Answers
The answers f(g(x)) = x and g(f(x)) = x tell us that the two functions f(x) and g(x) are inverses of each other.
What is inverse?The inverse of a function is a second function that "undoes" the effect of the first function. More specifically, if f is a function that maps elements from a set A to a set B, then its inverse function, denoted as f^(-1), maps elements from B back to A.
According to question:(a) To find f(g(x)), we need to substitute the expression for g(x) into f(x):
f(g(x)) = g(x) / (6 + g(x))
Substituting the expression for g(x) yields:
f(g(x)) = (6x / (1 - x)) / (6 + (6x / (1 - x)))
This equation can be made simpler by first locating a common denominator:
f(g(x)) = (6x / (1 - x)) / ((6(1 - x) / (1 - x)) + (6x / (1 - x)))
f(g(x)) = (6x / (1 - x)) / ((6 - 6x + 6x) / (1 - x))
f(g(x)) = (6x / (1 - x)) / (6 / (1 - x))
f(g(x)) = 6x / 6
f(g(x)) = x
To find g(f(x)), we need to substitute the expression for f(x) into g(x):
g(f(x)) = 6f(x) / (1 - f(x))
Substituting the expression for f(x) yields:
g(f(x)) = 6(x / (6 + x)) / (1 - (x / (6 + x)))
To simplify this expression, we can first find a common denominator:
g(f(x)) = 6(x / (6 + x)) / (((6 + x) / (6 + x)) - (x / (6 + x)))
g(f(x)) = 6(x / (6 + x)) / ((6 + x - x) / (6 + x))
g(f(x)) = 6(x / (6 + x)) / (6 / (6 + x))
g(f(x)) = x
(b) The answers f(g(x)) = x and g(f(x)) = x tell us that the two functions f(x) and g(x) are inverses of each other. This means that when we apply one function and then the other, we get back to the original input value. Specifically, if we apply f(x) to x and then apply g(x) to the result, we get x back, and if we apply g(x) to x and then apply f(x) to the result, we also get x back. This is a useful property when analyzing functions and their relationships.
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Related Questions
T
A
D
√
5
B
C
Ĉ
X
Figure ABCD is a square. The side lengths are all 1.
Estimate the length of the diagonal.
Submit
Answers
Therefore, the estimated length of the diagonal is approximately 1.414 units.
What is square root?The square root of a number is a value that, when multiplied by itself, gives the original number. In other words, the square root of a number "x" is a value "y" such that y multiplied by y equals x. The symbol for square root is √.
For example, the square root of 25 is 5, because 5 multiplied by 5 equals 25. Similarly, the square root of 64 is 8, because 8 multiplied by 8 equals 64. The square root of a negative number is not a real number, as there is no real number that, when squared, equals a negative number.
Using the Pythagorean theorem, we can find the length of the diagonal of the square.
If we draw a diagonal from one corner of the square to the opposite corner, we create a right triangle. The two legs of the right triangle are the sides of the square, which have a length of 1. The hypotenuse, which is the diagonal of the square, is the side we are trying to find.
Using the Pythagorean theorem, we can write:
[tex]d^2 = 1^2 + 1^2[/tex]
where d is the length of the diagonal.
Simplifying, we get:
[tex]d^2 = 2[/tex]
Taking the square root of both sides, we get:
d ≈ 1.414
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A wire 2.5 meters long was cut in a ratio of 1:4, find the measure of the longer part of the wire after cutting?
Answers
The wire can be divided into five equal parts, where one portion is one-fifth of the total length and the other four parts are four-fifths of the total length. the measure of the longer part of the wire after cutting is 2 meters.
What is the measure of the longer part of the wire?If the wire was cut in a ratio of 1:4, then the total length of the wire can be divided into 5 parts, where one part is 1/5 of the total length, and four parts are 4/5 of the total length. Let's call the length of one part "x".
So, the total length of the wire is:
[tex]5x = 2.5[/tex] meters
To find the length of the longer part of the wire, we need to find how many parts are in the longer portion. Since the wire was cut in a 1:4 ratio, the longer portion has four parts.
