Triangle Angle Sum Practice

Answers Only

The correct answer is option B) 1/3.

To evaluate the algebraic expression for the given value(s) of the variable(s), we need to substitute the value of x into the expression and simplify.

The expression is: x+4/4x+8

Substitute x = 3 into the expression:

3 + 4/(4*3) + 8

Simplify:

3 + 4/12 + 8

3 + 1/3 + 8

Combine the terms:

11 + 1/3

Convert to a common denominator:

33/3 + 1/3

Simplify:

34/3

Divide:

11 1/3

So the answer is 1/3, which is option B.

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A school fundraising event sold a total of 223 tickets and generated a total revenue of $955.90, 89 adult tickets and 134 child tickets were sold.

Let's use the following variables to represent the number of tickets sold:

Using this, we can write the following system of equations:

a + c = 223 (the total number of tickets sold is 223)

6.60a + 2.75c = 955.90 (the total revenue generated is $955.90)

To solve this system of equations, we can use any method of our choice, such as substitution or elimination. Here, we will use the elimination method:

(2.75a + 2.75c) - (6.60a + 2.75c) = 613.25 - 955.90

-3.85a = -342.65

Therefore, 89 adult tickets and 134 child tickets were sold during the fundraising event. We were given the total number of tickets sold and the total revenue generated by the school fundraising event. We used this information to set up a system of equations that describes the relationship between the number of adult tickets sold, the number of child tickets sold, and the total revenue generated. We solved this system of equations using the elimination method to find the number of adult and child tickets sold. The final answer is 89 adult tickets and 134 child tickets sold.

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sin 2x = -4sqrt(3)/49, cos 2x = 47/49, and tan 2x = -8/15.

Answer:

Given the information sin x = 1/7, x in QII. The second quadrant (QII) lies between 90° and 180°.Therefore, we know the following about the values of the trigonometric functions:cos x is negative because it is in the second quadrant.tan x is negative because it is in the second quadrant.Therefore, cos x = -sqrt(48)/7 and tan x = -1/4.To find sin 2x, cos 2x, and tan 2x, we will use the following trigonometric identities: sin 2x = 2sin x cos x cos 2x = cos² x - sin² x tan 2x = 2tan x / (1 - tan² x)Substituting in the values for sin x and cos x, we get:sin 2x = 2(1/7)(-sqrt(48)/7) = -4sqrt(3)/49cos 2x = (-sqrt(48)/7)² - (1/7)² = 47/49tan 2x = 2(-1/4) / (1 - (-1/4)²) = -8/15Therefore, sin 2x = -4sqrt(3)/49, cos 2x = 47/49, and tan 2x = -8/15.

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The answer is 7x^(8) + 9x^(7) - 2x^(3) + 2x^(2) - 5x + 9.

To add (7x^(8)-6x^(3)+2x^(2)+9) and (9x^(7)+4x^(3)-5x), we need to combine the like terms in both expressions. Like terms are terms with the same variable and exponent.

Combine the like terms in the first expression:
7x^(8) + 0x^(7) - 6x^(3) + 2x^(2) + 0x + 9

Combine the like terms in the second expression:
0x^(8) + 9x^(7) + 4x^(3) + 0x^(2) - 5x + 0

Add the two expressions together:
(7x^(8) + 0x^(8)) + (0x^(7) + 9x^(7)) + (-6x^(3) + 4x^(3)) + (2x^(2) + 0x^(2)) + (0x - 5x) + (9 + 0)

Simplify the expression:
7x^(8) + 9x^(7) - 2x^(3) + 2x^(2) - 5x + 9

Write the answer in descending order:
7x^(8) + 9x^(7) - 2x^(3) + 2x^(2) - 5x + 9

Therefore, the answer is 7x^(8) + 9x^(7) - 2x^(3) + 2x^(2) - 5x + 9.

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The statements that correctly interpret the scatter plot on the ages of the used cars is :

The scatter plot shows the number of miles that a car of a certain age would be expected to have driven. The data has a positive association to show that the older a car is, the more miles it would have driven.

There is a cluster of data points at 2 , 23 to show that a car of the age 2 would have been driven the same amount of miles in general.

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a) The simplified expression is \frac{(x-4)}{2(x+2)}. b) The solution to the equation is x = -7.

a) To simplify the expression \frac{7}{x^2\ -\ 4}\div\frac{14}{x^2\ -6x\ +8}, we need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by flipping the numerator and denominator. So, the reciprocal of \frac{14}{x^2\ -6x\ +8} is \frac{x^2\ -6x\ +8}{14}.

