Complete the proof of the exterior angle theorem.


Given: angle ACD is an exterior angle of triangle BCD.


Prove: angle B + angle C = angle ADC


Statements:

1. angle ADC is an exterior angle of triangle BCD

2.

3. angle ADC and angle BCD form a linear pair

4.

5.

6.

7. angle ADC = angle B + angle C


Reasons:

1. Given

2. Triangle Sum Theorem

3.

4.

5.

6.

7. Subtraction Property of Equality


(i need the blanks filled in)

The equation of the line perpendicular to line AB, which is a vertical line with an undefined slope, and passing through the point (2, 4) is y = 4.

Determine the equation of the perpendicular line:

Find the slope of line AB:

Since both points A and B have the same x-coordinate (-6), the line AB is a vertical line.

Vertical lines have an undefined slope.

Determine the slope of the perpendicular line:

The slope of a line perpendicular to a vertical line is 0. This means the perpendicular line will be a horizontal line.

Use the point-slope form to find the equation of the perpendicular line:

The point-slope form of a linear equation is: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

We know the slope (m = 0) and we have a point on the line (2, 4).

Plugging these values into the point-slope form, we get: y - 4 = 0(x - 2)

Simplify the equation:

Since the slope is 0, the term 0(x - 2) becomes 0.

Therefore, the equation simplifies to: y - 4 = 0

Add 4 to both sides to isolate y:

y = 4

Therefore, the equation of the line perpendicular to line AB and passing through the point (2, 4) is y = 4.

Complete question:

Line AB passes through points A(-6,7) and B(-6,-3) as shown on the coordinate grid . below. Determine the equation of a line that is perpendicular to line AB and passes through the point (2,4).

Answer:

155/2

Step-by-step explanation:

7/7=1

1*1/2=1/2

61-1/2+17

122/2-1/2+17

121/2+17=155/2

Answer:

$200,000

Step-by-step explanation:

Base Salary = 40,000

Total Salary = 70,000

So, the rest, Allison earns from commissions. How much??

70,000 - 40,000 = 30,000

So, 15% of total sales is equal to 30,000

Let total sales be "t", so we can write an equation as:

15% * t = 30,000

15% in decimal is:

15/100 = 0.15

So, equation to solve is:

0.15t = 30,000

Finding t:

t = 30,000/0.15 = 200,000

Allison's total sales were $200,000

Answer:

The mileages for which company A charge less than company B when , y [tex]>[/tex] 50 .

Step-by-step explanation:

Given as :

The two truck companies have following prices

Company A

The initial fee = i = $110

And unlimited mileage

Company B

The initial fee = I = $65

Additional charge = $0.90 per mile driven

Let it driven x mile

Let The mileages for which company A charge less than company B = y

Now, According to question

Charges of company A = charges of company B

i.e i = I + 0.90 × x

Or, $110 = $65 + 0.90 × x

Or, $110 - $65 = 0.90 × x

Or, $45 = 0.90 × x

∴    x = [tex]\dfrac{45}{0.90}[/tex]

i.e  x = 50

i.e  Company B driven x = 50 miles

So, The mileages for which company A charge less than company B = y

And y [tex]>[/tex] x

i.e y [tex]>[/tex] 50

Hence, The mileages for which company A charge less than company B when , y [tex]>[/tex] 50 . Answer

Answer:

Therefore the area of parallelogram  is 32 cm².

Step-by-step explanation:

Given:

[] ABCD is a Parallelogram .........Given

AB = 4 cm = Base

BC = 10 cm

CE  = Height of Parallelogram  

To Find:

area of parallelogram  = ?

