Common ratio of the geometric sequence 9/8, 3/4, 1/2,...,8/81 is 2/3 and the number of all terms in this sequence is 7.
As we know that,
Common ratio of any G.P. is a constant number that is multiplied by the previous term to obtain the next term.
So, r= (n+1)th term / nth term
where r ⇒ common ratio
(n+1)th term⇒ succeeding term
nth term⇒ preceding term
According to the given question, r = (9/8) / (3/4)
r = (2/3)
We also know,
Any term of a G.P. [nth term] can be obtained by the formula:
Tₙ= a[tex]r^{n-1}[/tex]
where, Tₙ= nth term
a= first term of G.P.
r=common ratio
Since last term of the G.P. is given to be 8/81; putting this in the above formula will yield us the total number of terms.
Tₙ= a[tex]r^{n-1}[/tex]
⇒ (8/81) = (9/8) x ([tex]2/3^{n-1}[/tex])
⇒ (64/729)= ([tex]2/3^{n-1}[/tex])
⇒[tex](2/3)^{6}[/tex] = ([tex]2/3^{n-1}[/tex])
⇒ n-1 = 6
⇒ n = 7
∴ The total number of terms in G.P. is 7.
Therefore, Common ratio of the sequence 9/8, 3/4, 1/2,...,8/81 is 2/3 and the number of all terms in this sequence is 7.
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The Common Ratio for this geometric sequence is 2/3 and the total number of terms in the sequence is 6.
Explanation:The given mathematical sequence appears to be a geometric sequence, which is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the Common Ratio. In a geometric sequence, you can find the Common Ratio by dividing any term by the preceding term.
So in this case, the second term (3/4) divided by the first term (9/8) equals 2/3. Therefore, the Common Ratio for this geometric sequence is 2/3.
To find the total number of terms in this sequence we use the formula for the nth term of a geometric sequence: a*n = a*r^(n-1), where a is the first term, r is the common ratio, and n is the number of terms. This gives us: 8/81 = (9/8)*(2/3)^(n-1). Solving this for n gives us n = 6. Therefore, the total number of terms in this sequence is 6.
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Given ABCD, what is the measure of
145
A. 90°
B. 35°
C. 10°
D. 145°
E. 55°
F. 235°
Answer: D. 145°
Step-by-step explanation:
Since it is a parallelogram given by the symbol, then angle B is equal to angle D which is 145°.
which of the following are like radicals? Check all
of the boxes that apply.
3x√√xy
-12x√√xy
-2x√√xj
x-√4x2²
-x√x²y
2√xy
Answer:
the first 2
Step-by-step explanation:
let me know if it is wrong
what best describes the relationship between the computed mean of 52.4 and the actual mean of 52.7
The computed mean of 52.4 and the actual mean of 52.7 suggest a close relationship in terms of central tendency.
A computed mean is a statistical measure calculated by summing up a set of values and dividing by the number of observations. In this case, the computed mean of 52.4 implies that when the values are averaged, the result is 52.4.
The actual mean of 52.7 refers to the true average of the population or data set being analyzed. Since it is higher than the computed mean, it indicates that the sample used for computation might have slightly underestimated the true population mean.
However, the difference between the computed mean and the actual mean is relatively small, with only a 0.3 unit discrepancy.
Given the proximity of these two values, it suggests that the computed mean is a reasonably accurate estimate of the actual mean.
However, it's important to note that without additional information, such as the sample size or the variability of the data, it is difficult to draw definitive conclusions about the relationship between the computed mean and the actual mean.
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1cm on a picture of a swimming pool represents 1200cm of the actual swimming pool. The length of the pictured swimming pool is 4.5cm and the width is 3cm. What is the perimeter of the actual swimming pool? Express your answer in meters.
Answer:
180 meters
Step-by-step explanation:
To find the perimeter of the actual swimming pool, you need to first find the length and width of the actual swimming pool by multiplying the length and width of the pictured swimming pool by the scale factor of 1200 cm.
Length of actual swimming pool = 4.5 cm × 1200 cm = 5400 cmWidth of actual swimming pool = 3 cm × 1200 cm = 3600 cmPerimeter of actual swimming pool = (5400 cm + 3600 cm) × 2 = 18000 cm.Now that we know that the perimeter of the actual pool is 18000 centimeters, we need to convert that to meters! Keep in mind that:
100cm = 1mNow we can divide 18000 by 100:
18000 cm ÷ 100 = 180 m
Therefore, the perimeter of the actual swimming pool is 180 m.