Now, we can multiply the two fractions: \frac{7}{x^2\ -\ 4} \times \frac{x^2\ -6x\ +8}{14} = \frac{7(x^2\ -6x\ +8)}{14(x^2\ -\ 4)}

Next, we can simplify the expression by canceling out any common factors. In this case, both the numerator and denominator have a factor of 7, so we can cancel them out:

\frac{7(x^2\ -6x\ +8)}{14(x^2\ -\ 4)} = \frac{x^2\ -6x\ +8}{2(x^2\ -\ 4)}

Finally, we can factor the numerator and denominator to see if there are any other common factors that can be canceled out:

\frac{x^2\ -6x\ +8}{2(x^2\ -\ 4)} = \frac{(x-4)(x-2)}{2(x+2)(x-2)}

Since both the numerator and denominator have a factor of (x-2), we can cancel them out:

\frac{(x-4)(x-2)}{2(x+2)(x-2)} = \frac{(x-4)}{2(x+2)}

So, the simplified expression is \frac{(x-4)}{2(x+2)}.

b) For the second part of the question, we need to solve the equation \frac{6}{x\ +1}\ =\ \frac{3}{x\ +\ 4}. To do this, we can cross-multiply and then simplify:

6(x+4) = 3(x+1)

6x + 24 = 3x + 3

3x = -21

x = -7

So, the solution to the equation is x = -7.

Note: The question is incomplete. The complete question probably is: a) Simplify the expression  \frac{7}{x^2\ -\ 4}\div\frac{14}{x^2\ -6x\ +8} . b) Solve the equation \frac{6}{x\ +1}\ =\ \frac{3}{x\ +\ 4}.

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If cos tetha = -5/17 and tetha is in Q III, then sin tetha can be found using the Pythagorean identity: sin^2 tetha + cos^2 tetha = 1.

Substituting the given value for cos tetha and solving for sin tetha gives:

sin^2 tetha + (-5/17)^2 = 1

sin^2 tetha = 1 - 25/289

sin^2 tetha = 264/289

sin tetha = √(264/289)

sin tetha = ±√264/17

Since tetha is in Q III, where sin is negative, the correct answer is:

sin tetha = -√264/17


If sin tetha = 1/4 and tan tetha = √3, then cos tetha can be found using the definition of tan tetha: tan tetha = sin tetha/cos tetha.

Substituting the given values for sin tetha and tan tetha and solving for cos tetha gives:

√3 = (1/4)/cos tetha

cos tetha = (1/4)/√3

cos tetha = √3/12


If sec tetha = -4 and tetha is in Q II, then cos tetha can be found using the definition of sec tetha: sec tetha = 1/cos tetha.

Substituting the given value for sec tetha and solving for cos tetha gives:

-4 = 1/cos tetha

cos tetha = -1/4

Once we have cos tetha, we can find sin tetha using the Pythagorean identity: sin^2 tetha + cos^2 tetha = 1.

Substituting the value for cos tetha and solving for sin tetha gives:

sin^2 tetha + (-1/4)^2 = 1

sin^2 tetha = 1 - 1/16

sin^2 tetha = 15/16

sin tetha = ±√(15/16)

Since tetha is in Q II, where sin is positive, the correct answer is:

sin tetha = √15/4

Once we have sin tetha and cos tetha, we can find the other trigonometric function values using their definitions:

tan tetha = sin tetha/cos tetha = (√15/4)/(-1/4) = -√15

cot tetha = 1/tan tetha = -1/√15 = -√15/15

csc tetha = 1/sin tetha = 4/√15 = 4√15/15

sec tetha = 1/cos tetha = -4 (given)

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The product of the two expressions can be set as the one in option 1.

The process of multiplying two given expressions with the aid of variables and constants is known as algebraic multiplication. The terms factors and multiplicands refer to the two expressions that make up an algebraic product, which is the combination of two expressions.

The two expressions can be set as follows:

4x³ + 0x² -3x + 4

                 2x - 5

As the x² term is not present we use the coefficient as 0.

Hence, the product of the two expressions can be set as the one in option 1.

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The area of the shaded region is 96ft²

The space enclosed by the boundary of a plane figure is called its area. The area of a figure is the number of unit squares that cover the surface of a closed figure. Area is measured in square units like cm² and m².

