Using the figure below, select the two pairs of alternate exterior angles.



1 and 4
2 and 3
6 and 7
5 and 8

Answer:

2

Step-by-step explanation:

[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{x+2}[/tex]

[tex]\frac{3^6}{4^6} \cdot \frac{16^5}{9^5}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]\frac{3^6}{4^6} \cdot \frac{(4^2)^5}{(3^2)^5}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]\frac{3^6}{4^6} \cdot \frac{4^{10}}{3^{10}}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]\frac{3^6}{3^{10}} \cdot \frac{4^{10}}{4^6}=\frac{4^{x+2}}{3^{x+2}}[/tex]

[tex]3^{-4} \cdot 4^{4}=4^{x+2}3^{-(x+2)}[/tex]

This implies

x+2=4

and

-(x+2)=-4.

x+2=4 implies x=2 since subtract 2 on both sides gives us x=2.

Solving -(x+2)=-4 should give us the same value.

Multiply both sides by -1:

x+2=4

It is the same equation as the other.

You will get x=2 either way.

Let's check:

[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{2+2}[/tex]

[tex](\frac{3}{4})^6 \times (\frac{16}{9})^5=(\frac{4}{3})^{4}[/tex]

Put both sides into your calculator and see if you get the same thing on both sides:

Left hand side gives 256/81.

Right hand side gives 256/81.

Both side are indeed the same for x=2.

Answer:

Miles he drove on highway = 180 miles

Miles he drove in the city = 217 miles

Step-by-step explanation:

Lets assume that gallon used on highway = x

Miles driven on highway = y

(I) 36 miles per gallon on the highway.

36x = y    equation 1

(II) 31 miles per gallon in the city.He recently drove 397 miles on 12 gallons of gasoline

31(12-x)= 397-y    equation 2

372-31x= 397-y

Combine the constants:

-31x= 397-372-y

y-31x = 25

Now substitute the value of equation 1 in equation 2

y-31x=25

36x-31x=25

5x=25

Now divide both sides by 5

5x/5=25/5

x=5

Now substitute the value x=5 in equation 1

36x=y

36(5)=y

180= y

Now subtract the miles driven on highway from recently drove miles to get the miles driven in the city.

397-y = 389 - 180

= 217

Therefore,

Miles he drove on highway = 180 miles

Miles he drove in the city = 217 miles ....

Julio drove 0 miles on the highway and 2856 miles in the city.

To find out how many miles Julio drove on the highway and in the city, we can set up a system of equations. Let x represent the number of miles driven on the highway and y represent the number of miles driven in the city. We know that the total distance driven is 397 miles and the total amount of gas used is 12 gallons. Using the given miles per gallon ratings, we can set up the following equations:

x + y = 397

36x + 31y = 12

From the first equation, we can isolate x:

x = 397 - y

Substituting this value of x into the second equation, we can solve for y:

36(397 - y) + 31y = 12

14292 - 36y + 31y = 12

-5y = -14280

y = 2856

Now we can substitute the value of y back into the first equation to find x:

x + 2856 = 397

x = 397 - 2856

x = -2459

However, since we cannot have negative miles, we know that Julio did not drive any miles on the highway.

Therefore, Julio drove 0 miles on the highway and 2856 miles in the city.

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Answer:

31877 people with live in the town in 2016 if the population increases at a rate of 6% every 2 years

Step-by-step explanation:

The formula used will be

A(t) = P(1+r)^t

A(t) = Future value

P = population

r = rate

t = time

P= 20,000

r =6% or 0.06

t = 16

Since population is increased every 2 years, so t = 16/2 = 8

Putting value:

A(16) = 20,000(1+0.06)^8

A(16)= 20,000(1.06)^8

A(16) = 31876.9 ≈ 31877

So, 31877 people with live in the town in 2016 if the population increases at a rate of 6% every 2 years

Answer:

[tex]\frac{x-1}{x^{2} +2x-3}[/tex]

[tex]\frac{1}{x+3}[/tex]

Step-by-step explanation:

first, subtract the two fractions

You will have

[tex]\frac{x-1}{x^{2} +2x-3}[/tex] the un-simplified difference of those expressions

Lets try to simplify that

Lets find the roots of the polynomial from the bottom using the formula for quadratic equation, (-b +/- sqrt( b2-4*a*c)) /(2*a)