Dylan's mom told him that she would replace each one of his dimes with a quarter. If he uses all of his coins, determine if Dylan would then have enough money to buy a game priced at $20.98 if he must also pay an 8% sales tax.
PLEASE HELP
Suppose that the functions fand g are defined for all real numbers x as follows.
f(x) = 5x
g(x)=4x-4
Write the expressions for (g.f)(x) and (g-f)(x) and evaluate (g+f)(2).
(g•f)(x) =
(g-f)(x) =
(g+r) (2)=
A tour group has $83 to buy train tickets. Each ticket costs $18. How many train tickets can
the group buy?
Graph the function f(x)= 3+2 in x and its inverse from model 1.
The graph of the function and its inverse is added as an attachment
Sketching the graph of the function and its inverseFrom the question, we have the following parameters that can be used in our computation:
f(x) = 3 + 2ln(x)
Express as an equation
So, we have
y = 3 + 2ln(x)
Swap x and y in the above equation
x = 3 + 2ln(y)
Next, we have
2ln(y) = x - 3
Divide by 2
ln(y) = (x - 3)/2
Take the exponent of both sides
[tex]y = e^{\frac{x - 3}{2}}[/tex]
Next, we plot the graphs
The graph of the functions is added as an attachment
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Find the net area of the following curve on the interval [0, 2].
(SHOW WORK)
f(x) = ex - e
The net area of the curve represented by f(x) = ex - e on the interval [0, 2] is e2 - 1.
To find the net area of the curve represented by the function f(x) = ex - e on the interval [0, 2], we need to calculate the definite integral of the function over that interval. The net area can be determined by taking the absolute value of the integral.
The integral of f(x) = ex - e with respect to x can be computed as follows:
∫[0, 2] (ex - e) dx
Using the power rule of integration, the antiderivative of ex is ex, and the antiderivative of e is ex. Thus, the integral becomes:
∫[0, 2] (ex - e) dx = ∫[0, 2] ex dx - ∫[0, 2] e dx
Integrating each term separately:
= [ex] evaluated from 0 to 2 - [ex] evaluated from 0 to 2
= (e2 - e0) - (e0 - e0)
= e2 - 1
The net area of the curve represented by f(x) = ex - e on the interval [0, 2] is e2 - 1.
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GEOMETRY 100POINTSSS
Find x
Answer:
5.9
Step-by-step explanation:
sin Θ = opp/hyp
sin 36° = x/10
x = 10 × sin 36°
x = 5.88
Answer: 5.9
Suppose a finite population has 6 items and 2 items are selected at random without replacement,then all possible samples will be:
Select one:
a. 15
b. 2
c. 36
d. 6
e. 12
Note: Answer D is NOT the correct answer. Please find the correct answer. Any answer without justification will be rejected automatically.
When 2 items are selected without replacement from a population of 6 items, there are 15 possible samples that can be formed. Option A.
To determine the number of possible samples when 2 items are selected at random without replacement from a population of 6 items, we can use the concept of combinations.
The number of combinations of selecting k items from a set of n items is given by the formula C(n, k) = n! / (k! * (n-k)!), where n! represents the factorial of n.
In this case, we have a population of 6 items and we want to select 2 items. Therefore, the number of possible samples can be calculated as:
C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5 * 4!) / (2! * 4!) = (6 * 5) / (2 * 1) = 15. Option A is correct.
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Diseases tend to spread according to the exponential growth model. In the early days of AIDS, the growth factor (i.e. common ratio; growth multiplier) was around 1.9. In 1983, about 1600 people in the U.S. died of AIDS. If the trend had continued unchecked, how many people would have died from AIDS in 2003?
what is (0.3)0 in binominal distribution
Answer:
When p, the probability of success, is zero in a binomial distribution, the probability of getting exactly k successes in n trials is also zero for all values of k except when k is zero (i.e., when there are no successes).
So, in the case of (0.3)^0, the result would be 1, because any number raised to the power of 0 is equal to 1. Therefore, the probability of getting zero successes in a binomial distribution when the probability of success is 0.3 is 1.