The area of a parallelogram is expressed as base × height

The area of parallelogram = 12× 12

= 144ft²

The area of the rectangle = l×w

= 8×6

= 48 ft²

The area of the shaded region = area of parallelogram - area of rectangle

area of the shaded region = 144-48

= 96ft².

Therefore the area of the shaded region is 96ft²

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Therefore , the solution of the given problem of triangle comes out to be the HL Congruence Hypothesis cannot be used to demonstrate anything.

A triangular is a polygon because it has two or maybe more additional sections. It has a straightforward square form. Only the edges A, B, but also C distinguish a triangular from a parallelogram. When the sides are not exactly collinear, Euclidean geometry produces a singular plane instead of a cube. If a shape has three edges and three angles, it is said to be triangular.

Here,

According to the HL Congruence Theorem, a triangle is congruent if its hypotenuse and one of its legs are congruent with the hypotenuse and corresponding limb of another right triangle.

We must assess whether the two right triangles have the following information in order to decide whether the HL Congruence Theorem can be used to prove that the triangles are congruent:

Both of them are right circles.

Each triangle has a limb that is congruent.

Each triangle's hypotenuses are similar to one another.

The HL Congruence Theorem allows us to infer that the triangles are congruent if all three requirements are satisfied.

The HL Congruence Hypothesis cannot be used to demonstrate anything if any one of the prerequisites is not satisfied.

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Answer:

2/3 (-9+3) simplifies to 2/3 (-6) which is equal to -4.

Step-by-step explanation:

To simplify 2/3(-9+3), we need to start by simplifying the expression inside the parentheses, which is -9+3=-6. Therefore, we can rewrite the expression as 2/3(-6).

Now we can apply the distributive property of multiplication over addition/subtraction to simplify further. This property states that a(b+c) = ab + ac. So, 2/3(-6) can be written as:

2/3 x -6 = 2/3 x (-1 x 6) = 2/3 x (-1) x 6 = -2/3 x 6

Now we just need to multiply -2/3 by 6, which gives us -12/3 or -4. Therefore, 2/3(-9+3) simplifies to -4.

The number of minutes to complete 1 lap is found as 5 minutes.

Multiplication is the reciprocal of division. If 3 groups of 4 add up to 12, then 12 divided into three categories of equal size results in 4 in each group.

The stated question:

10 minutes for 2 laps

Applying unitary method:

= 10/2

= 5

Thus, number of minutes to complete 1 lap is found as 5 minutes.

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The complete question -

10 minutes for 2 laps. Then, find number of minutes for 1 lap.

In interval notation, the the solution set in interval notation. (5)/(y-4)<9 is   (41/9, ∞).

Inequality is a mathematical statement that two values or expressions are not equal. It is represented by symbols, like >, <, or ≠, and is used to compare quantities or values.

To solve the inequality (5)/(y-4)<9, we need to isolate the variable on one side of the inequality. We can do this by multiplying both sides of the inequality by (y-4) and then simplifying.

(5)/(y-4)<9

5 < 9(y-4)

5 < 9y - 36

41 < 9y

y > 41/9

This means that any value of y greater than 41/9 will satisfy the inequality.

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The following exponential rule is called the Product Rule.

What is exponential rule ?

Exponential rules are mathematical rules that govern the manipulation of exponential expressions, which involve terms that have a base raised to a power. Exponential rules are important in many areas of mathematics and science, particularly in algebra, calculus, and statistics.

According to the question:
The following exponential rule is called the Product Rule. It states that when multiplying two powers with the same base, you can add their exponents. The rule is:

[tex]a^m * a^n = a^(m+n)[/tex]

For example, [tex]2^3 * 2^4 = 2^(3+4) = 2^7 = 128[/tex]. This rule is fundamental in simplifying expressions and solving equations involving exponential functions.

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The function h(x) can be expressed as f(g(x)) where g(x) = (x-7) and f(x) = x.

The function h(x) = (x-7)1 can be expressed in the form f(g(x)) where g(x) = (x-7) and f(x) is defined as f(x) = x1.

To find f(x), we need to isolate x on one side of the equation. We can do this by multiplying both sides of the equation by the reciprocal of 1, which is 1.

f(x) = x1 * 11 = x

Therefore, the function h(x) can be expressed as f(g(x)) where g(x) = (x-7) and f(x) = x.

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The percentage of races with a finishing time between 64.5 and 65.5 seconds is given as follows:

68%.

