In the right △ABC with m∠C=90°, m∠B=75°, and AB=12 cm. Find the area of △ABC.

Don't use trig to solve, don't know how to.

The meaning of the sentence is that for an one-to-one function, each input makes an unique output.

Here we want to find the meaning of  "A function f is one-to-one if f(x) = f(y) implies x = y"

That is a definition of an one-to-one function.

In simpler words, an one to one function is a function that for each input x, the output y is unique.

So the sentence says that if:

f(x) = f(y)  (we have two equal outputs)

Then x = y  (then the inputs must be equal)

So that is the meaning of the sentence.

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Answer is in the file atttached.            

The coordinate of vertices of DEF are;

D = ( -4, -3)

E = ( -7, 4)

F =( -7, -3)

Here, we have,

Reflection in the y-axis: The reflection of point (x, y) across the y-axis, the y-coordinate remains the same, but the x-coordinate is taken to be the additive inverse  (x,y)→( -x,y) .

First, triangle ABC is reflected across the y axes.

So, we have:

(x,y)→( -x,y)

then, translated down 5 units.

so, transformation rule on ABC, we have:

(x,y)→( -x , y - 5)

so, the coordinate of vertices of DEF are;

D = ( - 4, 2 - 5) = ( -4, -3)

E = ( -7, 9 - 5) = ( -7, 4)

F = (-7 , 2 - 5) = ( -7, -3)

Hence, the coordinate of vertices of DEF are;

D = ( -4, -3)

E = ( -7, 4)

F =( -7, -3)

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Answer:

Step-by-step explanation:

You have two problems involving quadratic sequences. For the sequence that starts 20, 18, 14, 8, ..., you want the n-th term, the 12-th term, and the term number of -36. For the sequence with terms 2, 3, and 5 having values 1, -6, and -14, you want the second difference and the first term.

The first attachment shows the first and second differences of this sequence each begin with -2. The first of differences at level n can be put into a formula for the n-th term:

  ∆0 +(n -1)(∆1 +(n -2)/2(∆2 + ...))

We have (∆0, ∆1, ∆2) = (20, -2, -2), so the expression for the n-th term is ...

  tn = 20 +(n -1)(-2 +(n -2)/2(-2))

(a) tn = -n² +n +20

Listing the first 12 terms, we find the 12th term is ...

(b) t12 = -112

Locating the term -36 in the list, we find ...

(c) -36 is term 8

For a quadratic sequence the third differences are zero. The second attachment shows us the third differences for this sequence are ...

  -3(y +11) = 0   ⇒   y = -11

  y -x +21 = 0   ⇒   x = 10

That attachment also shows us the second differences are x-8 (or y+13):

  x -8 = 10 -8 = 2

(a) The second difference is 2.

(b) The first term is 10.

__

Additional comment

The ability of this free calculator app to perform arithmetic symbolically and to find differences of successive list elements is very helpful for solving questions related to sequences. The linear and quadratic regression capabilities can also be useful for some questions. It pays to know your tools.

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The coordinates of the Triangle RST after it is translated 4 units down and 3 units to the left is the option;

R¢(-7, -2), S¢(2, -1), T¢(-1, -9)

A translation transformation is one in which the location of the pre-image is transformed from its initial location to the destination location.

The coordinates of the vertices of the triangle RST are; R(-4, 2), S(5, 3), T(2, -5)

The translation transformation that is applied on the triangle RST includes;

A vertical translation 4 units downwards

An horizontal translation 3 units to the left

The translation transformation is therefore; T₍₋₃, ₋₄₎

Therefore, applying each transformation, we get;

R(-4, 2) ⇒ T₍₋₃, ₋₄₎ ⇒ R¢(-4 -3, 2 - 4) = R¢(-7, -2)

S(5, 3) ⇒ T₍₋₃, ₋₄₎ ⇒S¢(5 - 3, 3 - 4) = S¢(2, -1)

T(2, -5) ⇒ T₍₋₃, ₋₄₎ ⇒ T¢(2 - 3, -5 - 4) = T¢(-1, -9)

The correct option is therefore, the first option

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ABC forms a right-angled triangle

Let the distance between AB = x

Using Pythagoras theorem

Sin 30 = Opposite / Hypoteneuse

Sin 30 = 12 in/ x

However, From the equilateral triangle, all sides are = 2 and the angles are all = 60 degrees. Then we take a right-angled triangle out of it. This means that the base equals 1, and the top side is halved and equals 30 degrees.

