Is the centroid of a triangle, the center of the circle cirumdcribing it? ​

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Explanation:

Check out figure 1 which is one of the attached images below.

In this diagram, I have angle A as 75 degrees and angle B as 51 degrees.

Angle C is therefore, C = 180-A-B = 180-75-51 = 54 degrees.

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Point D is somewhere between A and B such that it is on segment AB.

Figure 2 (also attached as an image) shows segment CD forming two angles DCB and ACD.

These are the blue and red angles respectively, such that the blue angle is twice as large as the red angle.

blue angle = 2*(red angle)

This is what it means when it says the ratio of the two angles is 2:1.

I have 2x as the blue angle and x as the red angle. We don't know what x is yet, but we do know that the x and 2x combine back to angle C = 54 degrees.

So,

(angle DCB) + (angle ACD) = angle C

(2x) + (x) = 54

3x = 54

x = 54/3

x = 18

Since x = 18, this means 2*x = 2*18 = 36

Therefore,

angle DCB = 2x = 36 degrees

angle ACD = x = 18 degrees

--------

Focus solely on triangle DCB. We found angle DCB = 36 degrees and we know that angle DBC = 51

The remaining angle y = angle BDC is...

(angle BDC)+(angle DCB)+(angle DBC) = 180

(y)+(36)+(51) = 180

y+87 = 180

y+87-87 = 180-87

y = 93

angle BDC = 93 degrees

Figure 3 shows the angles we found (basically I replaced x, 2x and y with their respective numbers).

The size of angle BDC is 0 degrees.

To find the size of angle BDC, you can use the fact that the sum of the angles in a triangle is always 180 degrees. Since angle ACD is 1 part and angle DCB is 2 parts, we can express their sizes as:

Angle ACD = x degrees

Angle DCB = 2x degrees

Now, you know that the sum of the angles in triangle ABC is 180 degrees. So, you can write the equation:

x (angle ACD) + 2x (angle DCB) + 180 degrees (angle BDA) = 180 degrees

Now, simplify and solve for x:

x + 2x + 180 = 180

Combine like terms:

3x + 180 = 180

Now, subtract 180 from both sides of the equation:

3x = 0

Finally, divide by 3:

x = 0

Now that you know the value of x, you can find the size of angle BDC (2x):

Angle BDC = 2x = 2(0) = 0 degrees

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[tex]\text{Solve for x:}\\\\-4x-6=-5y+9\\\\\text{Add 6 to both sides}\\\\-4x=-5y+15\\\\\text{Divide both sides by -4}\\\\\boxed{x=\frac{5}{4}y-\frac{15}{4}}[/tex]

Answer: Order if operations

Step-by-step explanation:

15-10n+12n

15+12n

Answer:

x = 5/4

y = 15/16

Step-by-step explanation:

We are given y = 3/4x

and 5/2x +2y = 5

Substituting the value y=3/4 x in second equation we get

5/2x + 2(3/4)x = 5

8/2x = 5

4x = 5

and so , x = 5/4

substituting value of x in y = 3/4x we get,

y = 3/4 (5/4)

So, y = 15/16

which are the required values of x and y on solving the two given equations.

The solution to the equation is (5/4, 15/16)

Given the system of equations

Substitute equation 1 into 2 to have:

5/2x  + 2(3/4x) = 5

5/2x + 3/2x = 5

8/2x = 5

4xx = 5

x = 5/4

Snce y = 3/4x

y = 3/4(5/4)

y = 15/16

Hence the solution to the equation is (5/4, 15/16)

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Answer:

  12 pm noon

Step-by-step explanation:

The least common multiple of the two times is 240 minutes, or 4 hours. The buses will meet at the station again at noon.

__

60 = 20×3

80 = 20×4

The least common multiple is 20×3×4 = 240 minutes. There are 60 minutes in an hour, so this is 240/60 = 4 hours. 4 hours after 8 am is 12 pm, noon.