Solution:

In Right angle triangle BEC  by Pythagoras theorem we have

[tex](\textrm{Hypotenuse})^{2} = (\textrm{Shorter leg})^{2}+(\textrm{Longer leg})^{2}[/tex]

Substituting the values we get

[tex]10^{2}=6^{2} +CE^{2} \\\\CE^{2}=100-36=64\\ \\\therefore CE = 8\ cm\\[/tex]

Which is the Height of Parallelogram

Now, area of parallelogram  given by

[tex]\textrm{Area of Parallelogram} = Base\times Height[/tex]

Substituting the values we get

[tex]\textrm{Area of Parallelogram} = 4\times 8=32[/tex]

[tex]\textrm{Area of Parallelogram} = 32\ cm^{2}[/tex]

Therefore the area of parallelogram  is 32 cm².

Answer:

k=f(x)+2x+2

Step-by-step explanation:

f(x)=-2x-2+k

-2x-2=f(x)-k

f(x)-(-2x-2)=k

f(x)+2x+2=k

If bills new porch is rectangular with an area of 50 square feet if the length of the porch is two times the width. Then 30 feet is the perimeter of the porch

The area of Rectangle is length times of width.

Given,

bills new porch is rectangular with an area of 50 square feet

Area=50  square feet

Length of the porch is two times the width

Length=2Width

Area=Length×Width

50=2W×W

Divide both sides by 2

25=W²

Take square root on both sides

W=5ft

Hence width of bills new is 5 feet.

The perimeter of rectangle=2(Length+width)

=2(2W+W)

=2(2(5)+5)

=2(10+5)

=2(15)

=30

So perimeter of rectangle is 30 feet.

Hence 30 feet is the perimeter of the porch.

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The number 0.66666666666 rounded to the nearest whole number is 1, and rounded to the nearest tenth is 0.7, following standard rounding rules.

The number 0.66666666666, when rounded, will depend on what decimal place you are rounding to. For example:

These rounding rules apply consistently across various operations including multiplication and division, similar to those in addition and subtraction. Let's consider a few examples where these rules are used:

These examples demonstrate the standard rules of rounding numbers with precision.

Final answer:

To round 0.66666666666 to the nearest tenth, it rounds up to 0.7 because the digit after the tenths place is greater than 5.

Explanation:

When rounding the number 0.66666666666, we must first decide the decimal place to which we want to round.

If we round to the nearest tenth, the number would round up to 0.7 because the digit following the tenth's place is a 6, which is greater than 5.

These same rules apply for other numbers: for example, 0.028675 rounds up to 0.0287 as the dropped digit, a 7, is greater than 5.

It's important to consider the digit immediately following the place value you're rounding to, as it will determine whether you round up or down.

Answer:

Step-by-step explanation:

9.5+0.5c=t

Answer:

8.5C + 9.5

Step-by-step explanation:

Im pretty sure u meant to put 8.5

Answer:

Solution is x= 0 , x = \frac{1}{14}  , x = \frac{-1}{12}

Step-by-step explanation:

Given, equation is  \sqrt[3]{15x-1} + \sqrt[3]{13x+1} = 4\sqrt[3]{x} →→→ (1)

Now, by cubing the equation on both sides, we get

( \sqrt[3]{15x-1} + \sqrt[3]{13x+1} )³ = (4\sqrt[3]{x})³

⇒ (15x-1) + (13x+1) + 3×\sqrt[3]{15x-1}× \sqrt[3]{13x+1} (\sqrt[3]{15x-1} +  \sqrt[3]{13x+1}) = 64 x.

(since (a+b)³ = a³ + b³ + 3ab(a+b) ).

⇒ 28x + 3×\sqrt[3]{15x-1}× \sqrt[3]{13x+1} (\sqrt[3]{15x-1} +  \sqrt[3]{13x+1})  = 64x.        

(since from (1), \sqrt[3]{15x-1} + \sqrt[3]{13x+1} = 4\sqrt[3]{x} )

⇒ 12×\sqrt[3]{15x-1}× \sqrt[3]{13x+1} (\sqrt[3]{15x-1}×\sqrt[3]{x}= 36x.

⇒ 3x = [tex]\sqrt[3]{(15x-1)(13x+1)(x)}[/tex] .

Now, once again cubing on both sides, we get

(3x)³ = ([tex]\sqrt[3]{(15x-1)(13x+1)(x)}[/tex])³.

