The auditorium at P.S 104 has 28 rows in all each consists of 95 unfortunately, 30 seats are broken calculate the total number of seats that are broken in the auditorium

the answer of this question is 603.2yd2

Answer:Area of the shaded region is 603.2 yards^2

Step-by-step explanation:

The diagram contains two circles. The smaller circle has a radius of 8 yards.

The bigger circle has a radius of 16 yards.

The area of a circle is expressed as

Area of circle = πr^2

Where

r = radius of circle

π is a constant whose value is 3.142

The area of the smaller circle would be

3.142 × 8^2 = 201.088 yards^2

The area of the bigger circle would be

3.14 × 16^2 = 804.352 yards^2

Area of the shaded region would be area of the bigger circle - area of the smaller circle. It becomes

804.352 - 201.088 = 603.2 yards^2

Answer:

The answer to your question is letter A

Step-by-step explanation:

Process

1.- Find two points of the line

 A ( -2, 0)

 B ( 0, 3)

2.- Find the slope

     [tex]m = \frac{y2 - y1}{x2 - x1}[/tex]

            [tex]m = \frac{3 - 0}{0 + 2}[/tex]

                   [tex]m = \frac{3}{2}[/tex]

3.- Find the equation of the line

            y - y1 = m(x - x1)

            y - 0 = [tex]\frac{3}{2} (x + 2)[/tex]

            y = [tex]\frac{3}{2} (x + 2)[/tex]

Simplify

            y = [tex]\frac{3}{2} x + 3[/tex]

The store received 1725 pencils and 1350 pens as of the given conditions.

Given that,
A store receives a shipment of pens and pencils. Each box contains 75 items There are 23 boxes of pencils and 18 boxes of pens. How many pens and pencils the store receive is to be determined.

In mathematics, it deals with numbers of operations according to the statements. There are four major arithmetic operators, addition, subtraction, multiplication and division,

Here,
Each box contains 75 items,
The number of item in 23 pencil boxes = 75 × 23 = 1725 items,
The number of items in 18 pencil boxes = 18 × 75 = 1350 items.

Thus, the store received 1725 pencils and 1350 pens as of the given conditions.

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Answer:

The answer to your question is $43.15

Step-by-step explanation:

Current bill = 25, 789 kWh

Previous bill = 25, 350 kWh

Charge = $ 0.0983/ kWh

Process

1.- Subtract the current meter from the previous one

                   25, 789 - 25, 350 = 439 kWh

2.- Find the amount of the bill using a rule of three

                 $0.0983     -----------------   1 kWh

                      x             ----------------   439 kWh

                     x = (439 x 0.0983) / 1

                     x = $ 43.15

Answer:

The sprinkler must rotate by an angle of 107.48°.

Step-by-step explanation:

Given:

Area of strawberry patch( in shape of sector)  = 1500 square yards

Radius of circle = 40 yards

To find angle through which the sprinkler should rotate.

Solution.

In order to find the angle of rotation of sprinkler, we will apply the area of sector formula.

[tex]Area\ of\ sector\ = \frac{\theta}{360}\times \pi r^2[/tex]

where [tex]\theta[/tex] is the angle of the sector formed and [tex]r[/tex] is radius of the circle.

Thus, we can plugin the given values to find [tex]\theta[/tex] which would be the angle of rotation.

[tex]1500 = \frac{\theta}{360}\times \pi (40)^2[/tex]

Taking [tex]\pi=3.14[/tex]

[tex]1500 = \frac{\theta}{360}\times \ (3.14) (40)^2[/tex]

[tex]1500 = \frac{\theta}{360}\times 5024[/tex]

Dividing both sides by 5024.

[tex]\frac{1500}{5024} = \frac{\theta}{360}\times 5024\div 5024[/tex]

Multiplying both sides by 360.

[tex]\frac{1500\times 360}{5024} =\frac{\theta}{360}\times 360[/tex]

[tex]107.48=\theta[/tex]

∴ [tex]\theta= 107.48\°[/tex]

Angle of rotation of sprinkler = 107.48°

The strawberry farmer needs the sprinkler to rotate approximately 107.46 degrees to cover the 1500 square yard sector of the circle with a 40-yard radius.

