8(11 + 2r) = 126r + 3
first open the parenthesis
88 + 16r = 126r + 3
88 - 3 = 126r - 16 r
85 = 110r
divide both-side of the equation by 110
85/110 = r
r= 17/22
2. The Venn diagram shows the sets U, X and Y.UXY.34 246..9.512:31List the elements of the following sets:(a) X(b) Y(c) U(d) XUY(e) XnY(g) X\Y(h) Y\X(f) X'(1) (XY)2:31
Given the Venn diagram in the question, we can proceed to answer the questions as follow
[tex]\begin{gathered} X=\text{members of the subset X} \\ This\text{ gives: 1,2,3,4, and 5} \end{gathered}[/tex][tex]\begin{gathered} QuestionA\text{ } \\ X=1,2,3,4,and\text{ 5} \\ \end{gathered}[/tex]Question B
Y= members of subset Y
Y =2,4,6, and 8
Question C
U means that we should list all elements in the universal set
U = ALL members of the set
U = 1,2,3,4,5,6,7,8, and 9
Question D
This is the union of both sets X and Y. This means we will list all the members that are found in the 2 subsets
[tex]\text{XUY}=1,2,3,4,5,6,\text{ and 8}[/tex]Question E
[tex]\begin{gathered} \text{XnY means we are to find the elements that are common to both X and Y} \\ \text{XnY}=2\text{ and 4} \end{gathered}[/tex]Question F
X' means that we should find all members of the set except that of X
[tex]X^{\prime}=6,7,8,\text{ and 9}[/tex]Question G
X\Y means that we should list the elements of X that are not found in Y
X\Y= 1,3, and 5
Question H
Y\X means that we should list the elements of Y that are not found in X
Y\X= 6, and 7
Question I
To solve (XnY)' we will follow the steps below
Step 1: Find (XnY)
[tex]\text{XnY}=2\text{ and 4}[/tex]Step 2: Find (XnY)'
[tex]We\text{ will list all elements aside (XnY)}[/tex][tex](XnY)^{^{\prime}}\Rightarrow1,3,5,6,7,8,\text{and 9}[/tex]
there are about 6*10^24 molecules in a litre of water. it is estimated that a person drinks about 2.2 *10^3 litres of water a year. how many molecules of water does a person drink in a year?
As per the concept of multiplication, the amount of molecules of water does a person drink in a year is 13.2 x ²⁷ or 1.32 x 10²⁸.
Molecules:
Molecules are referred as the smallest particle of a substance that has all of the physical and chemical properties of that substance. It is made up of one or more atoms.
Given,
There are about 6 x 10²⁴ molecules in a liter of water. it is estimated that a person drinks about 2.2 x 10³ liters of water a year.
Here we need to find the amount of molecules of water does a person drink in a year.
To calculate the total amount of molecules consumption for the year we have to use the following formula,
That is,
Total molecules per year = amount of water per year x molecules in water.
Here we know that,
the amount of water consumption per year = 2.2 x 10³ liters
And the amount of molecules of one liter water = 6 x 10²⁴
When we apply the values on the formula, then we get,
=> total amount of molecules consumption = (2.2 x 10³) x (6 x 10²⁴)
=> (2.2 x 6) x (10³ x 10²⁴)
=> 13.2 x 10³⁺²⁴
=> 13.2 x ²⁷
Therefore, the amount of molecules of water does a person drink in a year is 13.2 x ²⁷ or 1.32 x 10²⁸.
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208 x 26 using long multiplication
Answer:
2 0 8
× 2 6
+ 1 2 4 8
+ 4 1 6
= 5 4 0 8
The Answer of 208 × 26 Is 5.408
Explanation.= 208 × 26
= (208 × 6) + (208 × 20)
= 1.248 + 4.160
= 5.408
__________________
Class: Elementary School
Lesson: Multiplication
[tex]\boxed{ \colorbox{lightblue}{ \sf{ \color{blue}{ Answer By\:CyberPresents}}}}[/tex]
Write 5^-15 with a positive exponent
Given:
[tex]5^{-15}[/tex]To change a negative exponent to a positive exponent, the variable will change from numerator to denominator and vice versa.