Answer:

The three points for the line  y = -6x + 12 ...Red color line

point A( x₁ , y₁) ≡ ( 0 , 12) (blue color point on the graph)

point B( x₂ , y₂) ≡ (2 , 0) (green color point on the graph)

point C(x₃ , y₃ ) ≡ (1 , 6) (purple color point on the graph)

The Graph is attached below.

Step-by-step explanation:

Given:

[tex]y = -6x +12[/tex]  ........... equation of a line

Let the points be point A, point B, and point C  

To Find:

point A( x₁ , y₁) ≡  ?

point B( x₂ , y₂) ≡ ?

point C(x₃ , y₃ ) ≡ ?

Solution:

For Drawing a graph we require minimum two points but we will have here three points.

For point A( x₁ , y₁)

Put x = 0 in the given equation we get

y = 0 + 12

y = 12

∴ point A( x₁ , y₁) ≡ ( 0 , 12)

For point B( x₂ , y₂)

Put y= 0 in the given equation we get

0 = -6x + 12

6x = 12

[tex]x=\dfrac{12}{6}=2[/tex]

∴ point B( x₂ , y₂)  ≡ (2 , 0)

For point C(x₃ , y₃ )

Put x = 1 in the given equation we get  

y = -6 × 1 + 12

y = 6

∴ point C(x₃ , y₃ )≡ (1 , 6)

Therefore,

The three points for the line  -2y = -x + 8 are

point A( x₁ , y₁) ≡ ( 0 , 12) (blue color point on the graph)

point B( x₂ , y₂) ≡ (2 , 0) (green color point on the graph)

point C(x₃ , y₃ ) ≡ (1 , 6) (purple color point on the graph)

The Graph is attached below..

Answer:

6.2

Step-by-step explanation:

sin 20 = x/18

solve for x

Answer:

Area of trapezoid = 144 cm²

Step-by-step explanation:

The length of parallel sides are,

a = 8 cm

b = 16 cm

h = 12 cm

Formula area of trapezoid:-

[tex]A=\dfrac{1}{2}(a+b)\times h[/tex]

[tex]A=\dfrac{1}{2}(8+16)\times 12[/tex]

[tex]A=144[/tex]

Hence, the area of trapezoid is 144 cm²

Answer:

Hence, the area of trapezoid is 12 cm²

Step-by-step explanation:

Answer:

[tex]k=54[/tex]

Step-by-step explanation:

Find the coordinates of points M and N in terms of k. Solve the system of two equations:

[tex]\left\{\begin{array}{l}y=3x^2\\ \\y=-3x^2+k\end{array}\right.\Rightarrow \left\{\begin{array}{l}y=3x^2\\ \\y=-y+k\end{array}\right.\\ \\2y=k\\ \\y=\dfrac{k}{2}\\ \\\dfrac{k}{2}=3x^2\\ \\x^2=\dfrac{k}{6}\\ \\x=\pm\sqrt{\dfrac{k}{6}}[/tex]

Two points have the coordinates [tex]M\left(\sqrt{\dfrac{k}{6}},\dfrac{k}{2}\right)[/tex] and [tex]N\left(-\sqrt{\dfrac{k}{6}},\dfrac{k}{2}\right)[/tex]

Find the distance between M and N:

[tex]MN=\sqrt{\left(\sqrt{\dfrac{k}{6}}-\left(-\sqrt{\dfrac{k}{6}}\right)\right)^2+\left(\dfrac{k}{2}-\dfrac{k}{2}\right)^2}=\sqrt{4\dfrac{k}{6}}=\sqrt{\dfrac{2k}{3}}[/tex]

Since MN = 6, you have

[tex]\sqrt{\dfrac{2k}{3}}=6\\ \\\dfrac{2k}{3}=36\\ \\2k=108\\ \\k=54[/tex]

Answer: -13x^2 + 10x + 4

Step-by-step explanation:

group all of the same variables together to simplify the equation.

-7 - 6 = -13 // the x^2 value

-5 + 15 = 10 // the x value

4 is the only number without an exponent so leave it alone.

Answer:

-13x²+10x+4

Step-by-step explanation:

You need to group all of the same variables together:

-7x² - 5x - 6x² + 4 + 15x

-7x²-6x²-5x+15x+4

-13x²+10x+4

Answer:

See explanation!

