A certain triangle has two 45° angles. What type of triangle is it?

A. Acute Triangle
B. Right triangle.
C. Right isosceles
D. Acute isosceles

Answer:

[tex]\large\boxed{\left(\dfrac{7}{2},\ \dfrac{27}{2}\right)}[/tex]

Step-by-step explanation:

The formula of a midpoint of a segment AB with endpoints at A(x₁, y₁) and B(x₂, y₂):

[tex]M_{AB}\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)[/tex]

We have the points (-3, 9) and (10, 18).

Substitute:

[tex]M(x,\ y)\\\\x=\dfrac{-3+10}{2}=\dfrac{7}{2}\\\\y=\dfrac{9+18}{2}=\dfrac{27}{2}[/tex]

On a horizontal number line, -6 is located to the (left) of -4. So, -6 is (less than) 4.

Answer:

On a horizontal number line, -6 is located to the (left) of -4. So, -6 is (less than) 4.

Step-by-step explanation:

For this case we must solve the following inequality:

[tex]-8 <2x-4 <4[/tex]

Adding 4 in the parts of the inequality we have:

[tex]-8 + 4 <2x-4 + 4 <4 + 4\\-4 <2x <8[/tex]

Dividing between 2 each part of the inequality:

[tex]\frac {-4} {2} <\frac {2x} {2} <\frac {8} {2}[/tex]

[tex]-2 <x <4[/tex]

Answer:

[tex]-2 <x <4[/tex]

To solve the inequality -8 < 2x - 4 < 4, add 4 to each part to get -4 < 2x < 8, then divide by 2 to find -2 < x < 4.

To solve the inequality -8 < 2x - 4 < 4, we must isolate x. We do this in two steps, addressing each part of the compound inequality separately.

Therefore, the solution set is all x values between -2 and 4.

Answer:

So I did most of you table for you:

First blank:4

Second blank:-4

Third blank:-8

Fourth blank: Left to you: Just plug in 6 into 1/2x^2-5x+4 (I will check)

Fifth blank: Left to you: Just plug in 8 into 1/2x^2-5x+4 (I will check)

The graph is D.

Step-by-step explanation:

So they have a table to fill in and they tell you in the first row what they want you to plug in

So the table is asking us to answer this:

What is h(0),h(2),h(4),h(6), and h(8).

h(0) means to replace x with 0 in [tex]\frac{1}{2} x^2-5x+4[/tex].

[tex]\frac{1}{2}(0)^2-5(0)+4[/tex]

[tex]0-0+4[/tex]

[tex]0+4[/tex]

[tex]4[/tex]

So the first blank is 4 since h(0)=4.

h(2) means to replace x with 2 in [tex]\frac{1}{2} x^2-5x+4[/tex].

[tex]\frac{1}{2} (2)^2-5(2)+4[/tex]

[tex]\frac{1}{2} (4)-10+4[/tex]

[tex]2-10+4[/tex]

[tex]-8+4[/tex]

[tex]-4[/tex]

So the second blank is -4 since h(2)=-4.

h(4) means to replace x with 4 in [tex]\frac{1}{2} x^2-5x+4[/tex].

[tex]\frac{1}{2}(4)^2-5(4)+4[/tex]

[tex]\frac{1}{2}(16)-20+4[/tex]

[tex]8-20+4[/tex]

[tex]-12+4[/tex]

[tex]-8[/tex]

So the third blank is -8 since h(4)=-8

Maybe you can try the last 2 blanks in the table part. That is try computing h(6) and h(8). I will check it for you.

Now the points I have so far are (0,4) from the h(0)=4, (2,-4) from the h(2)=-4, and (4,-8) from the h(4)=-8.

I'm looking for a graph that goes through these points (0,4) , (2,-4) , and (4,-8).

By the way the only graphs that are worth looking at is C and D because they are open up.  I know my curve for h(x)=1/2x^2-5x+4 should be a parabola open up because 1/2 is positive and 1/2 is the coefficient of x^2.

So Graph C has y-intercept (0,10) not (0,4) so Graph C is not right.

Graph D has y-intercept (0,4).  It also goes through (2,-4) and (4,-8).  

I don't know if you notice but the x-axis and y-axis are going up by two's in each graph.