For example:
[tex]\begin{gathered} P^{-1}\text{ = }\frac{1}{P} \\ \\ We\text{ know that:} \\ 5^{-15}=5^{(15)-1} \\ \\ 5^{(15)-1}\text{ = }\frac{1}{5^{15}} \end{gathered}[/tex]Therefore, we have:
[tex]5^{-15}\text{ = }\frac{1}{5^{15}}[/tex]ANSWER:
[tex]\frac{1}{5^{15}}[/tex]Business Mathematics question
Number system depends on two basic concepts are Binary and Decimal.
Given the statement is :
Number System depends on two basic concept.
Let's know the definition of number system:
What is Number System?
A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, multiplication and division.
Hence, Number system depends on two basic concepts are Binary and Decimal.
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The volume of a right circular cylinder with a radius of 4 in. and a height of 12 in. is ___ π in^3.
For the given right cylinder:
Radius = r = 4 in
Height = h = 12 in
The volume of the cylinder =
[tex]\pi\cdot r^2\cdot h=\pi\cdot4^2\cdot12=192\pi[/tex]So, the answer will be the volume is 192π in^3
Does the point (3,-1) lie on the circle (x + 1)2 + (y - 1)1)2 = 16?no; the point is not represented by (h, k) in the equationyes; when you plug the point in for x and y you get a true statementno; when you plug in the point for x and y in the equation, you do not get a trueyes; the point is represented by (h, k) in the equation
We are given an equation of a circle and a point. We are then asked to find if the point lies on the circle. The equation of the circle and the point is given below
[tex]\begin{gathered} \text{Equation of the circle} \\ (x+1)^2+(y-1)^2=16 \\ \text{Given point =(3,-1)} \end{gathered}[/tex]To find if the point lies in the circle, we can use the simple method of substituting the coordinates into the equation of the circle.
This can be seen below:
[tex]\begin{gathered} (3+1)^2+(-1-1)^2=16 \\ 4^2+(-2)^2=16 \\ 16+4=16 \\ \therefore20\ne16 \end{gathered}[/tex]Since 20 cannot be equal to 16, this implies that the point does not lie on the circle.
ANSWER: Option 3
How do you slove this promblem 207.4÷61
we have
207.4÷61
[tex]207.4\div61=\frac{207.4}{61}=\frac{2,074}{610}=\frac{1,830}{610}+\frac{244}{610}=3+\frac{244}{610}=3\frac{244}{610}[/tex]simplify
244/610=122/305=4/10=2/5
therefore
the answer is 3 2/5Find the equation of the line. Use exortumbers. st V = 2+ 9 8- 6+ 5+ -4 3+ 2+ 1+ T + -9-8-7-65 2 3 5 6 7 8 9 4 -3 -2 -2 + -3+ -4+ -5* -6+ -7+ -8+
We can see that the line passes by the points (0, -5) & (5, 0), using this information we proceed as follows:
1st: We find the slope(m):
[tex]m=\frac{0+5}{5-0}\Rightarrow m=1[/tex]2nd: We use one of the points from the line and the slope to replace in the following expression:
[tex]y-y_1=m(x-x_1)[/tex]That is (Using point (0, -5):
[tex]y+5=1(x-0)[/tex]Now, we solve for y:
[tex]\Rightarrow y=x-5[/tex]And that is the equation of the line shown.
A machine that makes
toy spinners operates for 8 hours each
day. The machine makes 7,829 toy
spinners in
day. About how
many toy
spinners does the machine make each
hour?
Using the unitary method, the number of toy spinners the machines will make in an hour is 2069.
The unitary approach is a strategy for problem-solving that involves first determining the value of a single unit, then multiplying that value to determine the required value.
A machine makes 7829 toy spinners in a day.
The machines operate for 8 hours each day to make the toy spinners.