Which are x=-3, and x=1

Thus, you have (x+3)*(x-1)

So

[tex]\frac{x-1}{x^{2} +2x-3} = \frac{x-1}{(x+3) (x-1)}= \frac{1}{x+3}[/tex]

Answer:

The correct answer is third option.  22 units

Step-by-step explanation:

Points to remember

If a right angled triangle with angles are 45°, 45° and 90° then the sides are in the ratio 1 : 1 : √2

To find the length of one leg of the triangle

From the figure we can see a right angled triangle with hypotenuse = 22√2 units

The other two legs are equal. Therefore the right angled triangle with angles 45°, 45° and 90°

Therefore the given triangle sides are in the ratio,

1 : 1 : √2 = x : x : 22√2

Therefore x = 22 units

The correct answer is third option.  22 units

Answer:

y = (x - [tex]\frac{1}{2}[/tex] )² - [tex]\frac{25}{4}[/tex]

Step-by-step explanation:

Given

y = (x + 2)(x - 3) ← expand factors

  = x² - x - 6

Use the method of completing the square

add/ subtract ( half the coefficient of the x- term )² to x² - x

y = x² + 2(- [tex]\frac{1}{2}[/tex]) x + [tex]\frac{1}{4}[/tex] - [tex]\frac{1}{4}[/tex] - 6

  = (x - [tex]\frac{1}{2}[/tex] )² - [tex]\frac{25}{4}[/tex]

Answer:

B

Step-by-step explanation:

The entry A is the result of multiplying - x and 3x, that is

- x × 3x = - 3x² → B

Answer:

-3x^2

Step-by-step explanation:

A  3x * -x = -3x^2  so A = -3x^2

We can also find the value of B

B  -x *5 = -5x

and C

C = 3x*2 = 6x

Answer:

100 degrees (first choice)

Step-by-step explanation:

So we can actually find the degree measure of arc KI because there is a theorem that says this twice the angle IJK.

75(2)=150 is the degree measure of arc KI.

So a full rotation around a circle is 360 degrees.

This means the following equation should hold:

m(arc)KJ+m(arc)IJ+m(arc)IK=360

Inserting the values given for the arc measures we know:

110         +m(arc)IJ+    150    =360

Add 110 and 150:

260     +m(arc)IJ                   =360

Subtract 260 on both sides:

             m(arc)IJ                   =100

So m(arc)IJ is 100 degrees.

Answer:

The area in factored form is [tex]A=(x+2)(x-5)[/tex].

The area in standard form is [tex]A=x^2-3x-10[/tex].

Step-by-step explanation:

The area of a rectangle is length times width.

So the area here is (x+2)(x-5).

They are probably not looking for A=(x+2)(x-5) because it requires too little work.

They probably want A in standard form instead of factored form.

Let's use foil:

First x(x)=x^2

Outer: x(-5)=-5x

Inner: 2(x)=2x

Last: 2(-5)=-10

---------------------Adding together:  [tex]x^2-3x-10[/tex].

The area in factored form is [tex]A=(x+2)(x-5)[/tex].

The area in standard form is [tex]A=x^2-3x-10[/tex].

Final answer:

The equation describing the area of the rectangle in terms of x, given the sides x + 2 and x - 5, is A = x² - 3x - 10.

Explanation:

To find the equation that describes the area, A, of a rectangle in terms of x, we use the formula for the area of a rectangle, which is length times width. Given that the length is x + 2 and the width is x - 5, the equation for the area A in terms of x is A = (x + 2)(x - 5).

To express A as a polynomial, we can expand this equation:

A = x(x - 5) + 2(x - 5)

A = x² - 5x + 2x - 10

A = x² - 3x - 10

Therefore, the equation that describes the area of the rectangle in terms of x is A = x² - 3x - 10.

Answer:

x = 40 deg

Step-by-step explanation:

Given that the line at the base of the triangle is a continuous straight line,

x + (4x-20) = 180 degrees

x + 4x  - 20 = 180

5x = 180 + 20

5x = 200

x = 40 deg

Answer:

f-¹(x) =(1-5x) /3x.

Step-by-step explanation:

f(x)=1/(3x+5)

Let y=1/(3x+5)

Exchanging x and y,

x=1/(3y+5)

3y+1=1/x

3y=1/x-5

3y=(1-5x) /x

y=(1-5x)/3x

f-¹(x) =(1-5x) /3x.