Answer:

[tex]-y=-2x+8\\y=2x-8[/tex]

Step-by-step explanation:

To solve, first subtract 2x from both sides.

[tex]2x-y=8\\-y=-2x+8[/tex]

You appear to want the solution for -y, so I've included it. Given that you also want the solution for y, divide both sides by -1.

[tex]-y=-2x+8\\y=2x-8[/tex]

Answer:

- y = 8 - 2x

Step-by-step explanation:

2x - y = 8

Subtracting 2x on both sides,

=> 2x - 2x - y = 8 - 2x

=> - y = 8 - 2x  

Answer:

h < 14

Step-by-step explanation:

2h +8 > 3h - 6

2h - 3h >  - 6 - 8

-h > -14  (multiply both sides by -1; remember to flip the inequality sign!)

h < 14

Answer:

True

Step-by-step explanation:

A normal distribution shows a dense center , which would be the mean. Given a population or sample, the bulk of the data would be found near the mean and and spreads out thinner towards the ends when the data is said to be normally distributed. When graphed you will see that the data form a bell-shaped curve.

Attached below is an example of how it would look:

[tex]\bf ~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$5000\\ P=\textit{original amount deposited}\dotfill&\$2000\\ r=rate\to r\%\to \frac{r}{100}\\ t=years\dotfill &3 \end{cases} \\\\\\ 5000=2000(1+r3)\implies \cfrac{5000}{2000}=1+3r\implies \cfrac{5}{2}=1+3r \\\\\\ 5=2+6r\implies 3=6r\implies \cfrac{3}{6}=r\implies 0.5=r\implies \stackrel{\textit{converting to percent}}{0.5\cdot 100\implies 50\%}[/tex]

Answer:

2

Step-by-step explanation:

The rate of change of f(x) in the closed interval [ a, b ] is

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

here [ a, b ] = [ 3, 5 }

and from the graph

f(b) = f(5 ) = 3 and f(a) = f(3) = - 1, hence

rate of change = [tex]\frac{3-(-1)}{5-3}[/tex] = [tex]\frac{4}{2}[/tex] = 2

The rate of change for a quadratic function isn't constant but can be averaged over an interval using the formula [f(b) - f(a)] / (b - a).

The rate of change over a given interval for a quadratic function is not constant and varies with x. However, the function's average rate of change can be determined for an interval. This can be found by taking the difference in the function's value at the two endpoints of the interval, divided by the difference in x-values.

Without the specific function, I can't calculate the rate over the interval for you, but you can use this formula:

In your case, a = 3 and b = 5. So, just substitute these values (and the values of your function at these points) into that formula to find the average rate of change.

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Step-by-step explanation:

The slope-intercept form of an equation of a line:

[tex]y=mx+b[/tex]

m - slope

b - y-intercept

[tex]61)\ m=3,\ b=-2\\\\\boxed{y=3x-2}\\\\62)\ m=\dfrac{4}{5},\ b=4\\\\\boxed{y=\dfrac{4}{5}x+4}\\\\63)\ m=-\dfrac{7}{4},\ b=-3\\\\\boxed{y=-\dfrac{7}{4}x-3}\\\\64)\ m=-\dfrac{3}{4},\ b=-2\\\\\boxed{y=-\dfrac{3}{4}x-2}[/tex]

Answer:

[tex]y=5x[/tex]

Step-by-step explanation:

we know that

A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form [tex]y/x=k[/tex] or [tex]y=kx[/tex]

In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin

we have

For x=0, y=0 -----> the line passes through the origin

For x=1, y=5 ----> [tex]k=y/x=5/1=5[/tex]

For x=2, y=10 ----> [tex]k=y/x=10/2=5[/tex]

For x=3, y=15 ----> [tex]k=y/x=15/3=5[/tex]

so

The constant of proportionality is k=5

The table represent a direct variation

The equation is equal to [tex]y=kx[/tex]

substitute the value of k

[tex]y=5x[/tex]

Answer:

10π/3 mm

Step-by-step explanation:

Arc length is:

s = rθ × π/180

where r is the radius and θ is the angle in degrees.

