Answer:
450
Step-by-step explanation:
270/3=90, 90*5=450
Create a pattern for the rule a +4.
As you can see, each number in the pattern is obtained by adding 4 to the previous number.
What is equation?An equation is a mathematical statement that shows the equality of two expressions, usually with an equal sign "=" in between them. An equation can contain variables, constants, and operators. The variables are represented by letters and can take on different values, while constants are fixed values that do not change. Operators include mathematical symbols like plus, minus, multiplication, and division, as well as exponents and logarithms.
Here,
Sure, here's a pattern for the rule a + 4:
a = 1: 1 + 4 = 5
a = 2: 2 + 4 = 6
a = 3: 3 + 4 = 7
a = 4: 4 + 4 = 8
a = 5: 5 + 4 = 9
a = 6: 6 + 4 = 10
a = 7: 7 + 4 = 11
a = 8: 8 + 4 = 12
a = 9: 9 + 4 = 13
a = 10: 10 + 4 = 14
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The lengths of the sides of a triangle are 6, 9, and 12. Classify the triangle as acute, right, or obtuse. Explain how you know
The triangle with side lengths 6, 9, and 12 is an obtuse triangle. To classify the triangle as acute, right, or obtuse, we need to determine the type of triangle based on the lengths of its sides.
Recall that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side for it to be a valid triangle.
Let's check if the given triangle with side lengths 6, 9, and 12 satisfies the triangle inequality:
1. Sum of two shorter sides: 6 + 9 = 15 (greater than the length of the longest side, 12). Valid.
2. Sum of the two longer sides: 9 + 12 = 21 (greater than the length of the remaining side, 6). Valid.
3. Sum of the longest side and the shortest side: 6 + 12 = 18 (greater than the length of the remaining side, 9). Valid.
Since all three inequalities are satisfied, the given side lengths form a valid triangle.
Now, to classify the triangle, we can use the Pythagorean theorem. In a right-angled triangle, the square of the length of the longest side is equal to the sum of the squares of the other two sides.
In this case, the side lengths are 6, 9, and 12. To determine if it is a right-angled triangle, we can check if:
[tex]12^2 = 6^2 + 9^2\\144 = 36 + 81\\144 \neq 117[/tex]
Since the Pythagorean theorem is not satisfied, the triangle is not a right-angled triangle.
Now, let's check if it is an acute or obtuse triangle. In an acute triangle, all three angles are less than 90 degrees, while in an obtuse triangle, one angle is greater than 90 degrees.
To determine this, we can use the law of cosines, which states that in any triangle:
[tex]c^2 = a^2 + b^2 - 2ab * cos(C)[/tex]
where c is the longest side (12 in this case), a and b are the other two sides (6 and 9), and C is the angle opposite the longest side.
Calculate the value of cos(C):
[tex]12^2 = 6^2 + 9^2 - 2 * 6 * 9 * cos(C)\\144 = 36 + 81 - 108 * cos(C)\\144 = 117 - 108 * cos(C)\\108 * cos(C) = 117 - 144\\108 * cos(C) = -27\\cos(C) = -27 / 108\\cos(C) = -0.25[/tex]
Since the value of cos(C) is negative, the angle C is obtuse. Therefore, the triangle is an obtuse triangle.
In summary, the triangle with side lengths 6, 9, and 12 is an obtuse triangle.
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There are 6 adults in this class. Mary wants to give 5/8 of a doughnut to each person. How many doughnuts will he need?
Step-by-step explanation:
Divide.
On a map 1/2-inch represents 4/5 Miles. what distance on the map represents one mile. Show your work.
The distance οn the map that represents οne mile is 5/8 inch.
Tο find οut hοw much distance οn the map represents οne mile, we need tο use a ratiο οf the distance οn the map tο the actual distance.
Let x be the distance οn the map that represents οne mile. Then we have:
1/2 inch : 4/5 miles = x : 1 mile
We can crοss-multiply tο sοlve fοr x:
(1/2 inch) × (1 mile) = (4/5 miles) × x
Simplifying the right-hand side:
(1/2) × 1 = (4/5) × x
1/2 = (4/5) × x
Multiplying bοth sides by 5/4:
(1/2) × (5/4) = x
5/8 = x
Therefοre, the distance οn the map that represents οne mile is 5/8 inch.