Answer:

y=3(x-5)

Step-by-step explanation:

Answer:

75 cubic centimeters

Step-by-step explanation:

Volume of a cone is the area of the base times a third of the height.

The base is a circle.

The formula for the area of a circle is [tex]\pi \cdot r^2[/tex].

We are given that we want to use [tex]3.14[/tex] for [tex]\pi[/tex] and

r=(diameter)/2=6/2=3 cm.

So the area of the base is [tex]3.14 \cdot 3^2=28.26[/tex].

Now the height of the cone is 8 cm.

A third of the height is 8/3 cm.

So we want to compute area of base times a third of the height.

Let's do that:

[tex]\frac{8}{3} \cdot 28.26[/tex]

75.36 cubic centimeters

To the nearest cubic centimeters this is 75

To calculate the volume of a cone, we use the formula for the volume of a cone, which is:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( V \) is the volume of the cone
- \( \pi \) is the constant Pi (approximated as 3.14)
- \( r \) is the radius of the cone's base
- \( h \) is the height of the cone
Given that the diameter of the cone is 6 cm, we find the radius by dividing the diameter by 2:
\[ r = \frac{diameter}{2} = \frac{6 cm}{2} = 3 cm \]
Now we have the radius and the height (which is given as 8 cm), we can substitute these values into the formula:
\[ V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times (3 cm)^2 \times 8 cm \]
First, square the radius:
\[ (3 cm)^2 = 9 cm^2 \]
Now, perform the multiplication:
\[ V = \frac{1}{3} \times 3.14 \times 9 cm^2 \times 8 cm \]
\[ V = 3.14 \times 3 cm^2 \times 8 cm \]
\[ V = 9.42 cm^2 \times 8 cm \]
\[ V = 75.36 cm^3 \]
Finally, you want to round the volume to the nearest cubic centimeter. Since \( 75.36 \) is already a decimal to the nearest hundredth and has no fractional part in cubic centimeters, we simply round it to the nearest whole number:
\[ V \approx 75 \]
So, the volume of the cone is approximately \( 75 \) cubic centimeters.

Answer:

-3/4 degree Fahrenheit

Step-by-step explanation:

The temperature is decreasing by 3/16 each day

We write that as -3/16

There are 4 days in which we are measuring

Total temperature change = rate of change* number of days

                                            =-3/16 * 4

                                            = -12/16

We can simplify this by dividing the top and bottom by 4

-12/4 =3

16/4 =4

                                         =12/16 = -3/4

Answer:

-3/4 degree Fahrenheit

Step-by-step explanation:

If the tempature is changing every day by -3/16 degree fahrenhiet, the change in the tempature after 4 days is -3/4 degree Fahrenheit.

Answer:

22.5°

Step-by-step explanation:

We are given that a bike has 16 spokes which are spaced evenly apart.

We are to find the measure of angles of rotation.

Spoke is basically the rod which comes from the center of the tire, connecting it to the round surface.

Since the total measure of angles for a circle is 360° and 16 spokes are places evenly, so each angle of rotation = [tex]\frac{360}{16}[/tex] = 22.5°

Answer:

correct answer is option D

Step-by-step explanation:

assume the speed of the unicorn for the first time to cover 50 mile be 'x'

we know,

distance = speed × time

    50  =  x  × time

     t₁ = 50 / x...............................(1)

when unicorn travel 300 mile with speed of '3x'

distance = speed × time

300  = 3 x  × time

    t₂ = 100/ x...............................(2)

dividing equation (2)/(1)

[tex]\dfrac{t_2}{t_1}  =  \dfrac{100/x}{50/x}[/tex]

    t₂ = 2 × t₁

hence, the time will be twice the first one.

correct answer is option D

Answer:

[tex]-\frac{3080}{81}[/tex]

Step-by-step explanation:

The given function is:

[tex]p(x)=3x^{5}+2x^{2}-5[/tex]

We have to find the value of the function at x = -5/3

In order to do this we need to replace every occurrence of x in the given function by -5/3. i.e.