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

The measures in this problem are within one standard deviation of the mean, as:

Hence the percentage is of 68%.

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Answer:

To find YZ and DE in the sequence WXYZ FEDC, we need to understand how the sequence is constructed. The sequence is likely made up of two smaller sequences: WXYZ and FEDC.

To find YZ, we need to look at the WXYZ sequence. Since Y and Z are the last two digits of this sequence, we can conclude that Z is the last digit and Y is the second to last digit. Therefore, YZ = 30.

To find DE, we need to look at the FEDC sequence. Since D and E are the last two digits of this sequence, we can conclude that E is the last digit and D is the second to last digit. Therefore, DE = 96.

So, YZ = 30 and DE = 96

Step-by-step explanation:

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Answer:

#7
AB = 4.1

CB = 4.1

#10
AC = 21.21

CB = 15

#11
AC = 2.828
CB =2

#12
AB =  2.616
CB = 2.616

#13

AC = 6
CB = 3√2

#14

AC = 5.94

CB = 4.2

Step-by-step explanation:

The triangle is a right isosceles triangle with AB = AC

This means
AC² = AB² + CB²
= AB² + AB²
= 2AB²
AC =  AB√2

and

AC = CB√2                (since CB= AB)

#7
Given  AC = 5.8

We know AC = AB√2
[tex]AB= \dfrac{AC}{\sqrt{2}} = \dfrac{5.8}{\sqrt{2}} = 4.1[/tex]

CB = AB  is also  4.1

#10
Given AB = 15

AC = AB√2 =  15√2 =
CB= AB = 15

#11

Given CB = 2

AB = CB= 2

AC = CB√2 = 2√2 = 2.828

#12

Given AC = 3.7

We know AC = AB√2
AB= AC/√2 = 3.7 ÷ √2 = 2.616

CB = AB  is also  2.616

#13

Given AB = 3√2

AC = AB√2 = 3 x √2 x √2 = 3 x 2 =6   (√2 x √2 = 2)
CB = 3√2

#14

Given AB= 4.2

AC = 4.2 x √2 = 5.94

CB = AB = 4.2

When the L shape attached is rotated about the origin (0, 0) by 90° counterclockwise it would look like option C.

A shows a rotation of 90°  clockwise about (0,0)

B shows a rotation of 180° about (0,0)

C shows a rotation of 90° anticlockwise about (0,0)

D shows a rotation of 90° anticlockwise, but the center of rotation is (1,1)

Recall that Rotation in math is a transformation that involves rotating a figure around a fixed point called the center of rotation by a certain angle in a clockwise or counterclockwise direction.

Thus, Option C above is the right answer.

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Answer: The median is the best measure of center for distributions that are skewed or have extreme values, since it is less affected by outliers than the mean. Therefore, the correct answers are:

- A bar graph that forms a gradual incline and a sharp decline of values.

- A bar graph that forms a sharp incline and a gradual decline of values.

Both of these graphs suggest that the data is skewed, and in these cases, the median is a better representation of the "typical" value than the mean.

The other two options do not provide enough information to determine whether the data is skewed or not, so we cannot say for certain whether the median is the best measure of center in those cases. The bar graph with values that hover around 10 y could have a symmetrical distribution with a mean and median that are the same, or it could have a skewed distribution where the median is a better measure of center. Similarly, the pyramid-shaped bar graph could have a symmetrical distribution, or it could be skewed in a way that the median is not a good measure of center.

Step-by-step explanation:

Based on the provided information The result of rounding to four integer places is sin 58° ≈ 0.8480.

Making a number easier while maintaining a value that is near to what it was is known as rounding. Although less precise, the outcome is simpler to use. For instance, 73 is 70 when adjusted to the closest ten, since 70 is closer to 73 than 80 is.

Using a calculator, we can approximate sin 58° as follows:

sin 58° ≈ 0.8480

Rounding to four decimal places, we get:

sin 58° ≈ 0.8480

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By the concept of vertically opposite angles, m∠MDH = 44°.

We know when a transversal intersects two parallel lines at two distinct points,

Two pairs of interior and alternate angles are formed such that the measure of interior angles are same and the measure of alternate angles is also the same.

We know, Pair of vertically opposite angles are equal.

Therefore, 2y + 130 = 3y + 127.

y = 3.

So, 2y + 130 = 136°

Again, ∠PDG = ∠MDH.