What is the solution, if any, to the inequality |3x|\ge0? all real numbers no solution x\ge0 x\le0
Answer:
all real numbers
Step-by-step explanation:
Try a negative number, a positive number and zero for x.
All of them work.
Answer: all real numbers
James wants to have earned $6,180 amount of interest in 28 years. Currently he finds
that his annual interest rate is 6.12%. Calculate how much money James needs to invest
as his principal in order to achieve this goal.
Answer:
$3606.44
Step-by-step explanation:
The question asks us to calculate the principal amount that needs to be invested in order to earn an interest of $6180 in 28 years at an annual interest rate of 6.12%.
To do this, we need to use the formula for simple interest:
[tex]\boxed{I = \frac{P \times R \times T}{100}}[/tex],
where:
I = interest earned
P = principal invested
R = annual interest rate
T = time
By substituting the known values into the formula above and then solving for P, we can calculate the amount that James needs to invest:
[tex]6180 = \frac{P \times 6.12 \times 28}{100}[/tex]
⇒ [tex]6180 \times 100 = P \times 171.36[/tex] [Multiplying both sides by 100]
⇒ [tex]P = \frac{6180 \times 100}{171.36}[/tex] [Dividing both sides of the equation by 171.36]
⇒ [tex]P = \bf 3606.44[/tex]
Therefore, James needs to invest $3606.44.
93-(15x10)+(160:16) =
Answer:
Step-by-step explanation:
Let's calculate the expression step by step:
93 - (15 × 10) + (160 ÷ 16)
First, we perform the multiplication:
93 - 150 + (160 ÷ 16)
Next, we perform the division:
93 - 150 + 10
Finally, we perform the subtraction and addition:
-57 + 10
The result is:
-47
Therefore, 93 - (15 × 10) + (160 ÷ 16) equals -47.
Given the function f(x) = 4 – 2x, find f(3r – 1).
Answer:
f(3r - 1) = -6r + 6
Step-by-step explanation:
To find f(3r - 1), we substitute 3r - 1 for x in the expression for f(x) and simplify:
f(x) = 4 - 2x
f(3r - 1) = 4 - 2(3r - 1)
= 4 - 6r + 2
= -6r + 6
So, f(3r - 1) = -6r + 6.
How would you describe the difference between the graphs of f (x) = 3x²
and g(x) = -2² ?
OA. g(x) is a reflection of f(x) over the line y = x.
B. g(x) is a reflection of f(x) over the line y = -1.
C. g(x) is a reflection of f(x) over the x-axis.
D. g(x) is a reflection of f(x) over the y-axis.
Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C).
The given functions are f(x) = 3x² and g(x) = -2².
To understand the difference between their graphs, let's examine the characteristics of each function individually:
Function f(x) = 3x²:
The coefficient of x² is positive (3), indicating an upward-opening parabola.
The graph of f(x) will be symmetric with respect to the y-axis, as any change in x will result in the same y-value due to the squaring of x.
The vertex of the parabola will be at the origin (0, 0) since there are no additional terms affecting the position of the graph.
Function g(x) = -2²:
The coefficient of x² is negative (-2), indicating a downward-opening parabola.
The negative sign will reflect the graph of f(x) across the x-axis, resulting in a vertical flip.
The vertex of the parabola will also be at the origin (0, 0) due to the absence of additional terms.
Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C). This means that g(x) is obtained by taking the graph of f(x) and flipping it vertically. The reflection occurs over the x-axis, causing the parabola to open downward instead of upward.
Therefore, the correct answer is option C: g(x) is a reflection of f(x) over the x-axis.
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the population of a certain state can be estimated by the equation p=80.7t+18,312.3, where p represents the population of the state in thousands of people t years since 2010
The estimated population of the state in the year 2022 is 19,280,700 people.
The given equation represents the population of a certain state as a function of time, where p is the population in thousands of people and t is the number of years since 2010.
The equation is given as p = 80.7t + 18,312.3.
To estimate the population of the state, we substitute the value of t into the equation. For example, if we want to estimate the population in the year 2022 (12 years since 2010), we substitute t = 12 into the equation:
p = 80.7(12) + 18,312.3
= 968.4 + 18,312.3
= 19,280.7.
The estimated population of the state in the year 2022 is 19,280,700 people.
We can estimate the population for any given year by substituting the corresponding value of t into the equation.
It's important to note that the population is given in thousands of people, so we multiply the final result by 1,000 to obtain the population in actual numbers.
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Solve the missing element . use 3.14 for pi and Area = pi r2 ; C= pi D
We can solve for the missing elements as follows:
1. Radius - 10 inches
Diameter - 20
Circumference - 62.8
Area - 314
2. Radius - 6ft
Diameter - 12
Circumference - 37.68
Area - 113.04
3. Radius - 18
Diameter - 36 yards
Circumference - 113.04
Area - 1017.36
4. Radius 15
Diameter - 30 cm
Circumference 94.2
Area - 706.5
5. Radius - 5 mm
Diameter 10
Circumference 31.4
Area -78.5
6. Radius 20
Diameter - 40 inches
Circumference 125.6
Area -1256
How to solve for the valuesTo solve for the given values, we will use the formulas for area, circumference. Also, we can obtain the radius by dividing the diameter by 2 and the diameter is 2r. So we will solve for the values this way:
1. radius = 10 inches
diameter = 20
circumference = 2pie*r 2 *3.14*10 = 62.8
Area = 314
2. radius = 6ft
diameter = 12
circumference = 37.68
Area = 113.04
3. radius = 18
diameter = 36 yards
circumference = 113.04
Area = 1017.36
4. radius = 15
diameter = 30 cm
circumference = 94.2
Area = 706.5
5. radius = 5 mm
diameter = 10
circumference = 31.4
Area = 78.5
6. radius = 20 inches
diameter = 40 inches
circumference = 125.6
area = 1256
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50 PTS!!!!!!!!!!! I NEED HELP!!!!!
Answer this question based on the table above. Choose the right answer.
Is the statement true that between 1966 and 1976 the average number of miles flown per passenger increased by one-third. (Yes or no)
Answer:
No
Step-by-step explanation:
To determine if the average number of miles flown per passenger increased by one-third between 1966 and 1976, we need to compare the increase in miles flown during that period.
According to the given table:
In 1966, the average number of miles flown per passenger was 711 miles.In 1976, the average number of miles flown per passenger was 831 miles.To find the increase in miles flown, subtract the 1966 value from the 1976 value:
[tex]\begin{aligned}\sf Increase\; in\; miles\; flown &= \sf 831 \;miles - 711\; miles\\&= \sf 120\; miles\end{aligned}[/tex]
Therefore, the average number of miles flown per passenger between 1966 and 1976 increased by 120 miles.
To check if the increase is one-third of the initial value, we need to calculate one-third of the 1966 value:
[tex]\begin{aligned}\sf One\;third \;of \;711 \;miles &= \sf \dfrac{1}{3} \times 711\; miles\\\\ &= \sf \dfrac{711}{3} \; miles\\\\&=\sf 237\;miles\end{aligned}[/tex]
Since the increase in miles flown (120 miles) is not equal to one-third of the initial 1966 value (237 miles), the statement that the average number of miles flown per passenger increased by one-third between 1966 and 1976 is not true.
PLEASE HELPPPPPPP NEED NOW
Answer:
BC = 24 units
Step-by-step explanation:
This is an isosceles triangle which always has:
two legs that are congruent to each other (i.e., equal),and two angles that are congruent to each other.In this triangle, the legs CA and BA are congruent so CA = BA and the angles C and B are congruent to each other so angle C = angle B.
Thus, we can find x by setting CA and BA equal to each other:
(3x - 15 = x + 33) + 15
(3x = x + 48) - x
(2x = 48) / x
x = 24
Thus, x = 24
Since the length of BC is x and x = 24, BC is 24 units long.
45% of the Walton High School student body are male. 90% of Walton females love math, while only 60% of the males love math. What percentage of the student body loves math?
Approximately 76.5% of the student body at Walton High School loves math.
To determine the percentage of the student body that loves math, we need to consider the proportions of males and females in the Walton High School student body and their respective percentages of loving math.
Given that 45% of the student body are males, we can deduce that 55% are females (since the total percentage must add up to 100%). Now let's calculate the percentage of the student body that loves math:
For the females:
55% of the student body are females.
90% of the females love math.
So, the percentage of females who love math is 55% * 90% = 49.5% of the student body.
For the males:
45% of the student body are males.
60% of the males love math.
So, the percentage of males who love math is 45% * 60% = 27% of the student body.
To find the total percentage of the student body that loves math, we add the percentages of females who love math and males who love math:
49.5% + 27% = 76.5%
As a result, 76.5% of Walton High School's student body enjoys maths.
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Dewan’s bank account balance is -$16.75. He deposits checks totaling $23.59. What is his new balance? -$1.08
Answer:
$6.84
Step-by-step explanation:
This is quite a simple question, simply add the new deposited amount into the original balance to get your answer.
Original balance: -$16.75Deposit: $23.59New balance: -$16.75 + $23.59 = $6.84Circumference of circle inscribed or circumscribed polygon
Hint: you will need to find the diameter of the circle, use Pythagorean Theorem)
ind then I out of the 3 problems.
Find the exact circumference of each circle by using the given inscribed or circumscribed polygon.
8 cm
15 cm
The exact circumferences of the inscribed and circumscribed circles for the given polygons are 8π cm and 15π cm, respectively.
To find the exact circumference of a circle inscribed or circumscribed by a polygon, we can use the Pythagorean theorem to determine the diameter of the circle.
In the case of an inscribed polygon, the diameter of the circle is equal to the diagonal of the polygon. Let's consider the polygon with a diagonal of 8 cm. If we draw a line connecting two non-adjacent vertices of the polygon, we get a diagonal that represents the diameter of the inscribed circle.
Using the Pythagorean theorem, we can find the length of this diagonal. Let's assume the sides of the polygon are a and b. Then the diagonal can be found using the equation: diagonal^2 = a^2 + b^2. Substituting the given values, we have 8^2 = a^2 + b^2. Solving this equation, we find that a^2 + b^2 = 64.
For the circumscribed polygon with a diagonal of 15 cm, the diameter of the circle is equal to the longest side of the polygon. Let's assume the longest side of the polygon is c. Therefore, the diameter of the circumscribed circle is 15 cm.
Once we have determined the diameter of the circle, we can calculate its circumference using the formula C = πd, where C is the circumference and d is the diameter.
For the inscribed circle, the circumference would be C = π(8) = 8π cm.
For the circumscribed circle, the circumference would be C = π(15) = 15π cm.
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Find the limit (if the limit exists). Solve in two different ways.
The limit of the trigonometric expression is equal to 0.
How to determine the limit of a trigonometric expression
In this problem we find the case of a trigonometric expression, whose limit must be found. This can be done by means of algebra properties, trigonometric formula and known limits. First, write the entire expression below:
[tex]\lim_{\Delta x \to 0} \frac{\cos (\pi + \Delta x) + 1}{\Delta x}[/tex]
Second, use the trigonometric formula cos (π + Δx) = - cos Δx to simplify the resulting formula:
[tex]\lim_{\Delta x \to 0} \frac{1 - \cos \Delta x}{\Delta x}[/tex]
Third, use known limits to determine the result:
0
The limit of the trigonometric function [cos (π + Δx) + 1] / Δx evaluated at Δx → 0 is equal to 0.
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A village P is 12 km from village Q. It takes 3 hours 20 minutes to travel from Q to P and back to Q by a boat. If the boat travels at a speed of 6 km/h from P to Q and (6 + x) km/h back to P, find the value of x.
Answer:
Hope this helps and have a nice day
Step-by-step explanation:
To find the value of x, we can use the formula:
Time = Distance / Speed
Let's calculate the time taken to travel from Q to P and back to Q.
From Q to P:
Distance = 12 km
Speed = 6 km/h
Time taken from Q to P = Distance / Speed = 12 km / 6 km/h = 2 hours
From P to Q:
Distance = 12 km
Speed = (6 + x) km/h
Time taken from P to Q = Distance / Speed = 12 km / (6 + x) km/h
Given that the total time taken for the round trip is 3 hours 20 minutes, we can convert it to hours:
Total time = 3 hours + (20 minutes / 60) hours = 3 + (1/3) hours = 10/3 hours
According to the problem, the total time is the sum of the time from Q to P and from P to Q:
Total time = Time taken from Q to P + Time taken from P to Q
Substituting the values:
10/3 hours = 2 hours + 12 km / (6 + x) km/h
Simplifying the equation:
10/3 = 2 + 12 / (6 + x)
Multiply both sides by (6 + x) to eliminate the denominator:
10(6 + x) = 2(6 + x) + 12
60 + 10x = 12 + 2x + 12
Collecting like terms:
8x = 24
Dividing both sides by 8:
x = 3
Therefore, the value of x is 3.
Answer:
x = 3
Step-by-step explanation:
speed = distance / time
time = distance / speed
Total time from P to Q to P:
T = 3h 20min
P to Q :
s = 6 km/h
d = 12 km
t = d/s
= 12/6
t = 2 h
time remaining t₁ = T - t
= 3h 20min - 2h
= 1 hr 20 min
= 60 + 20 min
= 80 min
t₁ = 80/60 hr
Q to P:
d₁ = 12km
t₁ = 80/60 hr
s₁ = d/t₁
[tex]= \frac{12}{\frac{80}{60} }\\ \\= \frac{12*60}{80}[/tex]
= 9
s₁ = 9 km/h
From question, s₁ = (6 + x)km/h
⇒ 6 + x = 9
⇒ x = 3
NEED NOW PLEASE HELP OUT
Answer:
x=50
Step-by-step explanation:
Make this equal to 180.
x+3x-35+x-35 = 180
5x = 180 + 70
5x=250
x=50
What else would need to be congruent to show that ABC=AXYZ by SAS?
A
B
OA. ZB=LY
B. BC = YZ
OC. C= LZ
OD. AC = XZ
с
X
Z
Given:
AB XY
BC=YZ
What is needed to be congruent to show that ABC=AXYZ is AC ≅ XZ. option D
How to determine the statementGiven that in ΔABC and ΔXYZ, ∠X ≅ ∠A and ∠Z ≅ ∠C.
We are to select the correct condition that we will need to show that the triangles ABC and XYZ are congruent to each other by ASA rule..
ASA Congruence Theorem: Two triangles are said to be congruent if two angles and the side lying between them of one triangle are congruent to the corresponding two angles and the side between them of the second triangle.
In ΔABC, side between ∠A and ∠C is AC,
in ΔXYZ, side between ∠X and ∠Z is XZ.
Therefore, for the triangles to be congruent by ASA rule, we must have AC ≅ XZ.
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Similar Triangles
Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to
prove the triangles similar? Explain your reasoning.
I need help on number 1 and 2
The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;
1. ΔAJK ~ ΔSWY according to the SAS similarity postulate
2. ΔLMN ~ ΔLPQ according to the AA similarity postulate
3. ΔPQN ~ ΔLMN
LM = 12, QP = 8
4. ΔLMK~ΔLNJ
NL = 21, ML = 14
What are similar triangles?
Similar triangles are triangles that have the same shape but may have different sizes.
1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;
24/16 = 3/2
18/12 = 3/2
The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.
2. The ratio of the corresponding sides in each of the triangles are;
MN/LN = 8/10 = 4/5
PQ/LQ = 12/(10 + 5) = 12/15 = 4/5
The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule
3. The alternate interior angles theorem indicates;
Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;
ΔPQN ~ ΔLMN by the AA similarity postulate
LM/QP = (x + 3)/(x - 1) = 18/12
12·x + 36 = 18·x - 18
18·x - 12·x = 36 + 18 = 54
6·x = 54
x = 54/6 = 9
LM = 9 + 3 = 12
QP = x - 1
QP = 9 - 1 = 8
4. The similar triangles are; ΔLMK and ΔLNJ
ΔLMK ~ ΔLNJ by AA similarity postulate
ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)
ML/NL = LK/LJ = (24 - 8)/24
(24 - 8)/24 = (6·x + 2)/((7·x + 7)
16/24 = (6·x + 2)/(7·x + 7)
16 × (7·x + 7) = 24 × (6·x + 2)
112·x + 112 = 144·x + 48
144·x - 112·x = 32·x = 112 - 48 = 64
x = 64/32 = 2
ML = 6 × 2 + 2 = 14
NL = 7 × 2 + 7 = 21
MN = 2 + 5 = 7
Learn more on similar triangles here: https://brainly.com/question/2644832
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