Therefore, the length of the longer part of the wire is:
[tex]4x = 4/5 \times 2.5 meters = 2 meters[/tex]
Therefore, the measure of the longer part of the wire after cutting is 2 meters.
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Determine the perimeter of the composite figure.
Answers
The perimeter of the figure is approximately 63.6 meters, since the lengths of all the sides add up to the figure's perimeter.
What is perimeter?Perimeter is the total length of the boundary of a closed 2-dimensional shape. In other words, it is the distance around the edge of a shape. The perimeter is usually measured in units such as meters, centimeters, feet, or inches depending on the measurement system used. The perimeter can be calculated by adding up the lengths of all the sides of the shape.
Since opposite sides of a parallelogram are equal in length, the length of the straight side of the first parallelogram is:
Length of first parallelogram = 9m
Similarly, the length of the straight side of the second parallelogram is:
Length of second parallelogram = 11m
Now, let's find the length of the straight side of the semicircle segment. We are given that the other side of each parallelogram gets half of the diameter, which is 17m. Therefore, the length of the straight side of the semicircle segment is:
Length of semicircle segment = 17m / 2 = 8.5m
The lengths of all the sides add up to the figure's perimeter. Let's use P to represent the perimeter. Then:
P = 2 (parallelogram length) + semicircle length
When we replace the values we discovered earlier, we obtain:
P = 2(9m + 11m) + π(17m)/2
= 38m + 8.5πm
≈ 63.6m (using π ≈ 3.14)
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A quadratic function passes through the points (-2,5) and (-8,5) and has a graph in the xy-plane that opens downward. Which of these could be the coordinates of the highest point of the graph?
O (5, 1)
O (5, 8)
O (5, 1)
O (5, 8)
Answers
The highest point of the quadratic equation that opens down can be (-5, 8)
Which of these could be the coordinates of the highest point of the graph?We have a quadratic equation whose graph opens downwards, and we know that it also passes through the points (-2,5) and (-8, 5).
The highest point will be of the form (x, y).
Such that y is larger than 5, and the value of x is right between the two values above:
x = (-2 - 8)/2
x = -5
So the point is of the form (-5, y) where again y > 5.
From the given options the only of this form is (-5, 8)
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The number of users of the internet in a town increased by a factor of 1.01 every year from 2000 to 2010. The function below shows the number of internet users f(x) after x years from the year 2000: f(x) = 3000(1.01)x Which of the following is a reasonable domain for the function?
Answers
Answer:
0 ≤ x ≤ 10
Step-by-step explanation:
The domain is the possible values that the variable can take.
The statement "the number of internet users f(x) after x years from the year 2000" is telling us that the x-axis represents the number of years, so the domain of the function is the number of years from 2000 to 2010.
Number of years = 2010 - 2000 = 10 years
So we know that is at most 10 years; therefore
Since years can't be negative, we can also infer that x must be 0 or bigger than 0, so
Now we can combine both inequalities to find the reasonable domain of the function:
please help !!!!
Consider the table, equation,
and graph. Which of them represents a proportional relationship?
Answers
The graph represents a proportional relationship.
What is proportional relationship?
A proportional relationship is a relationship between two variables where their ratio remains constant. This means that as one variable increases or decreases, the other variable changes proportionally in order to maintain a constant ratio. In other words, the two variables are directly proportional to each other. This relationship can be represented by a straight line passing through the origin on a graph. Proportional relationships are commonly used in various fields such as physics, finance, and engineering to analyze and predict outcomes.
Explaining the table, equation and graph to know which represents a proportional relationship :
The relationship between the values in the given table is not proportional. If it was a proportional relationship, then the ratio of y to x would be constant. However, in this case, the ratios are not equal. For example, the ratio of y to x for the first row is 3.6/3 = 1.2, but the ratio of y to x for the second row is 6/5 = 1.2, and the ratio of y to x for the third row is 7.2/8 = 0.9. Therefore, the ratios are not equal and the relationship is not proportional.