So,
8 hours = 7829
Then by using the unitary method the number of toy spinners the machines will make each hour will be:
8 hours = 7829
24 hours = x toy spinner
Toys in one hour = ( 7829/ 24 ) × 8
Toys in one hour = 326.20833 × 8
Toys in one hour = 2609.6667
Toys in one hour = 2069
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about 23% of people are at a higher risk of stroke due to other medical conditions like high blood pressure. their risk is about 9% of stroke compared with the general population's 3% chance of having a stroke in their lifetime.
Given a family with four children, find the probability of the event. All are boys. The probability that all are boys
Answer:
0.0625
Explanation:
The number of children in the family = 4
The possible combination of genders:
[tex]|\Omega|=2^4=16[/tex]The event that all are boys, |A|=1
Therefore, the probability that all are boys:
[tex]\begin{gathered} P(A)=\frac{1}{16} \\ =0.0625 \end{gathered}[/tex]Subtract the following polynomial. Once simplified, name the resulting polynomial. 5.) (10x² + 8x - 7) - (6x^2 + 4x + 5)
The given polynomial expression: (10x² + 8x - 7) - (6x^2 + 4x + 5)
[tex]\begin{gathered} (10x^2+8x-7)-(6x^2+4x+5) \\ \text{Open the brackets:} \\ (10x^2+8x-7)-(6x^2+4x+5)=10x^2+8x-7-6x^2-4x-5 \\ \text{Arrange the like term together:} \\ (10x^2+8x-7)-(6x^2+4x+5)=10x^2-6x^2+8x-4x-7-5 \\ \text{Simplify the like terms together:} \\ (10x^2+8x-7)-(6x^2+4x+5)=4x^2+4x-12 \end{gathered}[/tex]The resulting polynomial be:
[tex](10x^2+8x-7)-(6x^2+4x+5)=4x^2+4x-12[/tex]The highest degree of the polynomial is 2 so, the polynomial is Quadratic polynomial
Answer: 4x^2 + 4x - 12, Quadratic polynomial
Determine whether point (4, -3) lies on the line with equation y = -2x + 5 by using substitution and by graphing.
The equation of the line is:
[tex]y=-2x+5[/tex]And we need to find if the point (4,-3) lies on the line.
To solve by using substitution, we need to remember that an ordered pair always has the form (x, y) --> the first number is the x-value, and the second number is the y-value.
In this case (4,-3):
x=4
and
y=-3
So now, we substitute the x value into the equation, and if the y-value we get in return is -3 --> the point lies on the line. If not, the point does not lie on the line.
[tex]\begin{gathered} y=-2x+5 \\ \text{Substituting x=4} \\ y=-2(4)+5 \\ y=-8+5 \\ y=-3 \end{gathered}[/tex]We do get -3 as the y-value which matches with the indicated y value of the point.
Thus, by substitution, we confirm that the point lies on the line.
To check the result by graphing, we need to graph the line. The graph of the line is shown in the following image:
In the image, we can see that the marked point on the line is the point we were looking for (4,-3). So we confirm by the graphing method that the point lies on the line.
Answer: It is confirmed by substitution and graphing that the point (4,-3) lies on the line.
) - At a farming supply store 7 pounds of seed cost $141.96. If a farmer needed 4 pounds ofseeds, how much would it cost him?
Hello
From the question, we know that 7 pounds of the seeds cost $141.96.
4 pounds would be assumed to be x and we can solve for x.
[tex]\begin{gathered} 7\text{ pounds = 141.96} \\ 4\text{ pounds = x} \end{gathered}[/tex]Cross multiply both sides.
[tex]\begin{gathered} 7\times x=4\times141.96 \\ 7x=567.84 \end{gathered}[/tex]Divide both sides by the coefficient of x
[tex]\begin{gathered} 7x=567.84 \\ \frac{7x}{7}=\frac{567.84}{7} \\ x=81.12 \end{gathered}[/tex]From the calculation above, the cost of 4 pounds of the seeds is equal to $81.12
What is the measure of the angle at the bottom of home plate?