Using Pythagoras' theorem, Sin 30 = 1/ 2

Substituting back into the equation, we get:

1/ 2 = 12 in / x

Cross multiplying, we get:

x = 24 in

Recall x = AB

AB = 24 inches

To remember the angles and their respective formula, use SOHCAHTOA

Where sin x = Opposite / Hypoteneuse

Cos x = Adjacent / Hypoteneuse

Tan x = Opposite / Adjacent

Given the relationship, 9r + 8t = 9, the relationship as a function r = f(t) will be (9 - 8t) / 9.

The relationship 9r + 8t = 9 as a function r = f(t), one is needed to isolate the variable r first on one side of the equation.

Taking with the original equation:

9r + 8t = 9

Subtract 8t from both sides in order to isolate the term with r:

9r = 9 - 8t

Divide both sides by 9 in order to solve for r:

r = (9 - 8t) / 9

Therefore, the relationship 9r + 8t = 9 can be written as the function:

r = (9 - 8t) / 9.

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Your question seems incomplete, the probable complete question is:

Consider the relationship 9r + 8t = 9. a. Write the relationship as a function r = f(t).

The value of X is 7.

Given that the set of number are in pattern,
X, 5, 9, 27, 15, 21.

To find the value of X such that given numbers are in pattern.

The next three number are three times of first three numbers respectively.

Let a, b, c is the first three numbers then 3a, 3b, 3c are next three numbers.

Consider the given set of numbers,  X, 5, 9, 9 × 3, 5 × 3, 21.

By using the data and the pattern set gives,

3X = 21.

Divide by 7 on both sides,

X = 7.

Hence, the value of X is 3

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Answer:

Height = 64 feet

Step-by-step explanation:

(***It's very important to model these kinds of problems.  I attached a picture modeling the problem so show how modeling makes it easier to solve the problem)

We know that the building is vertical, and the ground is horizontal.  Thus, the ground and the building make a right angle.  

Therefore, the top of the building, the base of the building, and the surveyor form a right triangle, where

Because we're working with a right triangle, we can now find h, the heigh of the building using one of the trigonometric ratios.

We know that the angle of elevation is made by the ground and the distance between the surveyor and top of the building.

When the 65° (angle of elevation) is our reference angle, the height of the building is the opposite side and the distance between the base of the building and the surveyor is the adjacent side.

Thus, we can use the tangent ratio, which is:  tan (θ) = opposite / adjacent:

We can now plug in 65 for θ and 30 for the adjacent side.  This will allow us to find the opposite side, which is the heigh of the building:

tan (65) = h / 30

30 * tan(65) = h

64.33520762

64 = h

Thus, the height of the building to the nearest foot is 64 feet.

Answer:

B.  90 tickets

Step-by-step explanation:

For t tickets sold, we want the revenue equal to the cost.

25t = 450 +20t

5t = 450 . . . subtract 20t

t = 90 . . . . . divide by 5

90 tickets is the break-even point.

___

:D

a. The variables are x and y which represents 2 shots and 3 shots respectively.

b. The system of equations are;

x + y = 11

2x + 3y = 29

a. Let's define the variables to write the system of equations:

Let x be the number of two-point shots Noah made.

Let y be the number of three-point shots Noah made.

b. We can write a system of equations that model this problem.

Equation 1: x + y = 11

This equation represents the total number of shots Noah made, which is 11.

Equation 2: 2x + 3y = 29

This equation represents the total number of points Noah scored, which is 29. Since each two-point shot is worth 2 points and each three-point shot is worth 3 points, we can multiply the number of two-point shots (x) by 2 and the number of three-point shots (y) by 3 and sum them up to get the total score of 29.