Step-by-step explanation:

Let us first give some principle theory to aid our solution.

Considering two functions [tex]A(x)[/tex] and [tex]B(x)[/tex], in order to show that function

which is the inverse of [tex]y=ax[/tex].

Now let as solve our problem. We are given the following:

[tex]f(x)=3x-4\\g(x)=\frac{x+4}{3}[/tex]

Method A: Show that these are inverse functions by finding f^-1 (x) and showing that it is the same as g(x).

Let us take [tex][f(x)=y]=3x-4[/tex] and "exchanging" our variables we have

[tex]x=3y-4\\x+4=3y\\\\y=\frac{x+4}{3}[/tex]

which is exactly the same with our given function of [tex]g(x)=\frac{x+4}{3}[/tex], so proved!

Method B: Show that these are inverse functions by showing that when the output of one function is used for the input of the other function, the final output is equal to the original input value. (you may choose any initial input)

For this case we will use a simple input let us say [tex]x=1[/tex]. Thus taking the [tex]f(x)[/tex] function and plugging in we have:

[tex]f(x=1) = 3(1)-4\\f(1)=3-4\\f(1)=-1[/tex]

Now let us take the output of [tex]f(1)[/tex] which is [tex]-1[/tex] and use it the input to our second function of [tex]g(x)[/tex], so we have:

[tex]g(x=-1) = \frac{(-1)+4}{3}\\ \\g(-1)=\frac{3}{3}\\ \\g(-1)=1[/tex]

so the output of the second function is equal to the original input value of the first function, hence proved!

Method C: Verify that these are the inverse function by showing that f(g(x)) = x AND g(f(x)) = x.

Basically we are asked to prove that both [tex]f(g(x))=g(f(x))=x[/tex]

To do so, we just replace one function into the [tex]x[/tex] value of the other function as follow:

[tex]f(g(x))=3(\frac{x+4}{3} )-4\\\\f(g(x))=x+4-4\\\\f(g(x))=x[/tex]

Lets repeat now for the opposite as follow:

[tex]g(f(x))=\frac{(3x-4)+4}{3}\\ \\g(f(x))=\frac{3x}{3}\\ \\g(f(x))=x[/tex]

Hence proved!

Answer:

Option A    M<17   m is less than or equal to 17

Step-by-step explanation:

Answer:

A.

Step-by-step explanation:

Answer:

There are 24 gumballs, 6 of which are yellow.

P(yellow) = 6/24 = 1/4

[tex]\bf \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}\qquad \implies y = \stackrel{\stackrel{m}{\downarrow }}{-\cfrac{5}{2}}x\stackrel{\stackrel{b}{\downarrow }}{-2}[/tex]

Answer:

22.77

Step-by-step explanation:

39.95 * 1.14 (tax + markup) * 0.5 (discount) = 22.77 (rounded)

Answer:B

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

solve this equation 4/2.40

Answer:No

Step-by-step explanation:

Answer:

x=4

Step-by-step explanation:

10x-7x=12

3x=12

divide both sides by 3

x=4

Answer: x = 4

Step-by-step explanation: To solve this equation, we can first combine our like terms on the left side of the equation which are the x's.

10x - 7x = 3x so we now have the equation 3x = 12.

Now to solve for x, we want to get x by itself on the left side of the equation. Since x is being multiplied by 3, to get x by itself, we need to divide by 3 on the left side of the equation. Since we divided by 3 on the left side, we must also divide by 3 on the right side.

On the left side, notice that the 4's cancel out so we're simply left with x. On the right side we have 12 divided by 3 which is 4 so x = 4.

Finally, remember to check your answer by substituting a 4 back into the original equation.

So we have 10(4) - 7(4) = 12 or 40 - 28 = 12 or 12 = 12.

Since this a true statement, we know our answer is correct.

Remember to show all your work when solving equations.