We will ave the following:
*First: We will determine the sum of all internal angles of the polygon:
[tex](n-2)\cdot180\Rightarrow(5-2)\cdot180=3\cdot180[/tex][tex]=540[/tex]*Second: Now, that we know that the sum of all internal angles will be 540°, the following is true:
[tex]90+90+135+135+\alpha=540[/tex]Now, we solve for alpha [The angle]:
[tex]\Rightarrow\alpha=540-135-135-90-90\Rightarrow\alpha=90[/tex]So, the measure of the angle at the bottom is 90°.
Divide 8 1/8 by 7 1/12 simplify the answer and write as a mixed number
The division of 8 1/8 by 7 1/12 is 91/136.
What is division?Division simply has to do with reduction of a number into different parts. On the other hand, a mixed number is the number that's made up of whole number and fraction.
Dividing 8 1/8 by 7 1/12 will go thus:
8 1/8 ÷ 7 1/12
Change to improper fraction
65/8 ÷ 85/7
= 65/8 × 7/85
= 91/136
The division will give a value of 91/136.
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A randomly generated list of integers from 0 to 7 is being used to simulate an event, with the numbers 0, 1, 2, and 3 representing a success. What is the estimated probability of a success? O A. 25% OB. 50% O C. 80% O D. 43%
The total numbers of integers used is 7 + 1 = 8 (since we are using ubtewgers from 0 to 7).
If 0, 1, 2 and 3 represents succes, we have 4 integers of the 8 total that are success, thus, the theoretical probability is:
[tex]P=\frac{4}{8}=0.5[/tex]So the probability os success is 0.50 = 50%.
The following relation defines y as a one-to-one function of x x y3.0 7.45-8.4 -8.072.4 -9.16-1.5 7.45TrueFalse
One-to-one functions are the ones that each value of "y" is related to only one value of "x". So we need to check in the provided values if that applies.
We have a group of 4 different values of "y". For these the value y = 7.45 is related to the x values of 3 and -1.5, therefore it is not a one-to-one function.
six teachers share 4 packs of paper equally.how much paper does each teacher get
Six teachers share 4 packs of paper
Each teacher gets
[tex]\frac{4}{6}=\frac{2}{3}\text{ = 4 sixths of a pack, option C}[/tex]The answer is option C
A rectangle has a diagonal of length 10 cm and a base of length 8 cm . Find its height
Given:
The length of diagonal of rectangle is d = 10 cm.
The length of base is b = 8 cm.
Explanation:
The relation between length, height and diagonal of rectangle is given by pythagoras theorem. So
[tex]d^2=l^2+h^2[/tex]Substitute the values in the equation to obtain the value of h.
[tex]\begin{gathered} (10)^2=(8)^2+h^2 \\ 100=64+h^2 \\ h=\sqrt[]{100-64} \\ =\sqrt[]{36} \\ =6 \end{gathered}[/tex]So the height of rectangle is 6 cm.
Answer: 6 cm
A population of a certain species of bird is 22,000 animals and is decreasing by 700 birds per year..Write an equation for y, the population at time t (in years), representing the situation.y= How many birds are in the population after 7 years?
We can solve that problem using a linear function, we know that
[tex]y=mx+y_0[/tex]Where y0 is the initial population and m is the rate of decreasing, we know that for each "x" years we have -700 birds, therefore
[tex]y=-700x+22000[/tex]Let's use t instead of x
[tex]y=-700t+22000[/tex]That's the equation that represents the population at time t
[tex]\begin{gathered} y=-700t+22000\text{ \lparen t = 7\rparen} \\ \\ y=-700\cdot(7)+22000 \\ \\ y=-4900+22000 \\ \\ y=17100 \end{gathered}[/tex]Therefore after 7 years, the population will be 17100 birds.
Which value of x makes the equation true 3x-6/3= 7x-3/6
The value of x that makes the equation true is - 3 / 8.