To determine the angle through which the sprinkler should rotate, we first need to find the area of the sector of the circle. The formula for the area of a sector is:

A = 0.5 × r² × θ

where A is the area, r is the radius, and θ is the angle in radians. Here, we know the area A is 1500 square yards and the radius r is 40 yards. Rearranging the formula to solve for θ gives:

θ = (2 × A) / r²

Substituting the given values:

θ = (2 × 1500) / 40²

θ = (3000) / 1600

θ = 1.875 radians

To convert this angle in radians to degrees, we use the conversion factor 180/π:

θ = 1.875 × (180 / π) ≈ 107.46 degrees

Therefore, the sprinkler should rotate through an angle of approximately 107.46 degrees.

Answer:

0.2288

Step-by-step explanation:

This is straightforward.

What we need to do is to divide the number of defective semiconductor chips over the total number of semiconductor chips.

Initially, the total number of semi conductor chips is 119 and the total number of defective semiconductor chips is 28.

After the first selections , we can infer that the total number of semiconductor chips is 118 while the number of defective ones is 27.

Hence on the second drawing, the probability that he will

Select a defective one is 27/118

Answer:

11 positive integers can be expressed.

Step-by-step explanation:

Consider the provided information.

The number of possible prime numbers are 5,7,11,and 13.

There are 4 possible prime numbers.

How many positive integers can be expressed as a product of two or more of the prime numbers, that means there can be product of two numbers, three number or four numbers.

The formula to calculate combinations is: [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

The number of ways are:

[tex]^4C_2+^4C_3+^4C_4=\frac{4!}{2!(4-2)!}+\frac{4!}{3!(4-3)!}+\frac{4!}{4!}[/tex]

[tex]^4C_2+^4C_3+^4C_4=\frac{4!}{2!2!}+\frac{4!}{3!}+1[/tex]

[tex]^4C_2+^4C_3+^4C_4=6+4+1[/tex]

[tex]^4C_2+^4C_3+^4C_4=11[/tex]

Hence, 11 positive integers can be expressed.

Good evening ,

Answer:

2x² - 18x + 36

Step-by-step explanation:

2(x-3)(x-6) = 2x²-18x+36.

:)

Answer:

[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:

[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]

Step-by-step explanation:

From the info given by the problem we need an integer defined as the smallest positive integer that is a multiple of 75 and have 75 positive integral divisors, and we are assuming that 1 is one possible divisor.

Th first step is find the prime factorization for the number 75 and we see that

[tex]75=3 5^2[/tex]

And we know that 3 =2+1 and 5=3+2 and if we replace we got:

[tex] 75 = (2+1)(4+1)^2 = (2+1)(4+1)(4+1)[/tex]

And in order to find 75 integral divisors we need to satisify this condition:

[tex]n= a^{r_1 -1}_1 a^{r_2 -1}_2 *......[/tex] such that [tex]a_1 *a_2*....=75[/tex]

For this case we have two prime factors important 3 and 5. And if we want to minimize n we can use a prime factor like 2. The least common denominator between 2 and 4 is LCM(2,4) =4. So then the need to have the prime factors 2 and 3 elevated at 4 in order to satisfy the condition required, and since 5 is the highest value we need to put the same exponent.

And then the value for n would be given by:

[tex]n=2^4 3^4 5^2 =32400[/tex] and then we have:

[tex]\frac{n}{75}=\frac{2^4 3^4 5^2}{3 5^2}=432[/tex]

The smallest positive integer that is a multiple of 75 is 32400 and

integral divisors are 432.

Positive integers are the whole number that is greater than zero and do not include decimal or fraction values.

Find the smallest positive integer that is a multiple of 75 that has exactly 75 positive integral divisors.

We know that

[tex]75 = 3*5^{2}[/tex]

So the value of n which has 75 divisors then the multiplication of power of prime factor should be 75. The formula is given by

[tex]n = 2^{x-1} *3^{y-1} *5^{z-1}[/tex]

then the multiplication of x, y, and z must be 75.

x, y, and z are 5, 5, and 3 will be the values.

[tex]n = 2^{4} *3^{4} *5^{2} = 32400[/tex]

And it is divisible by 75 also.

[tex]\dfrac{n}{3*5^{2} } = \dfrac{2^{4} *3^{4} *5^{2}}{3*5^{2} } = 432[/tex]

Thus,  the smallest positive integer that is a multiple of 75 is 32400 and integral divisors are 432.

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Answer:

[tex]h=\frac{SA-2s^2}{4s}[/tex]

Step-by-step explanation:

we know that'

The formula to calculate the surface area of a square prism is

[tex]SA=2s^2+4sh[/tex]

where

s is the length of the side of the square base

h is the height of the prism

Solve for h

That means ----> Isolate the variable h

so

subtract 2s^2 both sides

[tex]SA-2s^2=4sh[/tex]

Divide by 4s both sides

[tex]\frac{SA-2s^2}{4s}=h[/tex]

Rewrite

[tex]h=\frac{SA-2s^2}{4s}[/tex]

To find the height of a square prism when given the surface area and base side length, use the formula B. H = (SA - 2s^2) / (4s).

The question asks for a formula that can be used to find the height (h) of a square prism given the surface area (SA) and the side length of the base (s). The surface area of a square prism is calculated using the formula SA = 2s2 + 4sh. To solve for h, we need to re-arrange this equation:

Therefore, the correct formula to find h is B. H = (SA – 2s2) / 4s.

Answer:

Lines 'a' and 'c' are parallel. Line 'b' is non parallel to 'a' and 'c'.

Step-by-step explanation:

Picture is attached with the answer as a solution to it.

In the question it is give that line 'c' only intersect with line 'b' and not with, so the only way that two lines won't intersect here 'a' and 'c' is that lines are parallel. If the lines are not parallel then those lines will definitely meet.

Answer:

The remainder is 64

Step-by-step explanation:

Consider the function [tex]f(x)=x^3+3x^2-51x+91[/tex]

Use synthetic division to divide by x+9

x+9=0, x=-9

divide the given f(x) by -9 using synthetic division

-9               1           3       -51       91

                  0          -9     54        -27              

         --------------------------------------------------

                  1           -6      3         64

The remainder is 64

Total envelopes: 3 + 1 + 3 + 2 = 9 with a total of 3 green ones.

Probability of picking green first: 3 out of 9 = 3/9 = 1/3

There are 8 envelopes left, with 2 green ones.

Probability of picking a green one is 2 out of 8 = 2/8 = 1/4

There are 7 envelopes left and 1 green one.

Probability of picking green is 1 out of 7 = 1/7

Probability of picking all 3 = 1/3 x 1/4 x 1/7 = 1/84

Answer:

[tex]cos(x)=\frac{\sqrt{5}}{5}[/tex]

Step-by-step explanation:

we have that

[tex]tan(x)=-2[/tex]

The angle x is in quadrant IV

That means ---> The value of cos(x) is positive and the value of sin(x) is negative

Remember that

[tex]cos^2(x)+sin^2(x)=1[/tex] ----> equation A

[tex]tan(x)=\frac{sin(x)}{cos(x)}[/tex]

so

[tex]-2=\frac{sin(x)}{cos(x)}[/tex]

[tex]sin(x)=-2cos(x)[/tex] ----> equation B

substitute equation B in equation A

[tex]cos^2(x)+(-2cos(x))^2=1[/tex]

solve for cos(x)

[tex]cos^2(x)+4cos^2(x)=1[/tex]

[tex]5cos^2(x)=1[/tex]

[tex]cos^2(x)=\frac{1}{5}[/tex]

square root both sides

[tex]cos(x)=\pm\frac{1}{\sqrt{5}}[/tex]

but remember that the value of cos(x) is positive (IV quadrant)

[tex]cos(x)=\frac{1}{\sqrt{5}}[/tex]

simplify

[tex]cos(x)=\frac{\sqrt{5}}{5}[/tex]

Answer:

The answer is k=77

Step-by-step explanation:

It is given that n is an integer and greater than 2. Let us simplify k = (n + 2)(n – 2) to k=n^2-4.

In (1) it is stated that k is the product of two primes and in (2) it is stated that k<100. To solve the question we need to consider both cases (1) and (2).

Let say n=9 then k=n^2-4 becomes k=81-4=77. Well, 77 is the product of two prime numbers 7 and 11 (77=7*11). The answer is k=77

Answer:

Step-by-step explanation:

cos(c+d)=cos c  cos d-sin c sin d   ...(1)

cos(c-d)=cos  c  cos d+sin c sin d  ...(2)

add (1) and (2)

cos (c+d)+cos (c-d) =2 cos c cos d  ...(3)

put c+d=x

c-d=y

add

2c=x+y

c=(x+y)/2

subtract

2d=x-y

d=(x-y)/2

substitute in (3)

[tex]cos~ x+cos~ y=2 cos (\frac{x+y}{2}) ~cos (\frac{x-y}{2})[/tex]

To prove the identity cos x + cos y = 2cos((x+y)/2) cos((x-y)/2), we need to show that x+y/2+x-y/2=x and find a similar expression using x+y/2 and x-y/2 that equals y. Once we have these expressions, we can use them to prove the identity.

To prove the identity cos x + cos y = 2cos((x+y)/2) cos((x-y)/2), we need to show that x+y/2+x-y/2=x and find a similar expression using x+y/2 and x-y/2 that equals y. Once we have these expressions, we can use them to prove the identity.

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Answer:

[tex]P(X\geq 5)=1-P(X< 5)=1-[0.000649+0.00703+0.0343+0.0991+0.1878]=0.6712[/tex]

Step-by-step explanation:

1) Previous concepts  

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".  

2) Solution to the problem  

Let X the random variable of interest, on this case we now that:  

[tex]X \sim Binom(n=10, p=0.52)[/tex]  

The probability mass function for the Binomial distribution is given as:  

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]  

Where (nCx) means combinatory and it's given by this formula:  

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]  

What is the probability that at least 5 of the students in your study group of 10 have studied in the last week?

[tex]P(X\geq 5)=1-P(X< 5)=1-[P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)][/tex]

[tex]P(X=0)=(10C0)(0.52)^0 (1-0.52)^{10-0}=0.000649[/tex]  

[tex]P(X=1)=(10C1)(0.52)^1 (1-0.52)^{10-1}=0.00703[/tex]  

[tex]P(X=2)=(10C2)(0.52)^2 (1-0.52)^{10-2}=0.0343[/tex]  

[tex]P(X=3)=(10C3)(0.52)^3 (1-0.52)^{10-3}=0.0991[/tex]  

[tex]P(X=4)=(10C4)(0.52)^4 (1-0.52)^{10-4}=0.1878[/tex]  

[tex]P(X\geq 5)=1-P(X< 5)=1-[0.000649+0.00703+0.0343+0.0991+0.1878]=0.6712[/tex]

To find the probability that at least 5 of the students in a study group of 10 have studied in the last week, use the binomial probability formula and calculate the respective probabilities for each case. Add these probabilities together to get the final probability.

To calculate the probability that at least 5 of the students in your study group of 10 have studied in the last week, we can use the binomial probability formula. Let's denote the probability that a randomly selected teenager studied at least once during the week as p = 0.52. We want to find P(X >= 5) where X represents the number of teenagers in the study group who studied.

Using the binomial probability formula, P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10). We can calculate each of these individual probabilities using the formula: [tex]P(X = k) = C(n, k) * p^k * (1-p)^(^n^-^k^),[/tex] where C(n, k) is the combination of n items taken k at a time.

Once we have calculated each of these probabilities, we can add them together to find the final probability.

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Answer:

Step-by-step explanation:

Mrs Cain coleslaw recipe calls for 1/3 cup of oil, 1/2 cup of vinegar, 1/4 cup of sugar. This means that the ratio of the number of cups of oil to vinegar to sugar is 1/3 : 1/2: 1/4

If she has 1 cup of vinegar, the ratio that would correspond to the given ratio would be determined by multiplying the given ratio by 2. It becomes, 2/3 : 2/2 : 2/4 = 2/3 : 1 : 1/2

Therefore, she would require 2/3 cups if oil and 1/2 cups of sugar to make 1 batch.

Answer:

[tex]a_{n+1}=0.2a_n[/tex] for all n>0, [tex]a_1=16[/tex]

Step-by-step explanation:

Let [tex]\{a_n\}=\{16,3.2,0.64,0.128,\cdots \}[/tex] be the sequence described.

A geometric sequence has the following property: there exists some r (the ratio of the sequence) such that [tex]\frac{a_{n+1}}{a_n}=r[/tex] forr all n>0.

To find r, note that

[tex]\frac{3.2}{16}=\frac{32}{10(16)}=\frac{2}{10}=\frac{1}{5}=0.2[/tex]

Similarly

[tex]\frac{0.64}{3.2}=\frac{64}{10(32)}=\frac{1}{5}=0.2[/tex]

[tex]\frac{0.128}{0.64}=\frac{1}{5}=0.2[/tex]

Thus [tex]a_{n+1}=r a_n=\frac{a_n}{5}=0.2a_n[/tex] for all n>0, and [tex]a_1=16[/tex]

first term = 16

average ratio = 0.2

Answer:

(1) Insufficient data

(2) Sufficient data

Step-by-step explanation:

We need to check whether the given data is sufficient or not to prove that a rectangular region has perimeter P inches and area A square inches, is the region square.

Assume that the given conditions are

1. [tex]P=\frac{4}{3}A[/tex]

2. [tex]P=4\sqrt{A}[/tex]

Area of square is

[tex]A=a^2[/tex]

Taking square root on both sides.

[tex]\sqrt{A}=a[/tex]             .... (1)

where a is side length.

Perimeter of square is

[tex]P=4a[/tex]            ... (2)

From (1) and (2) we get

[tex]P=4\sqrt{A}[/tex]

It means second condition is sufficient to prove that the rectangle reason is square, because it is true for all.

For a=3,

A=9 and P=12

[tex]12=\frac{4}{3}(9)[/tex]

[tex]12=12[/tex]

LHS=RHS

For a=6,

A=36 and P=24

[tex]24=48[/tex]   (False statement)

The first condition is insufficient because it may or may not be true.

Y is a bisector of the two sides, so would be half the length of the base.

Y = 36 / 2

y = 18

The term quadratic comes from the word quadrate meaning square or rectangular. Similarly, one of the definitions of the term quadratic is a square. In an algebraic sense, the definition of something quadratic involves the square and no higher power of an unknown quantity; second degree. So, for our purposes, we will be working with quadratic equations which mean that the highest degree we'll be encountering is a square. Normally, we see the standard quadratic equation written as the sum of three terms set equal to zero. Simply, the three terms include one that has an [tex]x^{2}[/tex], one has an x, and one term is "by itself" with no [tex]x^{2}[/tex] or x.

Answer:the football team played 25 games in all

Step-by-step explanation:

Let x represent the total number of matches that the football team played.

The football team won 10 matches out of the total number of matches they played. if their win percentage was 40, it means that

10/x × 100 = 40

1000 = 40x

x = 1000/40 = 25

Answer:

[tex]I_2=9[/tex]

Step-by-step explanation:

We have been told that the ratios are inversely proportional in our given problem. We are asked to find the missing value.

We know that two inversely proportional quantities are in form [tex]y=\frac{k}{x}[/tex], where, y is inversely proportional to x and k is the constant of proportionality.

Let us find constant of proportionality using [tex]R_1 = 6[/tex] and [tex]I_1 = 12[/tex] in above equation.

[tex]6=\frac{k}{12}[/tex]

[tex]6*12=\frac{k}{12}*12[/tex]

[tex]72=k[/tex]

Now, we will use [tex]72=k[/tex] and [tex]R_2 = 8[/tex] in our equation to find [tex]I_2[/tex] as:

[tex]8=\frac{72}{I_2}[/tex]

[tex]I_2=\frac{72}{8}[/tex]

[tex]I_2=9[/tex]

Therefore, the value of [tex]I_2[/tex] is 9.

The question is about the mathematical concept of inverse proportionality. The missing value of [tex]I_2[/tex] is found by applying the property of inversely proportional quantities, yielding the result [tex]I_2 = 9[/tex].

In an inverse proportionality relationship, the product of the values in one set is equal to the product of the corresponding values in the other set. This can be represented as:

[tex]\[R_1 \cdot I_1 = R_2 \cdot I_2\][/tex]

Given that [tex]\(R_1 = 6\)[/tex], [tex]\(R_2 = 8\)[/tex], and [tex]\(I_1 = 12\)[/tex], you can solve for [tex]\(I_2\)[/tex]:

[tex]\[6 \cdot 12 = 8 \cdot I_2\][/tex]

Now, simplify the equation:

[tex]\[72 = 8 \cdot I_2\][/tex]

To isolate [tex]\(I_2\)[/tex], divide both sides by 8:

[tex]\[I_2 = \frac{72}{8}\][/tex]

[tex]\[I_2 = 9\][/tex]

So, the value of [tex]\(I_2\)[/tex] is 9. In this inverse proportionality relationship, when [tex]R_1[/tex] is 6, [tex]\(R_2\)[/tex] is 8, and [tex]\(I_1\)[/tex] is 12, [tex]\(I_2\)[/tex] is 9. This means that as one variable increases, the other variable decreases in such a way that their product remains constant.

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At 8 hours of service, the two limousine rental companies charge the same amount.

Step-by-step explanation:

Given,

Initial fee of Executive Limousine rental = $200

Per hour charges of service = $40

Let,

x be the number of hours.

E(x) = 40x+200

Initial fee of Jet Limousine rental = $120

Per hour charges of service = $50

J(x) = 50x + 120

For the charges to be same;

E(x) = J(x)

[tex]40x+200=50x+120\\200-120=50x-40x\\80=10x\\10x=80[/tex]

Dividing both sides by 10

[tex]\frac{10x}{10}=\frac{80}{10}\\x=8[/tex]

At 8 hours of service, the two limousine rental companies charge the same amount.

Keywords: function, addition

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Answer:

Heather pay will be 290 for 40 hours of work.

Step-by-step explanation:

Given:

Amount he gets paid weekly =174

Number of hours of work =24

we need to find the amount of pay for 40 hours of work.

Also Given:

weekly pay is directly proportional to the number of hours.

Framing in equation form we get;

Amount Of Pay ∝ Number of hours of work

Hence Amount of Pay = k × Numbers of hours of work.

where k is constant.

Substituting the values we will find the value of k

[tex]174 = k \times 24\\\\k = \frac{174}{24} = 7.25[/tex]

Now using this we will find the amount of pay when hours of work is 40.

Amount Of Pay = [tex]7.25\times 40 = 290[/tex]

Hence Heather pay will be 290 for 40 hours of work.

Answer: confidence interval = 27.5 +/- 1.68

= ( 25.82, 29.18)

Step-by-step explanation:

Given;

Number of samples n = 64

Standard deviation r = 6mi/gallon

Mean x = 27.5mi/gallon

Confidence interval of 97.5%

Z' = t(0.0125) = 2.24

Confidence interval = x +/- Z'(r/√n)

= 27.5 +/- 2.24(6/√64)

= 27.5 +/- 1.68

= ( 25.82, 29.18)

Answer: the number of new light bulbs that they put in is 36

Step-by-step explanation:

The Marino's put new light bulbs in all the fixtures the new house.

Each fixture used 2 light bulbs and each room had 3 fixtures. This means that the number of bulbs in each room would be 3×2 = 6 bulbs.

The new house had 6 rooms. This means that the total number of bulbs in 6 rooms would be

6 × 6 = 36 bulbs

Answer:

Step-by-step explanation:

1. Correct. Corresponding vertices are (A, D), (B, E), and (C, F). The triangle names must list corresponding vertices in the same order. Thus the answer is the one you have chosen.

__

2. Look again. As in the first problem, corresponding vertices are in the same order. Line segment EF is named by the first two vertices in the triangle name, so the corresponding segment is LM, also named by the first two vertices of that triangle's name.

Segment FG is named by the last two vertices of the triangle's name, so the corresponding segment in the other triangle is also named by the last two vertices of the triangle's name: MN. Corresponding segments are proportional, so EF/LM = FG/MN.

__

3. Correct. Segment DB is congruent to itself. The halves of the bisected angle at D are congruent to each other, and the segments DA and DC are shown as congruent. Hence, SAS is an appropriate choice for showing congruence of the triangles.

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4. The 1:1 ratio of these similar triangles tells you they are congruent, so ...

  AC = EC

  4x -3 = 2x +6 . . . . substitute the given expressions

  2x = 9 . . . . . . . . . . add 3-2x

  EC = 9+6 = 15 . . . substitute for 2x in the expression for EC

The question asks for AE, which is 2 times the length EC, so is ...

 AE = 2·EC = 2·15 = 30 . . . . feet

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The only indication that the units are feet is that the answer choice with the number 30 has feet as its units. (We should be seeing "(4x-3) feet" as the measure of AC, for example.)