Answer:

6(x+5)

Step-by-step explanation:

Answer: 6 x (x+5)

Step-by-step explanation: Since it is six times a number AND five, you want to do the addition equation first. To show this you place parentheses around a variable (x) plus 5. You add six in front of this because it comes first in the word problem.

Addie's speed was [tex]3\frac{1}{3}\ mph[/tex]

Step-by-step explanation:

Given,

Distance walked by Addie = [tex]2\frac{1}{2}=\frac{5}{2}\ miles[/tex]

Time taken by Addie = 45 minutes

Converting the time into hours;

1 hour = 60 minutes

45 minutes = [tex]\frac{45}{60}=\frac{3}{4}\ hour[/tex]

Distance = Speed * Time

Speed = Distance/Time

Speed = [tex]\frac{5}{2} / \frac{3}{4}[/tex]

Speed = [tex]\frac{5}{2}*\frac{4}{3} = \frac{10}{3}\ mph[/tex]

Speed = [tex]3\frac{1}{3}\ mph[/tex]

Addie's speed was [tex]3\frac{1}{3}\ mph[/tex]

Keywords: speed, distance

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Final answer:

A quadratic function with zeros at -5 and -6 can be written as f(x) = (x + 5)(x + 6), which expands to f(x) = x² + 11x + 30.

Explanation:

To write a quadratic function whose zeros are -5 and -6, you start by remembering that if α and β are the zeros of a quadratic function, the function can be written as f(x) = a(x - α)(x - β) where a is a nonzero coefficient. In this case, -5 and -6 are the zeros, so our function is f(x) = a(x + 5)(x + 6).

For simplicity, if we choose a to be 1, the quadratic function simplifies to

f(x) = (x + 5)(x + 6)

f(x) = x² + 11x + 30.

Therefore, the quadratic function with zeros at -5 and -6 is f(x) = x² + 11x + 30.

Answer:

5715x3=

Rs.17145

Step-by-step explanation:

The interest rate required for Rs.4500 to amount to Rs.5715 in 3 years is 9% per annum. This is obtained using the simple interest formula. Therefore, the annual rate is 9%.

To find the annual interest rate at which Rs.4500 amounts to Rs.5715 in 3 years, we use the formula for simple interest:

Final Amount (A) = Principal (P) + Interest (I)

Given:

First, we calculate the interest:

Next, we use the simple interest formula:

Rearranging for the rate (R), we get:

Substitute the known values:

Converting to a percentage:

Therefore, the rate per annum required for Rs.4500 to grow to Rs.5715 in 3 years is 9%.

Answer:

The width of the red tile would be [tex]2.25[/tex] inches.

Step-by-step explanation:

Given blue tile has a length of [tex]4.25[/tex] inches, and a width of [tex]6.75[/tex]inches.

Also, length of similar tile is [tex]12.75[/tex] inches.

Let us assume [tex]x[/tex] as the width of red tile.

It is given in the question that both tiles are similar.

So, their area will be the same.

Area of blue tile would be [tex]4.25\times 6.75=28.6875\ inch^2[/tex]

Area of red tile would be [tex]12.75\times x[/tex]

The area of blue tile would be equal to area of red tile.

[tex]28.6875=12.75\times x[/tex]

[tex]\frac{28.6875}{12.75}=x\\\\x=2.25[/tex]

So, the width of the red tile would be [tex]2.25[/tex] inches.

Answer:

vertex=?

Step-by-step explanation:

given data:

g(x)=-3x^2+18x+2

from the given equation

a=-3,b=18,c=2

x=-b/2a

x=-(18)/2(-3)

x=3

put value of x in the given equation

y=-3(3)^2+18(3)+2

y=-27+54+2

y=29

x=3    y=29

(3,29)

y=-2/3x+5

Step-by-step explanation:

From the scattered plot, it is evident that the line of best fit will have a negative slope from the way the plots are aligned.