How to solve equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Therefore, the value of x that makes the equation true is the value that makes the two sides of the equation equal.
Hence,
3x - 6 / 3 = 7x - 3 / 6
3x - 2 = 7x - 1 / 2
add 2 to both sides of the equation
3x - 2 = 7x - 1 / 2
3x - 2 + 2 = 7x - 1 / 2 + 2
3x = 7x - 1 / 2 + 2
3x = 7x + 3 / 2
subtract 7x from both side of the equation
3x - 7x = 7x - 7x + 3 / 2
- 4x = 3 / 2
cross multiply
- 8x = 3
x = - 3 / 8
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The value of x which makes the equation true is - 3 / 8.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given equation is 3x-6/3= 7x-3/6
Three x minus six divided by three equal to seven times of x minus three divided by six
3x-6/3= 7x-3/6
(9x-6)/3=(42x-3)/6
Apply cross multiplication
6(9x-6)=3(42x-3)
Apply distributive property
54x-36=126x-9
add 36 on both sides
54x=126x-9+36
54x=126x+27
-27=126x-54x
-27=72x
x=-27/72
x=-9/24=-3/8
Hence value of x is -3/8 for equation 3x-6/3= 7x-3/6.
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I need to find out what sine cosine and cotangent is, if this is my reference angle in the picture
We will use the following trigonometric identities
[tex]\begin{gathered} \tan \Theta=\frac{sin\Theta}{\cos \Theta} \\ \cot \Theta=\frac{1}{\tan \Theta} \end{gathered}[/tex]Using these identities we can identify
[tex]\begin{gathered} \tan \Theta=\frac{12}{5} \\ \sin \Theta=12 \\ \cos \Theta=5 \\ \cot \Theta=\frac{1}{\frac{12}{5}}=\frac{5}{12} \end{gathered}[/tex][tex]\begin{gathered} \Theta=\tan ^{-1}(2.4) \\ \Theta=67.38º \\ \sin \Theta=0.92 \\ \cos \Theta=0.38 \end{gathered}[/tex][tex]\begin{gathered} \tan \Theta=\frac{opposite}{\text{adjacent}}^{} \\ \text{opposite}=12 \\ \text{adjacent}=5 \\ \text{hippotenuse=}\sqrt[\square]{12^2+5^2} \\ \text{hippotenuse=}13 \end{gathered}[/tex][tex]\begin{gathered} \sin \Theta=\frac{12}{13} \\ \cos \Theta=\frac{5}{13} \end{gathered}[/tex]2. The area A of a rectangle is represented by the formula A = Lw, where Lis the length and wis the width. The length of the rectangle is 5. Write anequation that makes it easy to find the width of the rectangle if we knowthe area and the length.
1) Considering that the Area of a rectangle is given as:
[tex]A=lw[/tex]2) We can then write the following equation plugging into that the length= 5.
Say the area is "A", then we can find the width this way:
[tex]\begin{gathered} A=5ww \\ 5w=A \\ \frac{5w}{5}=\frac{A}{5} \\ w=\frac{A}{5} \end{gathered}[/tex]Note that we rewrote that to solve it for w (width).
All we need is to plug into the A the quantity of the area of this rectangle
Thus, the answer is w=A/5
I would like to know how to solve this answer.
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
k > 0
k * v
Step 02:
Scalars and Vectors:
k = scalar
v = vector
Scalar multiplication of a real vector by a positive real number multiplies the vector's magnitude, without changing its direction.