Here, r = 10 mm and θ = 60°.

s = (10) (60) (π/180)

s = 10π/3

Another way to calculate it is to find the entire circumference then divide by 6, since 60° is one-sixth of 360°.

Answer:

B. 4⁄5

Step-by-step explanation:

 7 1⁄5

– 6 2⁄5

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

We need to borrow from the 7  since 1/5 is less than 2/5

7 becomes 6 and the one becomes 5/5

 6  5/5+1⁄5

– 6 2⁄5

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

Combining the fraction

 6  6⁄5

– 6 2⁄5

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

      4/5

Answer:

(5x + 7y)(25x² - 35xy + 49y²)

Step-by-step explanation:

125x³ + 343y³ ← is a sum of cubes and factors in general as

a³ + b³ = (a + b)(a² - ab + b² )

125x³ = (5x)³ ⇒ a = 5x

343y³ = (7y)³ ⇒ b = 7y

125x³ + 343y³

= (5x + 7y)((5x)² - (5x × 7y) + (7y)²)

= (5x + 7y)(25x² - 35xy + 49y²) ← in factored form

The set of ordered pairs that defines a function is:  {(1,3),(5,2),(6,9),(1,12),(10,2)} (last option).

A set of ordered pairs that defines a function will have exactly one y-value that assigned to every x-value. In essence, it means none of its x-values can have two corresponding y-value.

All the sets of ordered pairs have exactly one y-value that corresponds to each of its x-value except {(1,3),(5,2),(6,9),(1,12),(10,2)}, which have two different y-values that corresponds to the x-value of 1.

Therefore, the set that doesn't define a function is:  {(1,3),(5,2),(6,9),(1,12),(10,2)} (last option).

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A set of ordered pairs defines a function if each element of the domain is mapped to a single, unique value in the range. Set D does not define a function because it contains two distinct ordered pairs with the same first element, (1,3) and (1,12).

A set of ordered pairs defines a function if each element of the domain is mapped to a single, unique value in the range. Looking at the given sets of ordered pairs, we can determine which ones define a function by checking if there are any repeated first elements in the pairs. If there are repeated first elements, then the set does not define a function.

Out of these sets, Set D does not define a function because it contains two distinct ordered pairs with the same first element, (1,3) and (1,12).

Answer:

Y = 1/3x + 2

Step-by-step explanation:

Since the line increases by 1 unit then goes to right 3 units the line equation must have a slope of 1/3 and must have a y intercept of + 2 since the line crosses the Y axis at +2

Answer: y=1/3x+2 is the correct answer just took the test

Step-by-step explanation:

Answer:

The 4th side on quadrilateral ABCD is [tex]11\frac{2}{3}\ ft[/tex]

Step-by-step explanation:

we know that

If two figures are similar, then the ratio of its corresponding sides is proportional and this ratio is called the scale factor

In this problem

The corresponding sides are

ABCD           EFGH

20 ft              ?

18 ft                ?

14 ft                6 ft

?                     5 ft

The length side of 14 ft in quadrilateral ABCD is the corresponding side to the length side of 6 ft in quadrilateral EFGH

so

the scale factor from quadrilateral ABCD to quadrilateral EFGH is

[tex]6/14=3/7[/tex]

therefore

To find the length of the 4th side on quadrilateral ABCD, divide the length of the 4th side on quadrilateral EFGH by the scale factor

so

[tex]5/(3/7)=35/3\ ft[/tex]

convert to mixed number

[tex]\frac{35}{3}\ ft=\frac{33}{3}+\frac{2}{3}=11\frac{2}{3}\ ft[/tex]

Answer:

C. 300

Step-by-step explanation:

EFG and EG is the total distance around the circle

EFG + EG = 360 degrees

EFG + 60 = 360

Subtract 60 from each side

EFG +60-60 = 360-60

EFG = 300

Answer:

C. [tex]\widehat{GFE}=300^{\circ}[/tex]

Step-by-step explanation:

We have been given an image of a circle. We are asked to find the measure of major arc EFG for our given circle.

First of all, we will find measure of arc GE.

We know that the measure of central arc is equal to its subtended arc. The measure of arc GS will be equal to measure of central angle GOE.

Since measure of central angle GOE is 60 degree, so measure of arc GE is 60 degrees as well.