[tex]p(-\frac{5}{3})=3(-\frac{5}{3})^{5}+2(-\frac{5}{3} )^{2}-5\\\\ p(-\frac{5}{3})=3(-\frac{3125}{243} )+2(\frac{25}{9} )-5\\\\p(-\frac{5}{3})=-\frac{3125}{81}+\frac{50}{9}-5\\\\ p(-\frac{5}{3})=-\frac{3080}{81}[/tex]

Thus, the value of the function at x =-5/3 is [tex]-\frac{3080}{81}[/tex]

Answer:

0.08

Step-by-step explanation:

A standard deck of 52 cards has four aces.

The sample space is:

n(S) = 52

Let A be the event that an ace is drawn from the deck.

Then,

n(A) = 4

So,

[tex]P(A) = \frac{n(A)}{n(S)}\\ =\frac{4}{52} \\=0.0769[/tex]

Rounding off to the nearest hundredth will give us:

0.08 ..

The theoretical probability of drawing an ace from a standard deck of 52 cards is 0.08. The calculation is based on dividing the number of aces (4) by the total number of cards (52).

The theoretical probability of an event occurring is calculated by dividing the number of ways an event can occur by the total number of possible outcomes. In this scenario, the question is asking about the theoretical probability of drawing an ace from a standard deck of 52 cards.

In a standard deck of cards, there are 4 aces. So, the number of ways the event 'drawing an ace' can occur is 4. The total number of possible outcomes, which is the total number of cards in the deck, is 52. Therefore, the probability of drawing an ace, P(Ace), is calculated as follows:

P(ace) = Number of aces in the deck / Total number of cards

P(ace) = 4 / 52

When you simplify this fraction or convert it into decimal form (rounded to the nearest hundredths), it gives:

P(ace) = 0.08

So, the theoretical probability of drawing an ace from a deck of 52 cards is 0.08.

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Answer:

Step-by-step explanation:

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

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

m - slope

b - y-intercept

The formula of a slope:

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

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

We have the points (-6, 4) and (2, 0).

Substitute:

[tex]m=\dfrac{0-4}{2-(-6)}=\dfrac{-4}{8}=-\dfrac{1}{2}[/tex]

Put the value of the slope and coordinates of the point (2, 0) to the equation of a line:

[tex]0=-\dfrac{1}{2}(2)+b[/tex]

[tex]0=-1+b[/tex]          add 1 to both sides

[tex]1=b\to b=1[/tex]

The equation of a line in the slope-intercept form:

[tex]y=-\dfrac{1}{2}x+1[/tex]

Convert to the standard form [tex]Ax+By=C[/tex]

[tex]y=-\dfrac{1}{2}x+1[/tex]           multiply both sides by 2

[tex]2y=-x+2[/tex]             add x to both sides

[tex]x+2y=2[/tex]

Answer:

Option 1.

Step-by-step explanation:

It is given that the line passes through the points (-6,4) and (2,0).

If a line passes through two points, then the equation of line is

[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]

Using the above formula the equation of line is

[tex]y-(4)=\dfrac{0-4}{2-(-6)}(x-(-6))[/tex]

[tex]y-4=\dfrac{-4}{8}(x+6)[/tex]

[tex]y-4=\dfrac{-1}{2}(x+6)[/tex]

Muliply both sides by 2.

[tex]2y-8=-x-6[/tex]

[tex]x+2y=-6+8[/tex]

[tex]x+2y=2[/tex]

Therefore, the correct option is 1.

Answer:

A

Step-by-step explanation:

Use the property

[tex]\sqrt[n]{a^m}=a^{\frac{m}{n}}[/tex]

The numerator can be simplified as

[tex]\sqrt[7]{x^2}=x^{\frac{2}{7}}[/tex]

The denominator can be simplified as

[tex]\sqrt[5]{y^3}=y^{\frac{3}{5}}[/tex]

Also remindr property

[tex]\dfrac{1}{a^n}=a^{-n}[/tex]

Thus, the expression is equivalent to

[tex]\dfrac{\sqrt[7]{x^2} }{\sqrt[5]{y^3} }=\dfrac{x^{\frac{2}{7}}}{y^{\frac{3}{5}}}=\left(x^{\frac{2}{7}}\right)\cdot \left(y^{-\frac{3}{5}}\right)[/tex]

Let P = Phil's age

P + 9 = 18

The correct equation representation for 'Phil's age increased by 9 years is 18 years' would be 'p + 9 = 18', which indicates that when you add 9 years to Phil's age, it equals 18. Hence, option D is the correct answer.