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need help with this , 13 × y z + 9 +2 × y z + 3
Answer:
simplified expression: 15yz + 12
i need help ASAP pls just say the answer
Answer:
the answer is c
Step-by-step explanation:
Given the piecewise function shown below, select all of the statements that
are true.
-x+1₁x<0]
f(x)= -2, x=0
[x²-1, x>0]
The correct statement regarding the numeric values of the piece-wise function is given as follows:
D. f(1) = 0.
How to obtain the numeric value of a function or of an expression?To obtain the numeric value of a function or of an expression, we substitute each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.
A piece-wise function is a function that has different definitions based on the input of the function, hence before obtaining the numeric value we must obtain the correct definition of the function in the interval.
Choosing the correct definitions of the function, the numeric values are given as follows:
f(-2) = -2 + 1 = -1 -> f(x) = -x + 1 for x = -2 as -2 < 0.f(-1) = -1 + 1 = 0.f(4) = 4² - 1 = 16 - 1 = 15 -> f(x) = x² - 1 for x = 4 as 4 > 0.f(1) = 1² - 1 = 0.Learn more about the numeric values of a function at brainly.com/question/28367050
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Write two numbers that multiply to the value on top and add to the value on bottom
Submit Answer
45
X
-14
Answer:
-9 and -5
Step-by-step explanation:
-9 + -5 = -14
-9 * -5 = 45
Answer:
- 9 and - 5
Step-by-step explanation:
- 9 × - 5 = 45 and - 9 + (- 5) = - 9 - 5 = - 14
Title: Averages
Things to remember:
Mixed
1) Range: 3, 4, 5, 7, 8, 8, 9
2) Median: 3, 4, 4, 6, 6,
3) Mean: 2, 9, 7, 5, 2, 2,
WORKING OUT & ANSWER
The values of the measures are;
Range = 6
Median = 4
Mean = 4.5
How to determine the valuesTo determine the values of the range, median and mean, we need to note that;
The range of a given set of values is the interval between the smallest number and the largest number in the set.
Given the data set;
3, 4, 5, 7, 8, 8, 9
The range = 9 - 3 = 6
The median of a given set of values is the middle number of the set when arranged in an ascending or descending form.
Given the set;
3, 4, 4, 6, 6,
The median is 4
The mean of a set of numbers is the average value of the numbers.
Given the set;
2, 9, 7, 5, 2, 2,
The mean = 2 + 9 + 7 + 5 + 2 + 2/6 = 27/6 = 4. 5
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Use the given information to find the unknown value.
2. y varies
1. y varies directly as the square root
inversely with the cube of x.
of x. When x = 16, then y = 4. Find y
y = 1. Find y when x = 1.
when x = 36.
When x = 3, then
3. The distances that an object falls varies directly with the
square of the time, t, of the fall. If an object falls 16 feet in one
second, how long for it to fall 144 feet?
4. The rate of vibration of a string under constant tension varies
inversely with the length of the string. If a string is 24 inches long
and vibrates 128 times per second, what is the length of a string
that vibrates 64 times per second?
The proportional and inverse relationships can be presented as follows;
1. The value of y when x = 36 is 6
2. The value of y when x = 1 is 27
3. The time of fall of an object that falls 144 feet is 3 seconds
4. The length of the string that vibrates 64 times per second is 48 inches long
What is a proportional relationship?A proportional relationship is one that can be expressed in the form; y = k × x.
The possible questions, obtained from a similar question posted online, are presented as follows;
1. y varies directly as the square root of x, when y = 4, x = 16, to find y when x = 36;
The value of y can be found by expressing the relationship between y and x using a proportional relation as follows;
y ∝ √x
y = c·√x
4 = c·√(16) = 4·c
Therefore;
c = 4/4 = 1
c = 1
When x = 36, we get;
y = 1 × √(36) = 6
Therefore, when x = 36, y = 62. The variation of y and x can be presented as follows;
y ∝ 1/x³
y = c/x³
y = 1, when x = 3, therefore;
1 = c/3³ = c/27
c = 27 × 1 = 27
c = 27
The value y when x = 1, can be found as follows;
y = 27/1³
y = 27/1³ =27
When x = 1, y = 273. The relationship between the distance and the time duration the ball can be presented as follows;
Let s represent the distance the ball falls, and let t represent the time the ball falls, we get;
s = k·t²
The distance an object falls in one second = 16 feet
Therefore, we get;
When s = 16, t = 1
16 = k × 1²
k = 16/1² = 16
k = 16
s = 16·t²
When s = 144 feet, we get;
144 = 16 × t²
t² = 144/16 = 9
t = √9 = 3
The time the object will take to fall 144 feet is 3 seconds4. The relationship between the vibration of a string and the tension in the string can be presented as follows;
Let f represent the vibration of the string, and let l represent the length in the string, we get;
f ∝ 1/l
f = c/l
When l = 24, f = 128, we get;
128 = c/24
c = 128 × 24 = 3,072
Therefore, if the string vibrates 64 times per second, we get;
64 = 3,072/l
l = 3,072/64 = 48
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There are 700 houses in Toby's town. Last summer, 651 of the houses were for sale. What percentage of the houses in the town were for sale last summer? Write your answer using a percent sign (%).