The equation y=2x+5 does not represent a proportional relationship. In a proportional relationship, there is a constant ratio between the two variables. However, in this equation, the ratio between y and x is not constant, but rather it increases as x increases.
The graph represents a proportional relationship as the ratio between y and x is constant.
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Parts of similar triangles
Find x
Answers
In the triangle, x has a value of 18.
Which triangle is on the right?A right triangle is one with a 90 degree inner angle. The hypotenuse, which is also the side of the right triangle that faces the right angle, is its longest side. The two arms of a right angle are made up of height and base.
We know that the respective sides of the two triangles in the diagram are proportional since they resemble one another. Namely, we have
AB / AE = BC / CD
Inputting the values provided yields:
x / 12 = 9 / 6
Finding x and simplifying results in:
x = 12 * (9/6) = 18
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The Thornton Street Block Association wants to raise $2000 to plant trees. Two weeks after starting its campaign, the association had raised 65% of its goal. How much more money does the association need to raise?
Answers
$ 2000 • 65 / 100 = $ 1300
We can subtract the amount of money raised from the goal amount:
$2000- $1300 = $ 700
Option D is correct
GEOMETRY TWO SIDES GIVEN WHAT IS POSSIBLE FOR THIRD 20 AND 15 ARE GIVEN PLEASE HELP
Answers
Using the triangle inequality theorem, the possible values for the third side of the triangle are: 5 < x < 35.
How to Apply the Triangle Inequality Theorem?The Triangle Inequality Theorem asserts that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Stated differently, the difference between the lengths of any two sides must be less than the length of the third side.
Thus, if you have the lengths of two sides of a triangle, as 20 and 15, we have:
20 - 15 < x < 20 + 15
5 < x < 35
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Me . Lake Bought four dozen apples for her band kids 1/4 of the apples were green and 3/4 or red. How many apples were red
Answers
Answer:
36
Step-by-step explanation:
3/4 of 48 is 36
69 POINTS NEED HELP ASAP QUESTION IS DOWN BELOW
Answers
Answer:
(a) 22 inches
(b) 770 inches
(c) 26,950 inches
Step-by-step explanation:
(a) To find the perimeter of the drawing, we add up the lengths of all four sides:
Perimeter of drawing = 7 + 4 + 7 + 4 = 22 inches
(b) The length and width of the actual garden are 35 times larger than the dimensions in the drawing. This means that the actual length is 7 x 35 = 245 inches and the actual width is 4 x 35 = 140 inches. To find the perimeter of the actual garden, we add up the lengths of all four sides:
Perimeter of actual garden = 245 + 140 + 245 + 140 = 770 inches
(c) When the dimensions of the garden are multiplied by 35, the perimeter of the garden will also be multiplied by 35. This is because each side will increase by a factor of 35, so the total length of all four sides will increase by a factor of 35 as well. Therefore, the new perimeter will be:
New perimeter = 35 x Perimeter of actual garden = 35 x 770 = 26,950 inches
If f(x)=3x-5, g(x)=7x-3, find (f+g)(x) and (f-g)(x
Answers
Answer:
Step-by-step explanation: To find (f+g)(x), we simply add the two functions:
(f+g)(x) = f(x) + g(x)
= (3x - 5) + (7x - 3)
= 10x - 8
Therefore, (f+g)(x) = 10x - 8.
To find (f-g)(x), we subtract g(x) from f(x):
(f-g)(x) = f(x) - g(x)
= (3x - 5) - (7x - 3)
= -4x - 8
Therefore, (f-g)(x) = -4x - 8.
QUICK PLS 30 POINTS
For a quadratic function, f, f(0) = 15.
Which equation could represent f?
A: f(x)=x²-8x+15
B: f(x)=(x-3)(x+5)
C: f(x)=x²+3x+5
D: f(x)=(x-15)²
Answers
f(0)=(0)^2-8(0)+15
= 15
Answer:
The answer is A
Step-by-step explanation:
You can use a graphing calculator (geogebra) It works
A student added the rational expressions as follows:
Problem:
Work:
5x
+
x+7
5x
+
x+7
Solution:
7
x
7(7)
x+7
5x+49
x+7
Describe and correct the error the student made. Solve the problem correctly.