⇒ 27x³ = (15x-1)(13x+1)(x).

⇒ 27x³ = 195x³ + 2x² - x

⇒ 168x³ + 2x² - x = 0

⇒ x(168x² + 2x -1) = 0

⇒ by, solving the equation we get ,

x = 0 ; x = \frac{1}{14}   ; x = \frac{-1}{12}

therefore, solution is x= 0 , x = \frac{1}{14}   ,  x = \frac{-1}{12}

Answer:

[tex]4\times \sqrt{11}\div \(11)[/tex]

Step-by-step explanation:

[tex]\sqrt{16}\div \sqrt{11}[/tex]

[tex]\sqrt{16}\times \sqrt{11}\div \sqrt{11}\times \sqrt{11}[/tex]

[tex]4\times \sqrt{11}\div \(11)[/tex]  answer

Answer:

0.3 lbs. of meat per hamburger

To solve for the number of adults going to the exhibition, you can set up a system of linear equations based on the given information about ticket costs and quantities, leading to the conclusion that there were 2 adults attending.

The problem is a classic example of a system of linear equations which can be solved using substitution or elimination.

Step-by-step Solution:

Let's denote the number of adults going to the exhibition as A and the number of children as C.

Therefore, there were 2 adults going to the exhibition.

The number of pennies in bank is 42

Solution:

Given that ratio of pennies to quarters in a piggy bank is 14 : 3

Let 14x be the number of pennies in piggy bank

Let 3x be the number of quarters in piggy bank

To find: number of pennies

Given that there are 51 quarters

number of pennies in piggy bank + number of quarters in piggy bank = 51

14x + 3x = 51

17x = 51

x = 3

number of pennies = 14x = 14(3) = 42

Thus the number of pennies in bank is 42

To find the number of pennies in the piggy bank, set up a proportion using the given ratio and the number of quarters. Cross-multiply and solve for x to find the number of pennies. There are 238 pennies in the piggy bank.

To find the number of pennies in the piggy bank, we need to use the given ratio and information. The ratio of pennies to quarters is 14:3. This means that for every 14 pennies, there are 3 quarters. We are given that there are 51 quarters in the bank. We can set up a proportion to find the number of pennies:

So, there are (14 * 51) / 3 = 238 pennies in the piggy bank.

Answer:

Second option: [tex]3 + 4 x\leq 2 (1 + 2x)[/tex]

Step-by-step explanation:

Let's solve for "x" from each inequality:

[tex]a.\ 6 (x + 2)>x-3\\\\6x+12>x-3\\\\6x-x>-3-12\\\\5x>-15\\\\x>\frac{-15}{5}\\\\x>-3[/tex]

[tex]b.\ 3 + 4 x\leq 2 (1 + 2x)\\\\3+4x\leq 2+4x\\\\3-2\leq 4x-4x\\\\1\leq 0\  (FALSE.\ IT\ HAS\ NO\ SOLUTION)[/tex]

[tex]c.\ -2 (x + 6)<x-20\\\\-2x-12<x-20\\\\-12+20<x+2x\\\\8<3x\\\\x>\frac{8}{3}[/tex]

[tex]d.\ x-9<3(x-3)\\\\x-9<3x-9\\\\x-3x<-9+9\\\\-2x<0\\\\x>0[/tex]

The inequality that has no solution is: 3 + 4x ≤ 2(1 + 2x).

The correct option is 2nd.

Given are inequalities, we need to check which of them has no solution.

Let's solve each inequality to determine if it has a solution:

1) 6(x + 2) > x - 3

First, distribute the 6:

6x + 12 > x - 3

Now, let's isolate the variable x on one side:

6x - x > -3 - 12

5x > -15

Finally, divide by 5 to solve for x:

x > -3

This inequality does have a solution, and x can take any value greater than -3.

2) 3 + 4x ≤ 2(1 + 2x)

First, distribute the 2 on the right side:

3 + 4x ≤ 2 + 4x

Now, subtract 4x from both sides:

3 ≤ 2

This is not true, and there is no value of x that satisfies this inequality.