k * v
The answer is:
k v is parallel and has same direction as v
5. The number of hours spent in an airplane on a single flight is recordedon a dot plot. The mean is 5 hours. The median is 4 hours. The IQR is 3hours. The value 26 hours is an outlier that should not have been includedin the data. When 26 is removed from the data set, calculate the following(some values may not be used):*H0 2 4 6 8 10 12 14 16 18 20 22 24 26 28number of hours spent in an airplane1.4 hours1.5 hours3 hours3.5 hoursWhat is themean?OWhat is themedian?оOOWhat is the IQR?OOOO
Solution
Since the outlier that is 26 has been removed
We will work with the remaining
Where X denotes the number of hours, and f represent the frequency corresponding to eaxh hours
We find the mean
The mean (X bar) is given by
[tex]\begin{gathered} mean=\frac{\Sigma fx}{\Sigma f} \\ mean=\frac{1(2)+2(2)+3(3)+4(3)+5(2)+6(2)}{2+2+3+3+2+2} \\ mean=\frac{2+4+9+12+10+12}{2+2+3+3+2+2} \\ mean=\frac{49}{14} \\ mean=\frac{7}{2} \\ mean=3.5 \end{gathered}[/tex]We now find the median
Median is the middle number
Since the total frequency is 14
The median will be on the 7th and 8th term in ascending order
[tex]\begin{gathered} median=\frac{7th+8th}{2} \\ median=\frac{3+4}{2} \\ median=\frac{7}{2} \\ median=3.5 \end{gathered}[/tex]Lastly, we will find the interquartile range
The formula is given by
[tex]IQR=Q_3-Q_1[/tex]Where
[tex]\begin{gathered} Q_3=\frac{3}{4}(n+1)th\text{ term} \\ Q_1=\frac{1}{4}(n+1)th\text{ term} \end{gathered}[/tex]We calculate for Q1 and Q3
[tex]\begin{gathered} Q_1=\frac{1}{4}(n+1)th\text{ term} \\ \text{n is the total frequency} \\ n=14 \\ Q_1=\frac{1}{4}(14+1)th\text{ term} \\ Q_1=\frac{1}{4}(15)th\text{ term} \\ Q_1=3.75th\text{ term} \\ Q_1\text{ falls betwe}en\text{ the frequency 3 and 4 in ascending order} \\ \text{From the table above} \\ Q_1=2 \end{gathered}[/tex][tex]\begin{gathered} Q_3=\frac{3}{4}(n+1)th\text{ term} \\ Q_3=\frac{3}{4}(14+1)th\text{ term} \\ Q_3=\frac{3}{4}(15)th\text{ term} \\ Q_3=11.25th\text{ term} \\ \text{From the table above} \\ Q_3=5 \end{gathered}[/tex]Therefore, the IQR is
[tex]\begin{gathered} IQR=Q_3-Q_1 \\ IQR=5-2 \\ IQR=3 \end{gathered}[/tex]Today's high temperature is 72°F. If yesterday's high temperature was 87"F, what was the change in high temperatures?
To find the change in temperature we have to do a subtraction:
Highest temperature-lowest temperature-
87-72= 15F
The change in high temperatures was 15 F .
help meeeeeeeeee pleaseee !!!!!
The solution to the composite function is as follows;
(f + g)(x) = x² + 3x + 5(f - g)(x) = x² - 3x + 5(f. g)(x) = 3x³ + 15x(f / g)(x) = x² + 5 / 3xHow to solve composite function?The composition of a function is an operation where two functions say f and g generate a new function say h in such a way that h(x) = g(f(x)).
If we are given two functions, it is possible to create or generate a “new” function by composing one into the other.
Composite functions are when the output of one function is used as the input of another.
In other words, a composite function is generally a function that is written inside another function.