Answer:
93%
Step-by-step explanation:
651/700 houses were for sale. Write that as a percentage.
651/700=0.93
0.93=93%
Which graph represents the function g(z)=√2-1+1?
In response to the given question, we can state that Because it is independent of z, the function g(z) is a constant function. In particular, g(z) equals 2 - 1 + 1 = 2.
what is function?Mathematicians examine numbers and complex variations, equations and associated structures, forms and their locations, and prospective positions for these things. The term "functioning" signifies the connection between a collection of inputs, each of which has a corresponding output. A function is a connection of inputs and results in which each input leads to a single, identifiable outcome. Each function is assigned a domain, a codomain, or a scope. The letter f is widely used this to denote functions (x). The symbol for admission is an x. The four primary types of usable functions are on operations, one-to-one capabilities, so multiple functionality, in capabilities, and then on functions.
Because it is independent of z, the function g(z) is a constant function. In particular, g(z) equals 2 - 1 + 1 = 2.
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Round up 64,325 to the nearest thousands
Answer: 64,000
Step-by-step explanation:
The fuel economy of a car, measured in miles per gallon, is modeled by the function ƒ(s) = –0.008s^2 + 1.4s, where s represents the average speed of the car, measured in miles per hour.
a. At what speed should the car travel to achieve its maximum fuel economy?
b. What's the maximum fuel economy of the car?
c. What's the fuel economy of the car if it travels at 60 miles per hour?
d. The point (40, 43.2) lies on the graph of the function. What does this mean in the context of the problem?
a. To find the speed at which the car achieves maximum fuel economy, we need to find the vertex of the function. The vertex of a quadratic function is given by the formula: (-b/2a, f(-b/2a)). In this case, a = -0.008 and b = 1.4. Substituting these values, we get:
Vertex = (-b/2a, f(-b/2a))
= (-1.4/(2*(-0.008)), f(1.4/(2*(-0.008))))
= (87.5, 6.125)
Therefore, the car should travel at a speed of 87.5 miles per hour to achieve maximum fuel economy.
b. To find the maximum fuel economy, we need to find the value of the function at the vertex. Substituting x = 87.5 in the function, we get:
f(87.5) = -0.008(87.5)^2 + 1.4(87.5) = 6.125
Therefore, the maximum fuel economy of the car is 6.125 miles per gallon.
c. To find the fuel economy of the car if it travels at 60 miles per hour, we need to evaluate the function at x = 60. Substituting x = 60 in the function, we get:
f(60) = -0.008(60)^2 + 1.4(60) = 7.6
Therefore, the fuel economy of the car if it travels at 60 miles per hour is 7.6 miles per gallon.
d. The point (40, 43.2) lies on the graph of the function. This means that when the car travels at a speed of 40 miles per hour, its fuel economy is 43.2 miles per gallon. However, this point does not provide any information about the maximum fuel economy or the speed at which it is achieved.
The width of a rectangle is 16 feet less than 3 times the length, and the area is 35 square feet.
Part a: Write an equation that can be used to determine the length and width of the rectangle. Express your answer as a quadratic equation set equal to zero
A rectangle has a width that is 16 feet shorter than its length and an area that is 35 square feet. The rectangle's length and breadth can be calculated using the equation [tex]3x^2[/tex] - 16x - 35 = 0.
Assume that the rectangle measures "x" feet in length. Then, according to the problem:
The width is 16 feet less than 3 times the length, so the width is 3x - 16 feet.