So, this inequality has no solution.

3) -2(x + 6) < x - 20

First, distribute the -2:

-2x - 12 < x - 20

Now, let's isolate the variable x on one side:

-2x - x < -20 + 12

-3x < -8

Finally, divide by -3 to solve for x (note the sign change when dividing by a negative number):

x > 8/3 or x > 2.67

This inequality has a solution, and x can take any value greater than 2.67.

4) x - 9 < 3(x - 3)

First, distribute the 3 on the right side:

x - 9 < 3x - 9

Now, move the variables to one side:

x - 3x < 9 - 9

-2x < 0

Finally, divide by -2 to solve for x (note the sign change when dividing by a negative number):

x > 0

This inequality has a solution, and x can take any value greater than 0.

Hence, the inequality that has no solution is: 3 + 4x ≤ 2(1 + 2x).

The correct option is 2nd.

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Let X = the number of weeks.

Jeanette will pass 3 per week, so 3x, and she already passed 8, so Jeanette would be 3x +8

Paul would be 4x + 3

Set them to equal to solve for x ( the number of weeks.)

3x +8 = 4x +3

Subtract 3x from both sides:

8 = x +3

Subtract 3 from both sides:

X = 5

So it will take 5 weeks to be the same.

Replace X with 5 to find the number of scales.

3x + 8 = 3(5) + 8 = 15 +8 = 23

They would have 23 scales.

Answer:

No. The differences between two elements should be same. For example 4-7-10-13 is a arithmetic sequence.

Step-by-step explanation:

Answer:

$486

Step-by-step explanation:

First you find the area of the window. The area of a square is equal to h×w, so we take 9×9 which equals 81. Since it costs $6 per square foot, we now take 81×6, which gives us an answer of $486.

The glass to replace the window will cost $486 to Linda.

The Area of a figure is the space inside its 2D perimeter.

To determine the area of a square we just square the length of its side.

Area of a square = (side)²

We are said that the window is in the shape of a square, with 9 feet as its side's length.

∴ Area of the window = 9² square feet = 81 square feet.

Cost of 1 square foot of glass = $6/square foot

∴ Cost of the glass for window = Area of the window * rate of glass

= $81*6 = $486.

∴ The glass to replace the window will cost $486 to Linda.

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Answer:

Benny's:

2.00 / 4 = $0.50 per sandwich

ABC:

1.50 / 2 = $0.75 per sandwich

Sandwich Hut:

2.00 / 2 = $1.00 per sandwich

Benny's has the best deal

Answer:

The answer for this question would be $3.00 for sandwiches

Step-by-step explanation:

Answer:

NO, It is non terminating / recurring decimal

Step-by-step explanation:

3/9= 0.33333333......

remainder= 3

Answer:

4x^2+20x+24

Step-by-step explanation:

(4x+8)(x+3)=4x^2+8x+12x+24=4x^2+20x+24

Answer:

First fund: $4000

second account: $2000

Step-by-step explanation:

invest x for first fund (A) and y for second fund (B)

interest for A: x * 0.05

interest for B: y * 0.09

0.05 x + 0.09 y = 380 ....(1)

x + y = 6000 ..... (2)

y = 6000 - x   ....substitute y in (1)

0.05 x + 0.09 (6000 - x) = 380

0.05x + 540 - 0.09x = 380

0.04 x = 160

x = 4000

y = 6000 -4000 = 2000

check: 4000 x 0.05 + 2000 x 0.09 = 200 + 180 = 380

Answer:

Mike = 3

Robert = 9

Step-by-step explanation:

R = Robert

M = Mike

Mike is 6 years younger than Robert

R = M + 6

Add 3 years to both sides

R + (3) = 2M + (3)

Represent that Robert is now twice as old as Mike

2M + 3 = M + 3 + 6

Solve + Subtract the 3 years

2M + 3 = M + 9

M = 6 - 3 = 3

R = 12 - 3 = 9

Hope this helps :)

Answer:

C - 19.43

Step-by-step explanation:

first you need to find how much 1 pound of jelly beans costs. to find that you do 14.74 ÷ 5.5 to get 2.68. Then you times that by 7.25 to get the answer, 19.43

The cost of the 7.25 pounds of the jelly bean will be $19.92. The correct option is B.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that Jacob bought 5.5 pounds of jelly beans for $14.74.