The system of equations that can be used to determine the number of two-point shots Noah made and the number of three-point shots he made is:

x + y = 11

2x + 3y = 29

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The formula for sin(a - b) is sin(a)cos(b) - cos(a)sin(b).

To find the formula for sin(a - b), we can use the identity for cosine of a sum of angles, which states that:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

We can rearrange this equation to solve for sin(a - b):

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

cos(a + b) = cos(a)cos(b) + (-1)(sin(a)sin(b)) [multiplying sin(b) by -1]

cos(a + b) = cos(a)cos(b) + sin(a)(-sin(b)) [rearranging terms]

cos(a + b) = cos(a)cos(b) - sin(a)sin(b) [sin(b) can be replaced by -sin(b)]

Comparing this equation with the given identity, we can see that:

sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Therefore, the formula for sin(a - b) is sin(a)cos(b) - cos(a)sin(b).

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Answer: The figure you are describing seems to be a rectangle with a length of 6 yards and a width of 4 yards. The area of a rectangle is calculated by multiplying its length by its width. So, the area of this rectangle would be 6 yd * 4 yd = 24 square yards.

Answer: Total Area = 36yd

Step-by-step explanation:

You have to find the area of both square and triangle.

Area Triangle = .5x5x8= 20yd
Area Square = 4x4 = 16yd

Triangle + Square - 20yd+16yd = 36yd

area of A = πr² = π7² = 153.9 m²

area of B = πr² = π9² = 254.5 m²

The statement "Since the y-values decrease as the x-values increase, the scatter plot shows a negative association" correctly explains the association of the scatter plot.

From the plot, we see that:

A scatter pIot is a type of graph that shows the reIationship between two variables. The variabIes are plotted on the x-axis and y-axis, and each data point is represented by a dot. The pattern of the dots can be used to infer the reIationship between the variables.

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The values of m and c are m = 1 and c = 5.

The function f(x) is given by: f(x) = x + 5

To find the values of m and c in the function f(x) = mx + c, we can use the given information that f(0) = 5 and f(1) = 6.

Let's substitute x = 0 into the function:

f(0) = m(0) + c = 0 + c = c

We know that f(0) = 5, so c = 5.

Now let's substitute x = 1 into the function:

f(1) = m(1) + c = m + c = m + 5 = 6

Subtracting 5 from both sides, we have:

m + 5 - 5 = 6 - 5

m = 1

Therefore, the values of m and c are m = 1 and c = 5.

The function f(x) is given by:

f(x) = mx + c

f(x) = 1x + 5

f(x) = x + 5

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Answer:

D. Alternate Interior Angles Converse

Edplanation:

13 ~= 12

13 altenated angle to nr.10

13 is parpendicular to nr.15 and 15 is alternate to 12

Answer:

∠ 1 and ∠ 3 = 180°

Step-by-step explanation:

∠ 1 and ∠ 3 are same- side interior angles and sum to 180° , that is

∠ 1 + ∠ 3 = 180

The Mayan number || :||- € is equivalent to the base-10 number 1021.

To convert a Mayan number to base-10, use the following steps:

In this case, the values of the digits are as follows:

Ones: 1

Twentys: 0

Four-hundreds: 2

Eight-thousands: 1

The place values are as follows:

Ones: 1

Twentys: 20

Four-hundreds: 400

Eight-thousands: 8000

The final value is calculated as follows:

1 × 1 + 0 × 20 + 2 × 400 + 1 × 8000 = 1021

Therefore, the Mayan number || :||- € is equivalent to the base-10 number 1021.

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Answer:

10

Step-by-step explanation:

[tex]2\frac{3}{8}[/tex]+[tex]1\frac{3}{4}[/tex]+[tex]5\frac{7}{8}[/tex]=

[tex]2\frac{3}{8} +1\frac{6}{8} +5\frac{7}{8}[/tex]=

[tex]8\frac{16}{8} =10[/tex]

Answer:

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

Given function:

We know that absolute value is non-negative, hence |x + 3| has a minimum value of 0 when x = - 3. Then the function has a minimum value of 2.