The area of the rectangle is 35 square feet, so we can write:
Area = Length x Width
35 = x(3x - 16)
To solve for x, we can simplify this quadratic equation by expanding the right-hand side and moving all the terms to one side:
35 = [tex]3x^2[/tex] - 16x
[tex]3x^2 - 16x - 35 = 0[/tex]
The rectangle's length and breadth can be calculated using the quadratic equation shown above.
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Pls help me solve whole page
Please help me with this math question!!
Answer:
The rocket splashes down after [tex]\boxed{24}[/tex] seconds
The rocket peaks at [tex]\boxed{769}[/tex] meters above sea-leval
Step-by-step explanation:
The relationship between height and time is given by the equation
[tex]h(t) = -4.9t^2+112t+129[/tex]
The splashdown will occur when [tex]h(t) = 0[/tex]
Therefore solve t for
[tex]-4.9t^2+112t+129 = 0[/tex]
This is a quadratic equation which can be solved with the aid of a quadratic formula calculator
The solution set for this equation is
[tex]t=-1.09894 \;and\;\:t=23.95609\right)[/tex]
Omitting the negative value we get
[tex]\text{t=23.95609 \;seconds}\\\\= 2\text{4 \;seconds \;rounded}}[/tex]
For the second part, the maximum value of h(t) can be found by finding the first derivative h'(t) and setting it equal to 0 and solving for t; then plug this value of t into h(t) to find the max height
[tex]h'(t) = \dfrac{d}{dt}\left(-4.9t^2+112t+129\right)\\\\= \dfrac{d}{dt}\left(4.9t^2\right)+\dfrac{d}{dt}\left(112t\right)+\dfrac{d}{dt}\left(129\right)\\\\= - 9.8t + 112 + 0\\\\= -9.8t + 112\\\\[/tex]
Setting the above expression to 0 gives
[tex]-9.8t+112=0\\\\-9.8t = -112\\\\t = \dfrac{-112}{-9.8}\\\\t = 11.4286 \;seconds[/tex]
Plugging this value back into h(t) gives
[tex]h(11.4286) = -4.9\cdot \:11.4286^2+112\cdot \:11.4286+129\\\\= 769 \;meters \;(rounded)[/tex]
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Isaac is mountain climbing with Rina and has just climbed a 6.2-meter vertical rock face.
Rina is standing at the bottom of the cliff, looking up at Isaac on a diagonal. If Rina is 10
meters away from Isaac, how far away from the cliff is Rina standing?
If necessary, round your answer to the nearest tenth.
meters
In a casde whereby saac is mountain climbing with Rina and has just climbed a 6.2-meter vertical rock face the distance from the cliff is Rina standing is 7.85m
How can the distance be calculated?Base on the given iformation, following the trigonometry rule it can be seen thatr the hypotenus of the triangle is the distance of Isaac to Rina which can be expressed a C = 10m
The height that was climbed by I saac = 6.2-m. a= 6.2m
the distance from Rina to cliff = b
Then base on pytagoras theorem
c^2= a^2 + b^2
10^2 = 6.2^2 + b^2
b^2= 10^2 - 6.2^2
b^2= 61.56
b = √61.56
b=7.85m
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What is the solution of the inequality z + 2 < 8? Ο Α. * < 4 B. 2 < 6 OC. 2-6 O D. 2 < 10
Answer:
[tex]\tt z < 6[/tex]Step-by-step explanation:
[tex]\tt z + 2 < 8[/tex]
Subtract 2 from both sides:-
[tex]\tt z + 2 -2 < 8-2[/tex]Simplify :-
[tex]\tt z < 6[/tex]Therefore, the solution of the inequality z + 2 < 8 is z < 6.
_______________________
Hope this helps!
Examine the following graph of the quadratic function f(x) = -(x + 1)² +8 and the linear function, g(x) = -x - 6.
For which of the following intervals is the average rate of change if f(x) faster than the average rate of change of g(x)? There is no than on correct answer. Select all that apply
The intervals for which the rate of change of the quadratic function, f(x) = -(x + 1)² + 8 is faster than the rate of change of g(x) = -x - 6 are; x ≤ -1.5 and x ≥ -0.5
What is a quadratic function?A quadratic function is a polynomial function with one or more variables in which the highest exponent is two.
The slope of f(x) is; f'(x) = -2·(x + 1)
The slope of g(x) = -1
When the rate of change of f(x) is faster than the rate of change of g(x), we get;
The slope of f(x) > The slope of g(x), therefore;
-2·(x + 1) > -1
From which we get;
|-2·(x + 1)| ≥ |-1|
2·(x + 1) ≥ 1
x + 1 ≥ 1/2
x ≥ 1/2 - 1 = -1/2
x ≥ -1/2 = -0.5
The absolute sign indicates that we get;
x + 1 ≤ -1/2
x ≤ -1/2 - 1 = -1 -1/2 = -3/2 = -1.5
x ≤ -1.5
The interval for which the average rate of change of f(x) is faster than the average rate of change of g(x) are; x ≥ -0.5 and x ≤ -1.5
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Suppose that the distance a car travels varies directly with the amount of gasoline it uses. A certain car uses 24 gallons of gasoline to travel 552 miles.
Write a direct variation equation to represent the relationship. Use d for the distance the car travels (in miles) and g for amount of gasoline it uses (in gallons)
[tex]\qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad \stackrel{\textit{constant of variation}}{y=\stackrel{\downarrow }{k}x~\hfill } \\\\ \textit{\underline{x} varies directly with }\underline{z^5}\qquad \qquad \stackrel{\textit{constant of variation}}{x=\stackrel{\downarrow }{k}z^5~\hfill } \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{D varies directly with G}}{D = k(G)}\hspace{5em}\textit{we also know that} \begin{cases} G=24\\ D=552 \end{cases} \\\\\\ 552=k(24)\implies \cfrac{552}{24}=k\implies 23=k\hspace{5em}\boxed{D=23G}[/tex]
If the perimeter of the isosceles triangle is at most 45 centimeters, which inequality could be used to find the value of p
Inequality to find the value of p is 7p - 12 ≤ 45. So correct option is C. The value of p is at most 8.14 centimeters.
Describe Inequality?Inequalities can be solved in a similar way to equations, but with some important differences. To solve an inequality, one must find the set of values that satisfy the inequality. This set of values is often expressed as an interval, which is a range of values between two endpoints. For example, the solution to the inequality x < 5 is the interval (-∞, 5), which includes all values of x that are less than 5. The solution to an inequality may also be expressed graphically on a number line or coordinate plane.
The perimeter of an isosceles triangle with sides of length 3p-6, 3p-6, and p is:
Perimeter = (3p-6) + (3p-6) + p
= 7p - 12
We are given that the perimeter is at most 45 centimeters. Therefore, we can write the following inequality to find the value of p:
7p - 12 ≤ 45
Adding 12 to both sides, we get:
7p ≤ 57
Dividing both sides by 7, we get:
p ≤ 8.14
Therefore, the value of p is at most 8.14 centimeters.
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Solve the equation 4x + 5y = 9 for x.
Answer:
x = -5y+9/4
Step-by-step explanation:
4x + 5y = 9
↓
(4x + 5y) + (-5y) = 9 + (-5y)
↓
4x + 5y - 5y = 9 - 5y
↓
4x = -5y + 9
↓
4x/4 = -5y + 9/4
↓
x = -5y + 9/4
Answer: x = 9/4 - 5y/4
Step-by-step explanation: Move all terms that don't contain x to the right side and solve.
I will mark you brainiest!
The sum of all of the exterior angles of an octagon is:
A) 180º.
B) 360º.
C) 1080º.
D) 36º.
Answer:
The sum of the exterior angles of any polygon, regardless of the number of sides, is always 360 degrees. Each exterior angle of a regular octagon measures 45 degrees, because the sum of the interior and exterior angles at each vertex is 180 degrees, and the octagon has 8 vertices. Therefore, the sum of all the exterior angles of an octagon is:
8 × 45º = 360º
So the answer is option B) 360º.
How do you determine area of a circle
Step-by-step explanation:
it's radius squared timesed by pi (π)
r^2×π
if u don't have a radius but a diameter then half the diameter to get the radius
Answer:
There are two common formulas that can be used to calculate the area of a circle. Here they are:
1. Diameter based formula: [tex]\frac{\pi d^{2} }{4}[/tex].Where [tex]\pi[/tex] is the universal value for all circles that expresses the ratio between the length of circumference and the diameter of any circle. It's a rational number and its value is about 3.141592653589793238462643383279502884197..., so you may want to just 3.14.
2. Radius based formula: [tex]\pi r^{2}[/tex].This is the easiest and simplest formula. Letter "r" represents the length of radius of the circle,
3. Circumference based formula: [tex]\frac{C^{2} }{4\pi }[/tex].Rarely used formula. In this equation, "C" represents the length of circumference of the circle.
Check the attached image to better understand the meaning of radius, diameter and circumference. Also, provided a second image with all the formulas for the area of a circle.
Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 6.6 per year.
a. Find the probability that, in a year, there will be 4 hurricanes.
b. In a 35-year period, how many years are expected to have 4 hurricanes?
c. How does the result from part (b) compare to a recent period of 35 years in which 3 years had 4 hurricanes? Does the Poisson distribution work well here?
The probability of having 4 hurricanes in a year is approximately 0.118 or 11.8%.
What is probability?It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
According to question:a. To find the probability of having exactly 4 hurricanes in a year, we use the Poisson distribution with a mean of 6.6:
P(X=4) = (e^-6.6)×(6.6^4)/(4!) ≈ 0.118
Therefore, the probability of having 4 hurricanes in a year is approximately 0.118 or 11.8%.
b. In a 35-year period, the expected number of years with 4 hurricanes is:
μ = λt = 6.635 ≈ 231
Therefore, we can expect to have 231 years with 4 hurricanes in a 35-year period.
c. If in a recent period of 35 years, only 3 years had 4 hurricanes, this is less than what the Poisson distribution would predict (231 years). However, this doesn't necessarily mean that the Poisson distribution doesn't work well here. The Poisson distribution is a theoretical model that assumes certain conditions, and it's possible that those conditions weren't met in this specific case. It's also possible that this was just a rare event that can happen due to chance. Further analysis would be needed to determine whether the Poisson distribution is a good fit for this data.
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2.5% of x is = 17
what is x??
Answer: 680
Step-by-step explanation:
Corresponding Angles are congruent. Which angle corresponds with 3? 1/2 3/4 fine 7/8 5 6 [?]
Answer:
5, 8 and 2
Because it has the same angel as 3 there for it's 5 , 8 and 2
Solve for the value of e.
(pls help me)
Answer:
e = 10
Step-by-step explanation:
Vertically opposite angles are the angles formed opposite to each other when two lines intersect.
Vertically opposite angles are equal.
9e - 8 = 8e + 2
Add 8 to both sides,
9e = 8e + 2 + 8
9e = 8e + 10
Subtract '8e' from both sides,
9e - 8e = 10
[tex]\boxed{\bf e = 10}[/tex]
Answer:
e = 10
Step-by-step explanation:
According to the Vertical Angle Theorem, when two straight lines intersect, the opposite vertical angles are congruent.
Since the two given angles are vertical angles, they are congruent.
Therefore, we can set them equal to each other:
[tex]\implies (9e - 8)^{\circ} = (8e + 2)^{\circ}[/tex]
[tex]\implies 9e - 8= 8e + 2[/tex]
To solve for e, begin by subtracting 8e from both sides of the equation:
[tex]\implies 9e - 8-8e= 8e + 2-8e[/tex]
[tex]\implies e - 8= 2[/tex]
Add 8 to both sides of the equation:
[tex]\implies e - 8+8= 2+8[/tex]
[tex]\implies e=10[/tex]
Therefore, the value of e is 10.
What is the meaning of "the group of rotations and symmetries of the configuration"?
The group of rotations and symmetries of a geometric configuration is the group of isometries.
Define the term "the group of rotations and symmetries of the configuration" in given paragraph?According to the above paragraph, "the group of rotations and symmetries of the configuration" refers to the group of isometries of a geometric configuration that preserve the configuration itself, without involving any translations. This group consists of rotations about a fixed point and symmetries about a straight line that pass through that point.
This group is formed by combining isometries of the configuration through the operation of composition, and it satisfies certain properties, such as closure, associativity, identity, and inverses. It is a useful tool for studying the symmetries and transformations of the configuration, and it is used in fields such as crystallography and quantum mechanics.
In summary, the group of rotations and symmetries of a geometric configuration is the group of isometries that preserve the configuration and can be composed with each other to obtain new isometries.
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