The cost of the 7.25 pounds of the jelly bans will be calculated as,

5.5 pounds ⇒ $14.74

1 pound ⇒ $(14.74/5.5)

7.25 pounds ⇒ (14.74x7.25)/5.5

7.25 pounds = $19.43

Therefore, the cost of the 7.25 pounds of the jelly bean will be $19.92. The correct option is B.

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Divide total miles by miles per gallon:

364 miles / 20 miles per gallon = 18.2 gallons

Answer:

y = (7/2)x - 2

Step-by-step explanation:

Use slope-intercept formula for linear equations y = mx + b

x and y show what points are on the line. We know (4, 12). x= 4 and y = 12

b is the y-intercept, which b = -2.

m is the slope.

To write the equation, you need the slope and y-intercept. Find the slope by putting x, y and b into the formula.

y = mx + b

12 = 4m - 2

14 = 4m

14/4 = m

m = 7/2

Put "m" into the formula with "b"

y = (7/2)x - 2

Answer: OPTION C.

Step-by-step explanation:

The equation of the line in Slope-Intercept form is:

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

Where "m" is the slope of the line and "b" is the y-intercept.

In order to verify which fucntion is not linear, you can solve for "y" from each function. Then:

[tex]A.\ y+1= x\\\\y=x-1[/tex]

This is a Linear function.

[tex]B.\ x=2y-3\\\\x+3=2y\\\\y=\frac{1}{2}x+\frac{3}{2}[/tex]

This is a Linear function.

[tex]C.\ xy=2\\\\y=\frac{2}{x}[/tex]

This is NOT a Linear function.

[tex]D.\ x+1=\frac{3}{4}y\\\\4x+4=3y\\\\y=\frac{4}{3}x+\frac{4}{3}[/tex]

This is a Linear function.

Final answer:

Option C) xy = 2 does not represent a linear function because it cannot be rewritten in the form y = mx + b, which is characteristic of linear equations. Instead, it represents a hyperbola.

Explanation:

The question asks: "Which does NOT represent a linear function?" with the options being A) y+1= x, B) x=2y-3, C) xy=2, and D) x+1=3/4y. To determine which equation does not represent a linear function, we must recall that linear functions can always be written in the form y = mx + b, where m is the slope and b is the y-intercept.

A) y+1 = x can be rewritten as y = x - 1, which is linear.

B) x = 2y - 3 can be rewritten as y = 1/2x + 3/2, which is linear.

C) xy = 2 cannot be rewritten in the form y = mx + b because it involves both x and y being multiplied together, which is characteristic of a hyperbola, not a linear equation.

D) x + 1 = 3/4y can be rearranged to y = 4/3x + 4/3, which is linear.

Therefore, the equation that does NOT represent a linear function is C) xy = 2.

Answer:

1. (x, y) → (x + 3, y - 2)

Vertices of the image

a) (-2, - 3)

b) (-2, 3)

c) (2, 2)

2. (x, y) → (x - 3, y + 5)

Vertices of the image

a) (-3, 2)

b) (0, 2)

c) (0, 4)

d) (2, 4)

3. (x, y) → (x + 4, y)

Vertices of the image

a) (-1, -2)

b) (1, -2)

c) (3, -2)

4. (x, y) → (x + 6, y + 1)

Vertices of the image

a) (1, -1)

b) (1, -2)

c) (2, -2)

d) (2, -4)

e) (3, -1)

f) (3, -3)

g) (4, -3)

h) (1, -4)

5. (x, y) → (x, y - 4)

Vertices of the image

a) (0, -2)

b) (0, -3)

c) (2, -2)

d) (2, -4)

6. (x, y) → (x - 1, y + 4)

Vertices of the image

a) (-5, 3)

b) (-5, -1)

c) (-3, 0)

d) (-3, -1)

Explanation:

To identify each IMAGE you should perform the following steps:

1. (x, y) → (x + 3, y - 2)

The rule means that every point of the preimage is translated three units to the right and 2 units down.