Therefore,
f(x) = x² + 5
g(x) = 3x
Hence, the composite function can be solved as follows:
(f + g)(x) = f(x) + g(x) = x² + 5 + 3x = x² + 3x + 5
(f - g)(x) = f(x) - g(x) = x² + 5 - 3x = x² - 3x + 5
(f. g)(x) = f(x) . g(x) = (x² + 5)(3x) = 3x³ + 15x
(f / g)(x) = f(x) / g(x) = x² + 5 / 3x
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Find the coordinates of the circumcenter of triangle PQR with vertices P(-2,5) Q(4,1) and R(-2,-3)
The given triangle has vertices at:
[tex]\begin{gathered} P(-2,5) \\ Q(4,1) \\ R(-2,-3) \end{gathered}[/tex]In the coordinate plane, the triangle looks like this:
There are different forms to find the circumcenter, we are going to use the midpoint formula:
[tex]M(x,y)=(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]Apply this formula for each vertice and find the midpoint:
[tex]M_{P,Q}=(\frac{-2+4}{2},\frac{5+1}{2})=(1,3)[/tex]For QR:
[tex]M_{Q,R}=(\frac{4+(-2)}{2},\frac{1+(-3)}{2})=(1,-1)[/tex]For PR:
[tex]M_{P,R}=(\frac{-2+(-2)}{2},\frac{5+(-3)}{2})=(-2,1)[/tex]Now, we need to find the slope for any of the line segments, for example, PQ:
We can apply the slope formula:
[tex]m=\frac{y2-y1}{x2-x1}=\frac{1-5}{4-(-2)}=\frac{-4}{6}=-\frac{2}{3}[/tex]By using the midpoint and the slope of the perpendicular line, find out the equation of the perpendicular bisector line, The slope of the perpendicular line is given by the formula:
[tex]\begin{gathered} m1\cdot m2=-1 \\ m2=-\frac{1}{m1} \\ m2=-\frac{1}{-\frac{2}{3}}=\frac{3}{2}_{} \end{gathered}[/tex]The slope-intercept form of the equation is y=mx+b. Replace the slope of the perpendicular bisector and the coordinates of the midpoint to find b:
[tex]\begin{gathered} 3=\frac{3}{2}\cdot1+b \\ 3-\frac{3}{2}=b \\ b=\frac{3\cdot2-1\cdot3}{2}=\frac{6-3}{2} \\ b=\frac{3}{2} \end{gathered}[/tex]Thus, the equation of the perpendicular bisector of PQ is:
[tex]y=\frac{3}{2}x+\frac{3}{2}[/tex]If we graph this bisector over the triangle we obtain:
Now, let's find the slope of the line segment QR:
[tex]m=\frac{-3-1}{-2-4}=\frac{-4}{-6}=\frac{2}{3}[/tex]The slope of the perpendicular bisector is:
[tex]m2=-\frac{1}{m1}=-\frac{1}{\frac{2}{3}}=-\frac{3}{2}[/tex]Let's find the slope-intercept equation of this bisector:
[tex]\begin{gathered} -1=-\frac{3}{2}\cdot1+b \\ -1+\frac{3}{2}=b \\ b=\frac{-1\cdot2+1\cdot3}{2}=\frac{-2+3}{2} \\ b=\frac{1}{2} \end{gathered}[/tex]Thus, the equation is:
[tex]y=-\frac{3}{2}x+\frac{1}{2}[/tex]This bisector in the graph looks like this:
Now, to find the circumcenter we have to equal both equations, and solve for x:
[tex]\begin{gathered} \frac{3}{2}x+\frac{3}{2}=-\frac{3}{2}x+\frac{1}{2} \\ \text{Add 3/2x to both sides} \\ \frac{3}{2}x+\frac{3}{2}+\frac{3}{2}x=-\frac{3}{2}x+\frac{1}{2}+\frac{3}{2}x \\ \frac{6}{2}x+\frac{3}{2}=\frac{1}{2} \\ \text{Subtract 3/2 from both sides} \\ \frac{6}{2}x+\frac{3}{2}-\frac{3}{2}=\frac{1}{2}-\frac{3}{2} \\ \frac{6}{2}x=-\frac{2}{2} \\ 3x=-1 \\ x=-\frac{1}{3} \end{gathered}[/tex]Now replace x in one of the equations and solve for y:
[tex]\begin{gathered} y=-\frac{3}{2}\cdot(-\frac{1}{3})+\frac{1}{2} \\ y=\frac{1}{2}+\frac{1}{2} \\ y=1 \end{gathered}[/tex]The coordinates of the circumcenter are: (-1/3,1).
In the graph it is: