Unit 4: Congruent Triangles Homework 5: Proving Triangles Congruent: SSS & SAS

If an expression for x that does not use an absolute value is y, rewrite |x-y| as x - y.

It is the same as rearranging one expression to plug it into another expression when rewriting algebraic expressions using structure. Solving for one of the variables and then plugging the resulting expression for that variable into the other expression is the initial step to take in these kinds of issues.

Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.

Here given :

|x−y| , if x then :

y, x - y is rewritten for  |x−y|

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Correct Question:

Rewrite each of the following expressions without using absolute value.

1. |x−y| , if x

The length of curve DE is equal to 26.25 units.

In Mathematics and Geometry, Pythagorean's theorem is modeled by the following mathematical expression:

x² + y² = z²

Where:

x, y, and z represents the length of sides or side lengths of any right-angled triangle.

In order to determine the length of the hypotenuse in this right-angled triangle, we would have to apply Pythagorean's theorem as follows;

AC² + BC² = AB²

20² + 15² = AB²

AB² = 400 + 225

AB = √625

AB = 25 units.

For the length of curve DE, we have:

DE = 105% of AB

DE = 1.05 × 25

DE = 26.25 units.

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The average temperature during the period from 9am to 9pm is approximately 32.51 degrees Fahrenheit

To find the temperature at 9am, we can simply plug in t=0 into the given function:

T(0) = 30 + 19 sin(0) = 30

So the temperature at 9am is 30 degrees Fahrenheit.

To find the temperature at 3pm, we need to find the value of t that corresponds to 3pm. Since 3pm is 6 hours after 9am, we have t=6:

T(6) = 30 + 19 [tex]sin((pi/12)*6[/tex]) = 30 + 19 s[tex]in(pi/2)[/tex] = 30 + 19 = 49

So the temperature at 3pm is 49 degrees Fahrenheit.

To find the average temperature during the period from 9am to 9pm, we need to find the average value of the function T(t) over the interval [0,12]. We can use the formula for the average value of a function:

avg(T) = (1/(b-a)) * ∫[a,b] T(t) dt

In this case, a=0 and b=12, so we have:

avg(T) =[tex](1/12) * ∫[0,12] (30 + 19 sin(pit/12[/tex])) dt

Integrating term by term, we get:

avg(T) = (1/12)[tex]* (30t - (19/12) *[/tex] ([tex]12cos(pit/12[/tex])) |[0,12]

Evaluating the expression at t=12 and t=0, we get:

[tex]avg(T) = (1/12)[/tex] [tex]* (3012 - (19/12) * (12cos[/tex][tex](pi)) - (300 - (19/12) * (12cos(0))))[/tex]

Simplifying, we get:

[tex]avg(T) = (1/12) *[/tex] (360 + 38.13) = 32.51

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The measure of the angle ∠CMD is 45.

We have,

From the figure,

∠FMD = 15

∠BMC = 30

Now,

∠BMF = 90

This can be written as,

∠BMC + ∠CMD + ∠FMD = 90

30 + ∠CMD + 15 = 90

∠CMD = 90 - 45

∠CMD = 45

Thus,

The measure of the angle ∠CMD is 45.

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The evidence at the 0.05 level of significance contradict claim that the bags should fail inspection.

We can use a one-tailed test with the null hypothesis that the standard deviation of the bags' weights is no more than 4 grams and the alternative hypothesis that the standard deviation is greater than 4 grams.

The test statistic for this hypothesis test is given by:
t = [tex]\frac{\frac{s}{\sqrt{n}}}{\frac{sigma}{\sqrt{n}}}[/tex]
where s is the sample standard deviation, n is the sample size, and sigma is the population standard deviation (which is assumed to be 4 grams).

Plugging in the given values, we get:
t = [tex]\frac{\frac{5.3}{\sqrt{27}}}{\frac{4}{\sqrt{ 27}}}[/tex] ≈ 1.325
Using a t-distribution table with 26 degrees of freedom (since we have a sample size of 27 and are estimating the population standard deviation), we can find the critical value for a one-tailed test at the 0.05 level of significance. The critical value is 1.705.

Since our calculated test statistic (1.325) is less than the critical value (1.705), we can support the null hypothesis and conclude that the bags of potato chips will not fail inspection.

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The  value of a= 17.5, b=-14 and c=-28.

A function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation

Given function:

f (x) = 3x – 1.

Also,

x        f(x)

-5       a

-2       7

4        b

8       c

Now, using the proportionality

k = y/x

k = 7 / (-2)

k = -3.5

So, -3.5 = a/ (-5)

-3.5 x (-5) = a

a= 17.5

again, -3.5 = b/4

b= -14

Lastly, -3.5 = c/8

c= -28.

Hence, the value of a= 17.5, b=-14 and c=-28.

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The angle is 86.8 degrees and its complement is 3.2 degrees.

let x be the angle and y be the Complementary angle.

If the angles are complementary, then their sum is 90 degrees.

x + y = 90................(1)

and, the angle measures 83.6 degrees more than its complement.

x = y + 83.6

y + 83.6 + y = 90

Solving the equation for y we get

2y + 83.6 = 90

2y = 90 - 83.6

2y = 6.4

y= 3.2

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The modification to a research study that will result in a narrower confidence interval is option D is correct choice

The modification to a research study that will result in a narrower confidence interval is option D: decreasing the confidence level and increasing the sample size. By decreasing the confidence level, we are willing to accept a lower level of certainty in our results, which can lead to a narrower interval. Increasing the sample size also leads to a narrower interval as it reduces the variability in our data and increases the precision of our estimates.

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The average sales for that period is 1375.

The average, also known as the mean, is a statistical measure that represents the central tendency of a set of values. It is calculated by summing up all the values in a dataset and dividing the sum by the total number of values.

Mathematically, the average (mean) is calculated as:

Average = (Sum of all values) / (Total number of values)

To calculate the average sales for the given period, you'll need to follow these steps:

1. Add up the daily sales figures: 1000 + 1200 + 1300 + 2000 = 5500

2. Count the number of days in the period: 4 days

3. Divide the total sales by the number of days to find the average: 5500 / 4 = 1375

So, the average sales for that period is 1375.

Therefore, the correct answer is: B) 1,375

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The 40% clearance discount is already factored into the clearance price of $30, so applying the 20% coupon only reduces the price further by 20% of $30, or $6. Therefore, you would pay $24 for the item with both discounts applied.

Let's break down the discounts and calculate the final price of the item using the terms "normal", "price", and "quantitatively".
The normal price of the item: $50

First, apply the 40% clearance discount:
40% off the normal price = 0.4 * $50 = $20

Subtract the clearance discount from the normal price:
New discounted price = $50 - $20 = $30

Now, apply the 20% off coupon to the discounted price:
20% off the new discounted price = 0.2 * $30 = $6

Quantitatively, the calculation would be:
Normal price = $50
Clearance price (40% off) = $30
Coupon discount (20% off clearance price) = 0.20 x $30 = $6
Final price = $30 - $6 = $24

Subtract the coupon discount from the discounted price:
Final price = $30 - $6 = $24

So, you should pay $24 for the item. It is not the same as taking 60% off the normal price because the discounts are applied sequentially, not combined. Quantitatively, you can see that taking 60% off the normal price would result in a $30 discount ($50 * 0.6), while the actual total discount here is $26 ($20 + $6).

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The value of x is determined as 1.

The value of x is calculated by applying the principle of similar triangles.

length SR ≅ length ST

length TU ≅ length RU

We will have the following equation, to solve for the value of x;

TU/SU = RU/SU

TU = RU

x + 9 = 10x

9 = 10x - x

9 = 9x

9/9 = xy

1 = x

Thus, the value of x is calculated by applying the principle of similar triangle.

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The surface area of the solid formed by the net = 150.72 in²

From the figure we can observe that the solid formed by the net is the net of a cylinder.

The cylinder bases are the 2 circles and the curved surface of the cylinder is the rectangle.

The surface area of the cylinder = Area of the 2 circles + area of the rectangle

Here, the diameter of circle is 4 in

So, the radius of circle = ½ × 4

                                     = 2 in

So, the area of the 2 circles would be,

2(πr²)

= 2 × 3.14 × 2²

= 25.12 in²

Here the width of the rectangle is 10 in. and the length is nothing but the circumference of the circle.

so, length L =  πd

                   = π × 4

                   = 12.56 in

Now the area of rectangle would be,

L × W

= 12.56 × 10

= 125.6 in²

The total surface area of net would be,

Area of the 2 circles + area of the rectangle

= 25.12 + 125.6

= 150.72 in²

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Answer: D.

Step-by-step explanation:

Using a 0.05 significance level to test a claim that UFO sightings occur in different seasons with the proportions listed in the table the test statistic x² needed to test the claim is 11.562. The correct option is B.

To test the claim that UFO sightings occur in different seasons with the proportions listed in the table, we can use a chi-square goodness-of-fit test.

The null hypothesis is that the observed frequencies in each season are equal to the expected frequencies based on the proportions listed in the table.

The expected frequency for each season can be calculated by multiplying the total number of sightings by the proportion listed in the table. For example, the expected frequency for spring is:

Expected frequency for spring = Total number of sightings × Proportion for spring

= 420 × 0.25

= 105

Similarly, the expected frequencies for summer, fall, and winter are 126, 210, and 105, respectively.

The chi-square test statistic can be calculated as:

χ² = ∑ [(O - E)² / E]

where O is the observed frequency and E is the expected frequency.

Using the observed frequencies from the table and the expected frequencies calculated above, we get:

χ² = [(150-105)²/105] + [(120-126)²/126] + [(100-210)²/210] + [(50-105)²/105]

   = 11.562

The degrees of freedom for the chi-square test is (number of categories - 1), which in this case is 4 - 1 = 3.

Using a chi-square distribution table with 3 degrees of freedom and a significance level of 0.05, the critical value is 7.815.

Since the calculated chi-square value (11.562) is greater than the critical value (7.815), we reject the null hypothesis and conclude that there is evidence of a difference in UFO sightings across seasons. Therefore, the test statistic x² needed to test the claim is 11.562.

The correct answer is (B) 11.562.

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(a) The function f(x) = 0.2, 1 < x < 6 is a probability distribution function (pdf) because it is non-negative for all x in its domain and the total area under the curve is equal to 1.

(b) The cumulative distribution function (cdf) F(x) for 1 < x < 6 is given by F(x) = 0.2(x-1), where F(x) = 0 for x ≤ 1 and F(x) = 1 for x ≥ 6.

(c) The probability P(2 < X < 4) is 0.4, which can be calculated by integrating the pdf f(x) = 0.2 over the interval [2, 4].

(d) The probability P(X > 4) is 0.6, which is obtained by subtracting the cumulative probability F(4) = 0.2(4-1) from 1.

(a) To show that f(x) = 0.2, 1 < x < 6 is a probability distribution function (pdf), we need to show that:

f(x) is non-negative for all x in its domain: f(x) = 0.2 is non-negative for all x between 1 and 6.

The total area under the curve of f(x) is equal to 1:

∫1^6 0.2 dx = 0.2(x)|1^6 = 0.2(6-1) = 1

Since both conditions are satisfied, f(x) is a pdf.

(b) The cumulative distribution function (cdf) F(x) is given by:

F(x) = ∫1^x f(t) dt

For 1 < x < 6, we have:

F(x) = ∫1^x 0.2 dt = 0.2(t)|1^x = 0.2(x-1)

For x ≤ 1, F(x) = 0, and for x ≥ 6, F(x) = 1.

(c) P(2 < X < 4) is given by:

P(2 < X < 4) = ∫2^4 f(x) dx = ∫2^4 0.2 dx = 0.2(x)|2^4 = 0.4

(d) P(X > 4) is given by:

P(X > 4) = 1 - P(X ≤ 4) = 1 - F(4) = 1 - 0.2(4-1) = 0.6

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Let A ∈ R^(nxn) and let C ∈ R^(nxm). We will prove the following:

(1) Assume A is positive semidefinite. We need to show that tr(A) = 0 if and only if A = 0.

Proof:

(i) If A = 0, then tr(A) = 0 since the trace of the zero matrix is 0.

(ii) Assume tr(A) = 0. Recall that A is positive semidefinite, which means that its eigenvalues are non-negative. Since the trace of a matrix is the sum of its eigenvalues, having tr(A) = 0 implies that all eigenvalues of A must be zero. Consequently, A is a diagonal matrix with all diagonal elements equal to 0. Therefore, A = 0.

Thus, we have shown that tr(A) = 0 if and only if A = 0.

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The domain of the function shown in the graph is the one in option A:

6 ≤ k ≤ 11

The domain of a function y = f(x) is the set of the inputs of the function. To identify the domain in a graph, we need to look at the horizontal axis (also called the x-axis).

On the graph we can see that it starts at x = 6 with a closed dot, and it ends at x = 11 also with a closed dot.

That means that these values belong to the domain, so we can write the domain as follows:

Domain = 6 ≤ k ≤ 11

(notice that the variable in the horizontal axis is k).

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The group described by 16% of the population is 73.

For a normal distribution curve, the population are often divided into 2% below the mean, 14 % below the mean, 34% below the mean, the mean, 34% above the mean, 16% above the mean and 2 % above the mean.

for 34% below the mean, the population = M - 1std

for 16% below the mean, the population = M - 2std

So the population represented by 16% is calculated as follows;

= M - 2std

where;

= 79 - 2 (3)

= 79 - 6

= 73

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(1) The probability of getting a soft chicken is 16.67%.

(2) The probability of getting a crunch beef is 16.67%.

(3) The probability of getting a fish  (crunchy or soft) is 33.33%.

The probability of getting a soft chicken is calculated as follows;

total outcome = 6

number of soft chicken = 1

Probability = 1/6 = 16.67%

The probability of getting a crunch beef is calculated as follows;

total outcome = 6

number of crunch beef = 1

Probability = 1/6 = 16.67%

The probability of getting a fish  (crunchy or soft) is calculated as follows;

total outcome = 6

number of soft fish = 1

number of crunch fish  = 1

P(soft or crunch) = 1/6 + 1/6 = 2/6 = 1/3 = 33.33%

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The equality statement is False (b) 7= -(-(7)).

The expression on the right side of the equation simplifies to -(-7), which is equal to 7, making the statement untrue. Therefore, 7=-(-7) should be used as the right equality declaration.

In other words, 7 is equal to the opposite of -(-7)

The area of mathematics known as algebra aids in the representation of circumstances or problems as mathematical expressions. Mathematical operations like addition, subtraction, multiplication, and division are combined with variables like x, y, and z to produce a meaningful mathematical expression.

The associative, commutative, and distributive laws are the three fundamental principles of algebra. They facilitate the simplification or solution of problems and aid in illustrating the connection between different number operations.

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The energy for vacancy formation per atom in the metal M is 0.91 eV/atom.

To calculate the energy (in eV/atom) for vacancy formation in the metal M, we can use the following formula:

E_v = RT * ln(N_v/N)

Where:
- E_v is the energy for vacancy formation per atom
- R is the gas constant (8.314 J/mol*K or 0.008314 eV/mol*K)
- T is the temperature in Kelvin (317°C = 590K)
- N_v is the equilibrium number of vacancies (6.67 × 10^23 m^-3)
- N is the number of atoms per unit volume, which can be calculated using the density and atomic weight of the metal as follows:

N = (6.40 g/cm^3) * (1 mol/27.00 g) * (6.022 × 10^23 atoms/mol) = 1.51 × 10^22 atoms/m^3

Plugging in these values, we get:

E_v = (0.008314 eV/mol*K) * (590 K) * ln(6.67 × 10^23 m^-3 / 1.51 × 10^22 atoms/m^3)
E_v = 0.91 eV/atom

Therefore, the energy for vacancy formation per atom in the metal M is 0.91 eV/atom.

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The rule for the nth term of the geometric sequence is given as follows:

[tex]a_n = 2^n[/tex]

Hence the 10th term of the sequence is given as follows:

1024.

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio q.

The explicit formula of the sequence is given as follows:

[tex]a_n = a_0q^{n}[/tex]

In which [tex]a_0[/tex] is the first term.

The parameters in this problem are given as follows:

Hence the rule is given as follows:

[tex]a_n = 2^n[/tex]

Hence the 10th term of the sequence is given as follows:

2^10 = 1024.

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Answer:

357=10.5*4*x

8.5x

Step-by-step explanation:

357=10.5*4*x

357=42*x

8.5=x

Answer:  B

Step-by-step explanation:

If you fold that bottom side up.  the door is not on the closer side to the window. so B is wrong because the door is near the window.

The translating point will be (x, y) --> (8-x, -5-y).

We have to translate a point 8 units right and 5 units down followed by a 180 degree rotation.

Now, the rule for 180 rotation is

(x, y)  --> (x, -y)

and, to shift 8 unit right apply (8-x)

and, to shift 5 unit down apply (5-y)

Then, the translating point will be (x, y) --> (8-x, -5-y).

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If Crunchy-Critters produces chips bags with mean weight as 16 oz, the the probability that weight of the bag is less than 15.4 oz is 0.0228.

We use the standard normal distribution to find the required probability. First, we need to standardize the value of 15.4 oz using the formula : z = (x - μ) / σ,

where x is = value we are interested in, μ is = mean weight, σ is = standard deviation, and z is the standardized score.

The mean-weight of the chips is (μ) = 16 oz,

The standard-deviation of weight (σ) is 0.3 oz,

Substituting the values we have, we get:

⇒ z = (15.4 - 16)/0.3,

⇒ z = -2, and

We know that, P(X < 15.4) = P(Z < -2) = 0.0228

Therefore, the required probability is 0.0228 or 2.28%.

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Answer:

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If x=2 and x=3 are roots of the equation 3x

2

−2kx+2m=0 then (k,m)=

Medium

Solution

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Correct option is A)

Since x=2 and x=3 are roots of the equation 3x

2

−2kx+2m=0

⇒12−4k+2m=0⇒2k−m=6 ...(i)

and ⇒27−6k+2m=0⇒6k−2m=27 ...(ii)

On multiplying (i) by 3 and subtracting (ii) from it, we get

6k−3m=18

−

6

k

+

−

2m=

−

2

7

−m=−9

∴m=9

On putting m=9 in (i), we get

2k=15⇒k=

2

15

∴(k,m)=(

2

15

,9)

Hence, Option A is correct.

Answer:

A. 2x+3 and B. 3y² - 2y + 4 and E. x²/2 - 3x + 7 are polynomial expressions.