Vertices of the preimage      Vertices of the image

a) (-5,2)                                   (-5 + 3, -1 - 2) = (-2, - 3)

b) (-5, 5)                                  (-5 + 3, 5 - 2) = (-2, 3)

c) (-1, 4)                                   (-1 + 3, 4 - 2) = (2, 2)

2. (x,y) → (x - 3, y + 5)

The rule means that every point of the preimage is translated three units to the left and five units down.

Vertices of the preimage      Vertices of the image

a) (0, -3)                                   (0 - 3, -3 + 5) = (-3, 2)

b) (3, -3)                                   (3 - 3, -3  + 5) = (0, 2)

c) (3, -1)                                    (3 - 3, -1 + 5) = (0, 4)

d) (5, -1)                                    (5 - 3, -1 + 5) = (2, 4)

3. (x, y) → (x + 4, y)

The rule represents a translation 4 units to the right.

Vertices of the preimage   Vertices of the image

a) (-5, -2)                               (-5 + 4, -2) = (-1, -2)

b) (-3, -5)                               (-3 + 4, -2) = (1, -2)

c) (-1, -2)                                (-1 + 4, -2) = (3, -2)

4. (x, y) → (x + 6, y + 1)

Vertices of the preimage      Vertices of the image

a) (-5, -2)                                  (-5 + 6, -2 + 1) = (1, -1)

b) (-5, -3)                                  (-5 + 6, -3 + 1) = (1, -2)

c) (-4, -3)                                   (-4 + 6, -3 + 1) = (2, -2)

d) (-4, -5)                                  (-4 + 6, -5 + 1) = (2, -4)

e) (-3, -2)                                  (-3 + 6, -2 + 1) = (3, -1)

f) (-3, -4)                                   (-3 + 6, -4 + 1) = (3, -3)

g) (-2, -4)                                  (-2 + 6, -4 + 1) = (4, -3)

h) (-2, -5)                                  (-2 + 3, -5 + 1) = (1, -4)

5. (x, y) → (x, y - 4)

This is a translation four units down

Vertices of the preimage      Vertices of the image

a) (0, 2)                                    (0, 2 - 4) = (0, -2)

b) (0,1)                                      (0, 1 - 4) = (0, -3)

c) (2, 2)                                     (2, 2 - 4) = (2, -2)

d) (2,0)                                     (2, 0 - 4) = (2, -4)

6. (x, y) → (x - 1, y + 4)

This is a translation one unit to the left and four units up.

Vertices of the pre-image     Vertices of the image

a) (-4, -1)                                   (-4 - 1, -1 + 4) = (-5, 3)

b) (-4 - 5)                                  (-4 - 1, -5 + 4) = (-5, -1)

c) (-2, -4)                                  (- 2 - 1, -4 + 4) = (-3, 0)

d) (-2, -5)                                 (-2 - 1, -5 + 4) = (-3, -1)

Answer:

$1039.55

Step-by-step explanation:

The total cost of the 2-year service of premium cable will be $344.6.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that a company offers premium cable for 39.95 per month plus a one-time setup fee. The total cost for setup and months of service was 264.70.

The total cost will be calculated by forming an equation as below,

C = 39.9m + 264.70

For 2 years the total cost will be,

C = (39.9 x 2) + 264.70

C = 79.9 + 264.70

C = $344.6

Therefore, the total cost of the 2-year service of premium cable will be $344.6.

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Answer:

16.72

Step-by-step explanation:

To find the answer divide 83.61 by 100 then multiply it by 20

20% of 83.61 is = 16.72