The student is seeking to translate a sentence into a mathematical equation. The sentence states: 'Phil's age increased by 9 years is 18 years.' Considering 'Phil's age' as 'p', the correct translation of the sentence into a mathematical equation would be 'p + 9 = 18' since this indicates that when we add 9 years to Phil's age, we get 18 years. Therefore, the correct answer among the provided choices is option D: p + 9 = 18.

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The complete question is here:

Write as an equation: Phil's age increased by 9 years is 18 years.

A. p+18=9

B. p-9=18

C. p=18+9

D. p+9=18

Answer:

k = 6

Step-by-step explanation:

i got it right on the test and i also graphed it

Please mark brainliest!

by the way when you need to graph something use a website called Desmos.  or just go onto the search bar on the top of your screen adn type in desmos. com

The transformation of a function may involve any change. The correct option is D.

The transformation of a function may involve any change.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Given that f(x) = 0.6x is replaced by the graph of g(x) = 0.6x + k. And g(x) is obtained by shifting f(x) up by 6 units. Therefore, the value of k is 6.

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Answer:

24

Step-by-step explanation:

The area of the given figure ABCD with respective coordinates is

24 square units.

The area of a quadrilateral is nothing but the region enclosed by the sides of the quadrilateral.

Given the coordinates of the quadrilateral as; A(−4, 1), B(2, 1), C(2, −5), and D(−4, −3).

By inspection, we see that the y coordinates of A and B are the same. Thus, their length will be the difference of their x coordinates. Thus;

AB = 2 - (-4)

AB = 6

Similarly, B and C have same x coordinates. Thus;

BC = -5 - 1 = -6

A and D have same x coordinate and as such;

AD = -3 - 1 = -4

AB and BC are perpendicular to each other because of opposite signs of same Number and since AD has a different length, then we can say that the figure ABCD is a rectangle.

Thus;

Area of figure = 6 × 4 = 24 square units.

The area of the given figure ABCD with respective coordinates is

24 square units.

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Answer:

The input value is x=1

Step-by-step explanation:

we know that

The intersection point both graphs is the point that  produces the same output value for the two functions

Observing the graph

The intersection point is (1,1)

therefore

For x=1 (input value)

The output value for the two functions is equal to 1

Answer:

ITS X=1

Step-by-step explanation:

Answer:

No. It is a constant function.

Step-by-step explanation:

The function f(x) = e^2 is not an exponential functional. Rather, it is a constant function. The reason for this is that in f(x) = e^2, there is no x involved on the right hand side of the equation. The approximate value of e is 2.718281, and the approximate value of 2.718281^2 is 7.389051. This means that f(x) = e^2 = 7.389051. It is important to note that for any value of x, the value of the function remains fixed. This is because the function does not involve the variable x in it. The graph of the function will be a line parallel to the x-axis, and the y-intercept will be 7.389051. For all the lines parallel to x-axis, the value of the function remains the same irrespective of the value of x. Also, the derivative of the function with respect to x is 0, which means that the value of the function is unaffected by the change in the value of x!!!

Answer:

The range is all real number y<3.

Step-by-step explanation:

[tex](u \circ v)(x)=u(v(x))[/tex]

So we have to have v(x) exist for input x.

Let's think about that.  v(x)=1/x so the domain is all real numbers except 0 since you cannot divide by 0.  v(x)=1/x will also never output 0 because the numerator of 1/x is never 0. So the range of v(x)=1/x is also all real numbers except y=0.

Now let's plug v into u:

[tex]u(x)=-2x^2+3[/tex]

[tex]u(v(x))=u(\frac{1}{x}[/tex]

[tex]u(v(x))=-2(\frac{1}{x})^2+3[/tex]

The domain of will still have the restrictions of v; let's see if we see any others here.

Nope, there are no, others, the only thing that is bothering this function is still the division by x (which means we can't plug in 0).

[tex]u(v(x))=\frac{-2}{x^2}+3[/tex]

Let's thing about what are y's value will not ever get to be.

Let's start with that fraction. -2/x^2 will never be 0 because -2 will never be 0.

So we will never have y=0+3 which means y will never be 3.

There is one more thing to notice -2/x^2 will never be positive because x^2 is always positive and as we know a negative divided by a positive is negative.

So we have (a always negative number) +  3 this means the range will only go as high as 3 without including 3.