Answers
The correct answer is (5x + 1) / (x+7).
What error did the student make?
The student made an error in adding the rational expressions. They attempted to add the numerators directly without finding a common denominator.
To add rational expressions, we must first find a common denominator, which is the Least Common Multiple (LCM) of the two denominators. Then, we can add the numerators.
Here is the correct solution:
Problem:
(5x) / (x+7) + 1 / (x+7)
First, note that both rational expressions have the same denominator (x+7). In this case, the LCM is simply (x+7). We can now add the numerators:
Work:
(5x) / (x+7) + 1 / (x+7) = (5x + 1) / (x+7)
Solution:
(5x + 1) / (x+7)
The correct answer is (5x + 1) / (x+7).
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help plsssss… i don’t understand & or know how to do it
Answers
Trigonometric ratios are mathematical functions that relate the angles of a right triangle to the lengths of its sides.
Trigonometric ratiosTan 45 = x/13
x = 13 Tan 45
x = 13
Sin 30 = x/4
x = 4 Sin 30
x = 2
Tan 60 = 21/y
y = 21/Tan 60
y = 12.1
Cos 45 = x/22
x = 22Cos 45
= 15.6
Tan 45 = 5/x
x = 5/Tan 45
x = 5
Sin 30 = 9/x
x = 9/Sin 30
9 = 18
Cos 60 = y/22
y = 22Cos 60
y = 11
Sin 45 = x/16
x = 16Sin 45
x = 11.3
Tan 30 = x/42
x = 42 Tan 30
x = 24.2
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PLEASE HELP!!!!!!!!!!!!!!
Answers
The quadratic function for the given graph is f(x) = -(x - 0.2)² + 6.2.
What is a quadratic function?
A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of second degree.
The graph crosses the y-axis at 6 so the graph has a y-intercept at (0,6).
The highest point in the graph is at (0.2,6.2) and the highest point is the maximum.
The graph crosses the x-axis at 2 and -1.5 so the graph has an x-intercept at (2,-1.5).
The graph's line of symmetry is x = 0.2 since the vertex is (0.2,6.2).
The zeros of a graph are its x-intercept so the graph has zeroes of 2 and -1.5.
The quadratic function is of the form a(x - h)² + k where (h, k) is the vertex.
Since, the graph is opening downwards so the value of a = -1 < 0.
The function f(x) = -(x - 0.2)² + 6.2 is the quadratic function of the graph.
Therefore, the function is obtained to be -(x - 0.2)² + 6.2.
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Select all the equations where x = 5 is a solution.
x-5=0
11-x=3
1+x=5
10=2x
(1/2)x=10
x^2=25
Answers
Answer:
x-5=0
10=2x
(1/2)x=10
x>2=25
Step-by-step explanation:
all of these are x=5
the ones that are wrong are 11-x=3 and 1+x=5
I need this done in 10 mins or less!!!
Answers
Answer:
To find the volume of Leah's new planter, we need to multiply the length, width, and height of the container:
V = 15 m x 4 m x 8 m x 11 m x 3 m
V = 11,880 cubic meters
Therefore, the total volume of Leah's new planter is 11,880 cubic meters.
Let's represent the volume of the large box as x cubic inches. Since the total volume of both boxes is 212 cubic inches, we can set up the following equation:
8 x 4 x 2 + x = 212
Simplifying the left side, we get:
64 + x = 212
Subtracting 64 from both sides, we get:
x = 148
Therefore, the volume of the large box is 148 cubic inches.
Find the interquartile range.
Answers
Answer:
3.7
Step-by-step explanation:
1.5, 2.2, 2.8, 3.4, 5, 5.6, 6.8, 7
Q1 = 2.2+2.8 = 5.0 divided 2 = 2.5
Q3 = 5.6+6.8 = 12.40 divided 2 = 6.2
IQR = 6.2 - 2.5 = 3.7
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1. What is the system of inequalities associated with the following graph?
