On the coordinate plane, the segment from D(-6, 3) to E(2, 3) forms one side of ADEF. The
triangle has an area of 28 square units. Select all of the points where F could be. (5,-4), (1, 10), (7,4),(-6, 10)
Can someone please help me
The cost of each shirt is $6 and the cost of each pant for the given condition is $9.
What is elimination method?The elimination approach involves taking one variable out of the system of linear equations by utilising addition or subtraction together with multiplication or division of the variable coefficients.
Let us suppose the cost of shirt = x.
Let us suppose the cost of pant = y.
Given that 3 shirts and 3 pants cost $45, this is represented as:
3x + 3y = 45 (equation 1)
1 shirt and 2 pants cost $24, this is represented as:
x + 2y = 24 (equation 2)
Using the elimination method to solve:
Multiply the second equation with 3 and change the signs of all the variables.
(x + 2y = 24) (3)
-3x - 6y = -72 (equation 3)
3x + 3y = 45
Comparing equation 1 and equation 3, the x term is cancelled, and the remaining equation after subtracting both equations is:
- 3y = -27
y = 9
Substitute the value of y in equation 2.
x + 2y = 24
x + 2(9) = 24
x + 18 = 24
x = 24 -18
x = 6
Hence, the cost of each shirt is $6 and the cost of each pant is $9.
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what is the y-interpect on this graph
Answer:
16
Step-by-step explanation:
It's the number where the line crosses the y-axis
It is a cold, dark, and stormy winter morning. Your boss has called a 5 a.m. meeting before she leaves to catch a flight to the Bahamas for her vacation, and there is a power outage. Your alarm does not go off and you sit up in bed with a fright at what you guess is about 4:45 a.m. You are trying frantically to get dressed and you need to reach into your sock drawer to get two matching socks in the dark. Naturally, your mind wanders to solving probability equations. Your sock drawer contains 20 total socks which consist of 10 matched pairs of socks. All of them are only black or white and they are loose, disorganized randomly, and not bound together. This means you have 10 black socks and 10 white socks in your disorganized drawer. You do not have a light source because your phone did not charge and you cannot find a flashlight or a candle. You are completely in the dark! You have to get dressed and go! You have to math fast! 1. Think for a minute and describe your optimal strategy for solving this problem in 50 words or less. 2. Let’s say for a moment that all 20 socks are different colors meaning that none match. What is the total number of different ways they could be combined in pairs? 3. Now consider all the socks are only black and white. What is the total number of different ways they could be combined in pairs? (Suggestion: You might consider listing the sample space of all possible outcomes.) For the remaining problems, let’s go back to the assumption that there are 20 total socks, 10 white and 10 black. This makes 10 total matching pairs of 5 pair of white and 5 pair of black.
1. Optimal strategy for solving this problem would be to feel for the socks, and grab two socks of the same color at random, as they have a 50% chance of being a match.
2. The total number of different ways they could be combined in pairs, assuming all 20 socks are different colors, is calculated by using the formula (nCr) = n! / (r! * (n-r)!)
So, the total number of different ways they could be combined in pairs would be (20*19)/2 = 190.
3. The total number of different ways they could be combined in pairs would be 10, as there are 10 black and 10 white socks, and they can only be paired with one another.
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10 pts.!!!!!
You select a marble without looking and then put it back. If you do this 7 times, what is the best prediction possible for the number of times you will pick a marble that is not orange?
ASAPPPPPPPPP PLSSS ITS GRADED
The best prediction possible for the number of times you will pick a marble that is not orange, is 7 times.
What is the probability?The Probability in mathematics is the possibility of an event in time. In simple words, how many times that incident is happening in any given time interval.
Given:
There are 4 pink marbles and 3 orange marbles.
The probability of not picking orange marbles,
= 4/(3 + 4)
= 4/7
If you do this 7 times,
the best prediction possible for the number of times you will pick a marble that is not orange,
= 4/7 x 7
= 4 times
Therefore, number of times is 4.
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The complete question:
There are 4 pink marbles and 3 orange marbles.
You select a marble without looking and then put it back. If you do this 7 times, what is the best prediction possible for the number of times you will pick a marble that is not orange?
Please help me find each measure!! Thank you
There exist two complex numbers c, say c1 and c2, so that 2+2i, 5+i, and c form the vertices of an equilateral triangle. Find the product c1c2 .enter image description here
So far, I have used the distance formula to get (a−2)2+(b−2)2=(a−5)2+(b−1)2 for c1(a,b). I expanded and simolified this to get b=3a−9. What do I do next?
The product of the two complex number c₁c₂ is thus 10 + 9i.
Let c₁ = a + ib and c₂ = c + id. Also z₁ represent 2 + 2i and z₂ represent 5 + i. The length of side of the equilateral triangle formed can be obtained by
|z₁ - z₂| = r = √((5 - 2)² + (1 - 2)² = √10
a² + b² - 4a - 4b + 8 = 10
a² + b² - 10a - 2b + 26 = 10
So 6a - 2b = 18 that is 3a - b = 9
Thus b = 3a - 9
Similarly c² + d² - 4c - 4d + 8 = 10
c² + d² - 10c - 2d + 26 = 10
So 3c - d = 9, that is d = 3c - 9
Midpoint of line joining c₁ and c₂ is same as that of z₁ and z₂, so
(c + a)/2 = ( 5 + 2)/ 2 and (d + b)/2 = (1 + 2)/2
So midpoint is 3.5 + 1.5i
Altitude of equilateral triangle of side a is √3a/2
That is √(a - 3.5)² + (b - 1.5)² = √3×√10/2
√(a - 3.5)² + (3a - 9 - 1.5)² = √30 /2
a² - 7a + 12.25 + 9a² - 63a + 110.25 = 30/4
On solving a = (7 ± √3)/ 2
So a = (7 - √3)/ 2, c = (7 + √3)/ 2
b = 3a - 9 = (3 - 3√3)/ 2, d = (3 + 3√3)/2
So c
c₁ = (7 - √3)/2 + i (3 - 3√3)/ 2
c₂ = (7 + √3)/2 + i (3 + 3√3)/ 2
So the product c₁c₂ is
= ((7 - √3)/ 2) * ((7 + √3)/ 2) - ((3 - √3)/ 2) * ((3 + √3)/ 2) + i (((7 - √3)/ 2) * ((3 + √3)/ 2) + ((7 + √3)/ 2) * ((3 - √3)/ 2))
= 10 + 9i
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