We know that the circumference of circle is equal to 360 degrees. So we can set an equation as:

[tex]\widehat{GE}+\widehat{GFE}=360^{\circ}[/tex]

[tex]60^{\circ}+\widehat{GFE}=360^{\circ}[/tex]

[tex]60^{\circ}-60^{\circ}+\widehat{GFE}=360^{\circ}-60^{\circ}[/tex]

[tex]\widehat{GFE}=300^{\circ}[/tex]

Therefore, the measure of arc GFE is 300 degrees and option C is the correct choice.

Answer:

The correct answer is third option

They are not congruent, because only one pair of corresponding sides is congruent.

Step-by-step explanation:

From the figure we can see that, two isosceles  triangles.

ΔFGH and ΔFJH

We get FG = GH and FJ = HJ

And side FH is common for both the triangles.

We can not say these two triangles are congruent.

Therefor the  correct answer is third option

They are not congruent, because only one pair of corresponding sides is congruent.

Final answer:

The congruency between Triangle FGH and Triangle FJH can be considered through Desargues's theorem, which relates congruency to parallel sides and intersecting vertex connections, and similarity can also be established by the AAA theorem, which involves congruent corresponding angles.

Explanation:

To determine whether Triangle FGH is congruent to Triangle FJH, one would need to employ the principles of Desargues's theorem, which states that if two triangles have their corresponding vertices connected by lines that meet at a point, and if the corresponding sides of the triangles are parallel, then the triangles are congruent. The information given suggests multiple instances where triangles are similar or congruent based on the congruency of angles or parallelism of lines, as dictated by the aforementioned theorem. For example, in the triangles ABC and FCE mentioned, the similarity is confirmed via the Angle Angle Angle (AAA theorem) because all corresponding angles of the triangles are congruent, which is a direct consequence of the vertical angles property and the alternate interior angles property. This similarity implies that there's a proportionality between the sides of the triangles, which could be a stepping stone in proving the congruency between Triangle FGH and Triangle FJH if similar conditions apply.

Answer:

C. Gas Station A sells gasoline at a lower rate. Its price is $3.05 per gallon

Step-by-step explanation:

To answer this, all you need to do is first figure out how much per gallon each of the gas stations offer.

Because Gas station B is practically given to you by the equation, I'll show you how to interpret it:

p = $3.08g

p is the price

g is the gallon

So the price you will pay will be $3.08 times the number of gallons. This means that 1 gallon, if you substitute it will be $3.08.

This would eliminate options A and B.

Now to figure out how much Gas station A charges, just choose one of the prices. All we need to do is to divide the price given by the number of gallons:

[tex]3 gallons = \$ 9.15\\\\\dfrac{3gallons} {3}=\dfrac{\$9.15}{3}=\$3.05[/tex]

So Gas station A is lower than Gas station B.

Step-by-step explanation:

To answer this, all you need to do is first figure out how much per gallon each of the gas stations offer. Because Gas station B is practically given to you by the equation, I'll show you how to interpret it:p = $3.08gp is the priceg is the gallonSo the price you will pay will be $3.08 times the number of gallons. This means that 1 gallon, if you substitute it will be $3.08.This would eliminate options A and B.Now to figure out how much Gas station A charges, just choose one of the prices. All we need to do is to divide the price given by the number of gallons:So Gas station A is lower than Gas station B.

Answer:

15.733kg

Step-by-step explanation:

33.2 - 21.4 = 11.8

3/4 of the bucket empty is 11.8kg.

11.8 divided by 3 is 3.933. 3.933 x 4 is 15.733

Answer:

Option C is correct.

Step-by-step explanation:

y = x^2+4x+5

We need to find the vertex of the above equation.

The above equation represents the parabola.

The slope of parabola can be found by taking derivative of the given equation

dy/dx = 2x+4

The slope of the parabola at the vertex is zero SO,

2x+4 = 0

2x = -4

x = -4/2

x = -2

Putting value of x =-2 to find the value of y

y = x^2+4x+5

y =(-2)^2+4(-2)+5

y = 4-8+5

y =9-8

y = 1

So, the vertex is (-2,1)

Option C is correct.