Answers
Answer:
y ≥ -x + 2
y < 2x - 1
Step-by-step explanation:
Solid line:
y ≥ -x + 2
Dashed line:
y < 2x - 1
The system of inequalities associated with the graph is y ≥ 2 and x < 3.
The system of inequalities associated with the graph can be determined by examining the shaded region on the graph. The shaded region represents all the possible solutions that satisfy the given inequalities. To determine the system of inequalities, we need to consider the boundaries of the shaded region. These boundaries are formed by the lines or curves on the graph.
1. Identify the lines or curves that form the boundaries of the shaded region. These lines or curves are usually represented by equations or inequalities. For example, if there is a solid line, it indicates that the boundary is included in the solution. If there is a dashed line, it indicates that the boundary is not included in the solution.
2. Determine the inequalities associated with each boundary line or curve. You can do this by examining the direction of the inequality symbol (<, >, ≤, or ≥) and the location of the line or curve in relation to the shaded region.
3. Write down the inequalities associated with each boundary line or curve. Make sure to include the correct inequality symbol and any necessary constants or variables.
4. Combine all the inequalities together to form the system of inequalities. This can be done by using logical operators such as "and" or "or" to connect the individual inequalities. The "and" operator is used when all the inequalities must be satisfied simultaneously, while the "or" operator is used when at least one of the inequalities must be satisfied. For example, let's say the graph has a solid horizontal line at y = 2, a dashed vertical line at x = 3, and the shaded region is above the horizontal line and to the left of the vertical line. In this case, the system of inequalities would be: - y ≥ 2 (since the shaded region is above the line) - x < 3 (since the shaded region is to the left of the line) Therefore, the system of inequalities associated with the graph is y ≥ 2 and x < 3.
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What is the value of angle b?
Answers
If they’re vertical angles
I thought of a number. Seven times the number decreased by 13 equals 5 times the number increased by 5. What was my number?
Answers
answer:
The number is 9
Step-by-step explanation:
9 × 7 = 63
63 - 13 = 50
9×5 = 45
45 + 5 = 50
needed help on this one too
Answers
Answer:
1st -pentagon -regular
2nd -hexagon -irregular
3rd -octagon -irregular
You pick a card at random. 5 6 7 What is P(odd or greater than 6)?
Answers
The answer to the question is 1/3.
There are three possible outcomes: 5, 6, or 7.
Out of these three outcomes, only two satisfy the condition of being odd or greater than 6: 7.
Therefore, the probability of picking a card that is odd or greater than 6 is 1/3, or approximately 0.333 or 33.3%.
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Which of the following shapes has exactly 1 line of symmetry?
Answers
Answer:
Isosceles triangle is the shape which has exactly one line of symmetry.
A heptagon has perimeter 88 feet. Four of the sides are the same length, and the remaining sides are half as long. How long are the shorter sides?
Answers
the other two angles of the triangle must also be equal, which means that the other two sides of the triangle must be equal as well. the length of the shorter sides is [tex]y = 16 feet[/tex] .
What are the remaining sides are half as long in heptagon?Let's start by using the information given to write equations for the perimeter of the heptagon in terms of the length of the sides:
Let x be the length of the four equal sides, and y be the length of the remaining three sides, which are half as long. Then, we have:
Perimeter [tex]= 4x + 3y[/tex]
We also know that the perimeter is 88 feet, so we can set up the equation:
[tex]4x + 3y = 88[/tex]
Now we need to solve for y, which represents the length of the shorter sides.
We can simplify the equation by substituting y = (1/2)x:
[tex]4x + 3(1/2)x = 88[/tex]
Simplifying this expression, we get:
[tex]7x/2 = 88[/tex]
Multiplying both sides by 2/7, we get:
[tex]x = 32[/tex]
Now that we know x, we can find y by substituting it into the equation y [tex]= (1/2)x[/tex]:
[tex]y = (1/2)(32) = 16[/tex]
Therefore, the length of the shorter sides is [tex]y = 16[/tex] feet.
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i’m begging someone help please i’ll give brainlist
Answers
Answer:
a-no for the first triangle, because the total of the inner angles must be 180, the last angle must be 90.
b-yes- a triangle with angles 90, 50, and 40 is possible
c-yes. 65+45+70=180.