The range is the set of output values. As we see given function has a minimum value of 2 and no maximum value.

It can be shown as:

The matching choice is C.

Answer:

A. 1.78cm².

B. 331.34 square meters.

Step-by-step explanation:

The area of the shaded region in a circle if the radius and central angle is given can be calculated using the following formula:

Area of shaded region = (θ/360) * πr²

Where:

A.

If the radius is 2 meters and the central angle is 51 degrees, then the area of the shaded region is:

Area of shaded region = (51/360)*π*2² = 0.357π m²

≈ 1.78 square meters

Therefore, the area of the shaded region is approximately 1.78square meters.

Therefore, the area of the shaded region is 1.78cm².

B.

If the radius is 12.5 meters and the central angle is 243 degrees, then the area of the shaded region is:

Area of shaded region = (243/360)*π*12.5² = 105.47π m²

≈ 105.47π square meters

Therefore, the area of the shaded region is approximately 331.34 square meters.

Answer: Area = = 38.5 - Perimeter - 29.6819.

Step-by-step explanation:

To find the perimeter and area of a triangle with vertices L(5,5), M(-2,-4), and N(5,-6), we can use the distance formula and the formula for the area of a triangle.

Perimeter:

The perimeter of a triangle is the sum of the lengths of its sides. We can find the lengths of each side using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Using this formula, we can calculate the lengths of LM, MN, and NL:

LM = √((-2 - 5)^2 + (-4 - 5)^2)

= √((-7)^2 + (-9)^2)

= √(49 + 81)

= √130

MN = √((5 - (-2))^2 + (-6 - (-4))^2)

= √((7)^2 + (-2)^2)

= √(49 + 4)

= √53

NL = √((5 - 5)^2 + (-6 - 5)^2)

= √((0)^2 + (-11)^2)

= √(0 + 121)

= √121

= 11

The perimeter of the triangle is the sum of these three sides:

Perimeter = LM + MN + NL

= √130 + √53 + 11 = 29.6819

Area:

To calculate the area of the triangle, we can use the formula for the area of a triangle using the coordinates of its vertices. The formula is given by:

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Using the coordinates of L(5,5), M(-2,-4), and N(5,-6), we can calculate the area:

Area = 0.5 * |5(-4 - (-6)) + (-2)(-6 - 5) + 5(5 - (-4))|

= 0.5 * |5(2) + (-2)(-11) + 5(9)|

= 0.5 * |10 + 22 + 45|

= 0.5 * |77|

= 38.5

Therefore, the perimeter of the triangle is given by √130 + √53 + 11 = 29.6819 units, and the area of the triangle is 38.5 square units.

The given histogram is bimodal, which is the last option/ option E, as bimodal distribution means that the data set has two distinct peaks or modes. In a bimodal histogram, the bars are grouped in a way that shows two prominent peaks or modes in the data.

Each bar in the histogram represents a specific range of values, and the height of the bar indicates the frequency or count of data points falling within that range. In a bimodal histogram, there will be two relatively high bars that correspond to the two modes of the data. The bimodal histogram is useful for visualizing and analyzing data sets that exhibit two distinct groups or clusters. It provides a clear visual representation of the two modes and their frequencies, which can help in identifying patterns and understanding the underlying characteristics of the data.

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The values of given x and y is,

x  = 6 and  y = 13/2

Since we know that,

Basic proportionality theorem,

The basic proportionality theory was developed by Thales, a great Greek mathematician; hence, it is also known as the Thales theorem. According to the great mathematician, the ratio of any two matching sides of any two equiangular triangles is always the same. The basic proportionality theorem (BPT) was suggested based on this premise.

Since we know that,

In the figure we that,

⇒ 2x + 1 = x +7

⇒        x  = 6

Similarly

⇒ 3y - 8 = y +5

⇒ y = 13/2

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