Answer:

[tex]\large\boxed{f(x)=6x^3+2x}[/tex]

Step-by-step explanation:

[tex]\text{If}\ f(-x)=f(x)\ \text{then}\ f(x)\ \text{is an even function.}\\\\\text{If}\ f(-x)=-f(x)\ \text{then}\ f(x)\ \text{is an odd function.}[/tex]

======================================================

[tex]f(x)=3x^2+x\\\\f(-x)=3(-x)^2+(-x)=3x^2-x\\\\f(-x)\neq f(x)\ \wedge\ f(-x)\neq-f(x)\\\\============================\\\\f(x)=4x^3+7\\\\f(-x)=4(-x)^3+7=-4x^3+7\\\\f(-x)\neq f(x)\ \wedge\ f(-x)\neq-f(x)\\\\============================\\\\f(x)=5x^2+9\\\\f(-x)=5(-x)^2+9=5x^2+9\\\\f(-x)=f(x)-\text{It's an even function}\\\\============================\\\\f(x)=6x^3+2x\\\\f(-x)=6(-x)^3+2(-x)=-6x^3-2x=-(6x^3+2x)\\\\f(-x)=-f(x)-\text{It's an odd function.}[/tex]

The function F(x)= 3x²+x is an odd function, option A is correct.

A relation is a function if it has only One y-value for each x-value.

To determine whether a function is odd or not, we need to check if f(-x) = -f(x) for all x in the domain of the function.

Let's check the function

F(x) = 3x²+ x

F(-x) = 3(-x)² + (-x) = 3x²- x

-f(x) = -(3x² + x) = -3x² - x

Since F(-x) = -f(x), this function is odd.

The other functions are even as they satisfy f(-x)=f(x)

Hence, the function F(x)= 3x²+x is an odd function, option A is correct.

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Answer:

The total length of all 8 movies is 740 minutes

Step-by-step explanation:

* Lets revise the arithmetic series

- In the arithmetic series there is a constant difference between  

 each two consecutive numbers  

- Ex:  

# 2 , 5 , 8 , 11 , ………………………. (constant difference is 3)

# 5 , 10 , 15 , 20 , ………………………… (constant difference is 5)

# 12 , 10 , 8 , 6 , …………………………… (constant difference is -2)

* General term (nth term) of an Arithmetic series:  

- If the first term is a and the common diffidence is d, then

 U1 = a , U2 = a + d , U3 = a + 2d , U4 = a + 3d , U5 = a + 4d  

 So the nth term is Un = a + (n – 1)d, where n is the position of the

  number in the series

- The formula to find the sum of n terms is

  Sn = n/2 [a + l] , where l is the last term in the series

* Lets solve the problem

- A new movie is released each year for 8 years to go along with a

 popular book series

∴ n = 8

- Each movie is 5 minutes longer than the last

∴ d = 5

- The first movie is 75 minutes long

∴ a = 75

- To find the total length of all 8 movies find the sum of the 8 terms

∵ Un = a + (n - 1)d

∵ The last term l is u8

∵ a = 75 , d = 5 , n = 8

∴ l = 75 + (8 - 1)(5) = 75 + 7(5) = 75 + 35 = 110

∴ l = 110

∵ Sn = n/2 [a + l]

∴ S8 = 8/2 [75 + 110] = 4 [185] = 740 minutes

* The total length of all 8 movies is 740 minutes

The total length of all 8 movies, using the arithmetic series formula, is 740 minutes. Each movie increases by 5 minutes, starting from 75 minutes. The last movie is 110 minutes long.

Calculating the Total Length of All 8 Movies Using an Arithmetic Series Formula

To determine the total length of the 8 movies, we can use the formula for the sum of an arithmetic series:

→ [tex]S_n[/tex] = n/2 × (a + l)

Where:

→ [tex]S_n[/tex] is the total length of all movies

→ n is the number of terms (movies)

→ a is the first term (length of the first movie)

→ l is the last term (length of the last movie)

Given:

→ a = 75 minutes (first movie)

→ d = 5 minutes (increase in length per movie)

→ n = 8 (total number of movies)

First, we find the length of the last movie using the formula for the nth term of an arithmetic series:

→ l = a + (n - 1) × d

Thus:

→ l = 75 + (8 - 1) × 5

    = 75 + 35

    = 110 minutes

Next, we use the sum formula:

→ [tex]S_n[/tex] = n/2 × (a + l)

→ [tex]S_8[/tex] = 8/2 × (75 + 110)

        = 4 × 185

        = 740 minutes

So, the total length of all 8 movies is 740 minutes.