Selecting the line of best fit that will cut the y axis at +5, the points can be estimated as;

(5,1.5) and (3.5,2.5)

The slope can be found as;

m=Δy/Δx

m=2.5-1.5/3.5-5 =1.0/ -1.5 = -0.67

Finding the equation as;

m=Δy/Δx

-0.66 = y-2.5/x-3.5

-0.66(x-3.5) = y-2.5

-0.66x+2.3=y-2.5

-0.66x+2.3+2.5=y

-0.66x+4.7=y

⇒⇒  y= -0.66 x + 4.7

⇒⇒y= -2/3 x + 5

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Answer:

b

Step-by-step explanation:

yes it b because it goes origion and it not a zero pair

The value of p that makes the equation true is -20

Step-by-step explanation:

To solve an equation of one variable

∵ The equation is [tex]\frac{3}{5}p+\frac{1}{5}(40 - p)=0[/tex]

- Simplify the left hand side

∵ [tex]\frac{1}{5}(40-p)=\frac{1}{5}(40)-\frac{1}{5}(p)[/tex]

∴ [tex]\frac{1}{5}(40-p)=8-\frac{1}{5}p[/tex]

- Substitute [tex]\frac{1}{5}(40-p)[/tex] in the equation by [tex]8-\frac{1}{5}p[/tex]

∴ [tex]\frac{3}{5}p+8-\frac{1}{5}p=0[/tex]

- Add the like terms

∴ [tex](\frac{3}{5}p-\frac{1}{5}p)+8=0[/tex]

∴ [tex]\frac{2}{5}p+8=0[/tex]

- Subtract 8 from both sides

∴ [tex]\frac{2}{5}p=-8[/tex]

- Divide both sides by [tex]\frac{2}{5}[/tex]

∴ p = -20

To check the answer substitute p by -20 in the equation, the left hand side is equal to the right hand side, then the answer is right

∵ left hand side = [tex]\frac{3}{5}(-20)+\frac{1}{5}(40-(-20))[/tex]

∴ left hand side = [tex]\frac{-60}{5}+\frac{1}{5}(40+20)[/tex]

∴ left hand side = [tex]\frac{-60}{5}+\frac{60}{5}[/tex]

∴ left hand side = 0

∵ Right hand side = 0

∴ Left hand side = Right hand side

∴ The value of p makes the equation true

The value of p that makes the equation true is -20

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Answer:

-20

Step-by-step explanation:

Answer:

x=3/8, y=-7/8. (3/8, -7/8).

Step-by-step explanation:

y=-5x+1

y=3x-2

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-5x+1=3x-2

-5x-3x+1=-2

-8x+1=-2

-8x=-2-1

-8x=-3

8x=3

x=3/8

y=3(3/8)-2=9/8-2=9/8-16/8=-7/8

a = amount invested at 9%

b = amount invested at 11%

[tex]\bf \begin{array}{|c|ll} \cline{1-1} \textit{a\% of b}\\ \cline{1-1} \\ \left( \cfrac{a}{100} \right)\cdot b \\\\ \cline{1-1} \end{array}~\hfill \stackrel{\textit{9\% of "a"}}{\left( \cfrac{9}{100} \right)a\implies 0.09a}~\hfill \stackrel{\textit{11\% of "b"}}{\left( \cfrac{11}{100} \right)b\implies 0.11b}[/tex]

we know she has a total of $40,000 to invest, so, if she invested "a" amount in one, the other quantity must be the slack left from 40,000 and "a", namely b = 40000 - a.

we also know that the yield or earned interest from both amounts is 4300, thus

[tex]\bf \stackrel{\textit{yield of "a"}}{0.09a}~~+~~\stackrel{\textit{yield of "b"}}{0.11b}~~=~~\stackrel{\textit{annual income from both}}{4300} \\\\\\ \stackrel{\textit{yield of "a"}}{0.09a}~~+~~\stackrel{\textit{yield of "b"}}{0.11(40000-a)}~~=~~4300 \\\\\\ 0.09a+4400-0.11a = 4300\implies -0.02a+4400 = 4300 \\\\\\ 4400=4300+0.02a\implies 100=0.02a\implies \cfrac{100}{0.02}=a\implies \boxed{5000=a} \\\\\\ \stackrel{\textit{we know that}}{b = 40000-a}\implies \boxed{b = 35000}[/tex]