Now, 2∠PDG = 360° - 2(136°)

2∠PDG = 360° - 272°.

∠PDG = 44° = ∠MDH.

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Answer:

I assume the inequality is

[tex]\sf{ \dfrac{2c}{3} + 1 > 7}[/tex]

[tex]\sf{ \dfrac{2c}{3} > 7-1}[/tex]

[tex]\sf{ 2c> 18}[/tex]

[tex]\sf{ c > 9}[/tex]

This distribution of inequities ensures that the total amount of ticket sales revenue stays within the budget, that no more than 150 tickets are sold overall, and that no negative ticket sales occur.

Let x represent the quantity of single tickets sold and y represent the quantity of pair tickets sold. The entire revenue from ticket sales, R, is then calculated as follows:

R = 45x + 70y

The entire amount of ticket sales revenue is limited by the budget to $5000:

45x + 70y ≤ 5000

No more than 150 tickets may be sold in total.

x + y ≤ 150

X and Y must both be non-negative since we cannot sell negative tickets:

x ≥ 0\sy ≥ 0

As a result, the system of disparities that best captures this circumstance is:

5000 x + y 45x + 70y 150x 0y 0

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The solution set for the inequality is (-1, ∞) in interval notation. To solve the inequality, we need to isolate the variable on one side of the inequality sign. We can do this by using the following steps:

Step 1: Divide both sides of the inequality by 8 to simplify the equation.
8(2x+3)>8 → (2x+3)>1

Step 2: Subtract 3 from both sides of the inequality to isolate the variable.
(2x+3)>1 → 2x>-2

Step 3: Divide both sides of the inequality by 2 to find the value of x.
2x>-2 → x>-1

Step 4: Write the solution set using interval notation. Since the inequality sign is ">", we use a parenthesis to indicate that the solution does not include the endpoint -1.
x>-1 → (-1, ∞)

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The greatest possible distance from  Austin to san Antonio is 335.77 mi, b. the Least possible distance to Austin to san Antonio is 120.03 mi.

Describe Distance?

Distance refers to the physical length or space between two points or objects. It is an important concept in mathematics, physics, and everyday life. Distance can be measured in various units such as meters, feet, kilometers, miles, or light-years, depending on the context and scale of the measurement.

In mathematics, distance is often measured using the Euclidean distance formula, which calculates the distance between two points in a plane or in three-dimensional space. The formula is given by:

[tex]d = √((x2 - x1)^2 + (y2 - y1)^2)[/tex]

where d is the distance between the two points, (x1, y1) and (x2, y2).

In physics, distance is often used to describe the separation between two objects or particles. The distance between two objects can affect the strength of gravitational or electromagnetic forces between them. Distance is also an important concept in special relativity, where it is affected by the relative motion of observers and the distortion of spacetime.

a. Greatest possible distance from  Austin to san Antonio= Distance from Austin →Houston → san Antonio.

Austin - Houston = 146.43 mi

Houston - San Antonio = 189.34 mi

Total  distance= 146.43+ 189.34= 335.77 mi

b. Least possible distance to Austin to san Antonio= the shortest distance = x

By pythagoras theorem

[tex](189.34)²= (146.43)²+ x²x²= (189.34)²- (146.43)²x²= 35849.63- 21441.744x²= 14407.89x=√14407.89x= 120.03[/tex]

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The complete question is:

B) The left end of the cable is 15 m above the roadway. The y-intercept of the graph represents the height of the left end of the cable, which is 15 m above the roadway.

Intercept is the point where a line intersects with an axis on a graph. It is the point at which a line crosses the x-axis or y-axis. Intercepts are important when analyzing relationships between variables, and they can help to draw conclusions about how two variables are related. For example, if a line has a negative slope, the y-intercept will be a positive number, indicating that as the x-value increases, the y-value will decrease. In linear regression, the intercept is the average value of the dependent variable when the independent variable is equal to zero. It is also used to determine the value of a linear equation when given the value of one of its variables.

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x1 = 1.3, x2 = 0.3, x3 = 0.7

The solution to the linear programming problem is: x1 = 1.3, x2 = 0.3, x3 = 0.7.

Using the Dual Simplex Algorithm (DSA), we can solve the problem by setting up the initial tableau:

x1 = −5x1 + x2 − 2x3

2x1 + x3 = 0

x1 − x2 ≥ 1

3x1 − x2 + x3 ≤ 3

Once the tableau is set up, we can use DSA to find the optimal solution.

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