Answer:

Step-by-step explanation:

The angle 30° and 2x are complementary angles.

Two angles are called complementary angles, if their sum is one right angle (90°).

Therefore we have the equation:

30 + 2x = 90          subtract 30 from both sides

2x = 60      divide both sides by 2

x = 30

The point slope of the line that passes through the points (-7, 2) and having a slope of 5 is y - 2 = 5 (x + 7).

Slope of a line is the ratio of the change in y coordinates to the change in the x coordinates of two points given.

Given that,

Slope of a line = 5

A point on the line = (-7, 2)

Point slope of a line having a slope of m and passing through a point (x', y') is,

y - y' = m(x - x')

Substituting the given slope and point,

y - 2 = 5 (x - -7)

y - 2 = 5 (x + 7)

Hence the required form of the line is y - 2 = 5 (x + 7).

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Answer:

a. 2.14% should have IQ scores between 40 and 60

b. 15.87% should have IQ scores below 80

Step-by-step explanation:

* Lets explain how to solve the problem

- For the probability that a < X < b (X is between two numbers, a and b),

 convert a  and b into z-scores and use the table to find the area

 between the two z-values.

- Lets revise how to find the z-score

- The rule the z-score is z = (x - μ)/σ , where

# x is the score

# μ is the mean

# σ is the standard deviation

* Lets solve the problem

- IQS are normally distributed with a mean of 100 and standard

 deviation of 20

∴ μ = 100 and σ = 20

a.

- The IQS is between 40 and 60

∴ 40 < X < 60

∵ z = (x - μ)/σ

∴ z = (40 - 100)/20 = -60/20 = -3

∴ z = (60 - 100)/20 = -40/20 = -2

- Use the z table to find the corresponding area

∵ P(z > -3) = 0.00135

∵ P(z < -2) = 0.02275

∴ P(-3 < z < -2) = 0.02275 - 0.00135 = 0.0214

∵ P(40 < X < 60) = P(-3 < z < -2)

∴ P(40 < X < 60) = 0.0214 = 2.14%

* 2.14% should have IQ scores between 40 and 60

b.

- The IQS is below 80

∴ X < 80

∵ z = (x - μ)/σ

∴ z = (80 - 100)/20 = -20/20 = -1

- Use the z table to find the corresponding area

∵ P(z < -1) = 0.15866

∵ P(X < 80) = P(z < -1)

∴ P(X < 80) = 0.15866 = 15.87%

* 15.87% should have IQ scores below 80

Answer:

112

Step-by-step explanation:

3(7-3) = 3 x 4 = 12

12^2 = 144

144 -4(6+2)

144-4(8)

144-32

112

[tex]\bf \stackrel{\mathbb{P~E~M~D~A~S}}{3(7-3)^2-4(6+2)}\implies 3(\stackrel{\downarrow }{4})^2-4(\stackrel{\downarrow }{8})\implies 3(\stackrel{\downarrow }{16})-4(8) \\\\\\ \stackrel{\downarrow }{48}-4(8)\implies 48-\stackrel{\downarrow }{32}\implies 16[/tex]

Answer:

(x-7) (x^2-5)

Step-by-step explanation:

x^3-7x^2-5x+35

Factor out an x^2 from the first two terms and -5 from the last 2 terms

x^2 (x-7) -5(x-7)

Factor out (x-7)

(x-7) (x^2-5)

Answer:

(x - 7) (x^2 - 5)

Step-by-step explanation:

Factor the following:

x^3 - 7 x^2 - 5 x + 35

Factor terms by grouping. x^3 - 7 x^2 - 5 x + 35 = (x^3 - 7 x^2) + (35 - 5 x) = x^2 (x - 7) - 5 (x - 7):

x^2 (x - 7) - 5 (x - 7)

Factor x - 7 from x^2 (x - 7) - 5 (x - 7):

Answer:  (x - 7) (x^2 - 5)