Step-by-step explanation:
100 POINTS + BRAINLIEST
Find a number between 100 and 200 which is also equal to a square number
multiplied by a prime number.
Answers
Answer:
Numbers between 100 and 200 which are also equal to a square number multiplied by a prime number are:
108, 112, 116, 117, 124, 125, 128, 147, 148, 153, 162, 164, 171, 172, 175, 176, 180, 188, 192Step-by-step explanation:
A square number is a number that has been multiplied by itself.
For example, 25 is a square number as 5 × 5 = 25.
The square numbers between zero and 200 are:
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196.A prime number is a whole number greater than 1 that cannot be made by multiplying other whole numbers (its only factors are 1 and itself).
Since the smallest square number is 4, and our final number needs to be between 100 and 200, we only need to list the primes numbers that are less than 50 as 4 × 50 = 200.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.Begin with the smallest prime number, 2. If we divide 100 and 200 by this prime number, we get 50 and 100. Therefore, to find a number between 100 and 200 that is equal to a square number multiplied by a prime number, the square number should be more than 50 and less than 100. Therefore:
64 × 2 = 12881 × 2 = 162The next smallest prime number is 3. If we divide 100 and 200 by 3 we get 33.33.. and 66.66... Therefore, the prime number 3 should be multiplied by a square number that is more than 33 and less than 67:
36 × 3 = 10849 × 3 = 14764 × 3 = 192The next prime number is 5. If we divide 100 and 200 by 5 we get 20 and 40. Therefore, the prime number 5 should be multiplied by a square number that is more than 20 and less than 40:
25 × 5 = 12536 × 5 = 180Continuing this way, all the numbers that are between 100 and 200, which are also equal to a square number multiplied by a prime number are:
64 × 2 = 12881 × 2 = 16236 × 3 = 10849 × 3 = 14764 × 3 = 19225 × 5 = 12536 × 5 = 18016 × 7 = 11225 × 7 = 17516 × 11 = 1769 × 13 = 1179 × 17 = 1539 × 19 = 1714 × 29 = 1164 × 31 = 1244 × 37 = 1484 × 41 = 1644 × 43 = 1724 × 47 = 188Number between 100 and 200 which is also equal to a square number multiplied by a prime number is 147.
Further Explanation:A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. In simpler terms, a prime number is a number that can only be evenly divided by 1 and itself.
To find this number, I first looked for square numbers between 100 and 200. The square numbers within this range are 121 (11 squared) and 169 (13 squared).
I then checked if either of these square numbers could be multiplied by a prime number to equal a number between 100 and 200. After some calculation, I found that 169 cannot be multiplied by a prime number to give a number between 100 and 200. However, 121 can be multiplied by 2 to give 242, which is greater than 200.
Finally, I checked if there were any other square numbers I missed. I found that 7 squared is equal to 49, which when multiplied by 3 (a prime number) gives 147, a number between 100 and 200 that satisfies the problem.
Therefore, the number between 100 and 200 which is also equal to a square number multiplied by a prime number is 147.
I hope this helps!
bob had 2 apples and he ate 1 how many does he have now?
Answers
Answer: He has one left
Step-by-step explanation: 2-1=1
What polygon pair is similar
Answers
Scale factor of the similar polygon pairs is:
5. 6
6. 5/12
7. 4/15
8. 1/7
Define scale factor?When both the original dimensions and the new dimensions are known, the scale factor may be determined.
The following is a basic formula to determine a figure's scale factor:
Scale factor is equal to the difference between the dimensions of the old and new shapes.
Here in the first figure,
The sides are in a ratio of 1:6. The sides have increased 6 times so the scale factor here is 6.
Similarly in the trapezium, the ratio of the sides is 5/12.
The scale factor here becomes 5/12.
In the next figure, the sides are in a ratio of 4/15. Scale factor for the pair here is 4/15.
In the last figure, the equivalent sides are in the ratio of 1/7. The scale factor between the pair is 1/7.
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