The range is all real number y<3.

Answer:

yes it is a solution

Step-by-step explanation:

o>-4

0<1

yes both work out to be equal

Answer:

It is a solution

Step-by-step explanation:

Check and see if (0,0) satisfies both inequalities.

The first inequality is:

[tex]y \geqslant {x}^{2} + x - 4 [/tex]

When we put x=0, and y=0, we get:

[tex]0\geqslant {0}^{2} + 0 - 4[/tex]

[tex] \implies0\geqslant- 4[/tex]

This part is true.

The second inequality is:

[tex]y \leqslant {x}^{2} + 2 x + 1[/tex]

We put x=0 and y=0 to get:

[tex]0 \leqslant {0}^{2} + 2 (0) + 1[/tex]

[tex]0 \leqslant 1[/tex]

This part is also true.

Since the (0,0) not satisfy both, inequalities, it is a solution.

Answer:

Clayton could use the relationship (x,y)→ (y,x) to find the points of the image

C’ will remain in the same location as C because it is on the line of reflection

The image and the pre-image will be congruent triangles

The image and pre-image will not have the same orientation because reflections flip figures

Step-by-step explanation:

Verify each statement

case A) Clayton could use the relationship (x,y)→ (y,x) to find the points of the image

The statement is true

we know that

The rule of the reflection of a point across the line y=x is equal to

(x,y)→ (y,x)

case B) Clayton could negate both the x and y values in the points to find the points of the image

The statement is false

case C) C’ will remain in the same location as C because it is on the line of reflection

The statement is true

If a point is on the line of reflection (y=x), then the point remain in the same location because the x and y coordinates are equal

case D) C’ will move because all points move in a reflection

The statement is false

Because C' is on the line of reflection (y=x), then the point remain in the same location

case E) The image and the pre-image will be congruent triangles

The statement is true

Because the reflection not change the length sides of the triangle or the measure of its internal angles. Reflection changes only the orientation of the figure

case F) The image and pre-image will not have the same orientation because reflections flip figures

The statement is true

Because in a reflection across the line y=x, the x coordinate of the pre-image becomes the y-coordinate of the image and the y-coordinate of the pre-image becomes the x-coordinate of the image

Answer:

f(x) = (x-1)² + 3

Step-by-step explanation:

f(x)  = 4+x²-2x   or f(x)  = x²- 2x + 4   -----> eq 1

consider the top-most choice

f(x) = (x-1)² + 3 = x² + 2·x·(-1) + 1²  +  3

f(x) = x² - 2x+ 1  +  3

f(x) = x² - 2x+ 4  -----> compare this with eq 1 above (they match!)

hence f(x) = (x-1)² + 3 is the answer

Answer:

ASA postulate

Step-by-step explanation:

The sum of the measures of interior angles of triangle is always 180°, so

∠LRT=180°-∠RTL-∠RLT=180°-∠2-∠4;

∠KST=180°-∠STK-∠SKT=180°-∠1-∠3.

Since

∠1≅∠2 and ∠3≅∠4, we have that ∠LRT ≅ ∠KST.

The ASA (Angle-Side-Angle) postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. (The included side is the side between the vertices of the two angles.)

In your case:

Triangles TKS and TLR are congruent by ASA postulate.

Answer:

4x - 3y = 5

Step-by-step explanation:

The equation of a line in standard form is

Ax + By = C ( A is a positive integer and B, C are integers )

First obtain the equation in point- slope form

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

Calculate m using the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (- 1, - 3) and (x₂, y₂ ) = (2, 1)

m = [tex]\frac{1+3}{2+1}[/tex] = [tex]\frac{4}{3}[/tex]

Using (a, b) = (2, 1), then

y - 1 = [tex]\frac{4}{3}[/tex] (x - 2) ← in point- slope form

Multiply both sides by 3

3y - 3 = 4(x - 2) ← distribute and rearrange

3y - 3 = 4x - 8 ( add 8 to both sides )

3y + 5 = 4x ( subtract 3y from both sides )

5 = 4x - 3y, so

4x - 3y = 5 ← in standard form

Sure, let's find the equation of the line that passes through the points (-1, -3) and (2, 1) step-by-step.
1. First, calculate the slope (m) of the line using the formula:
  \[
  m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-3)}{2 - (-1)}
  \]
  \[
  m = \frac{1 + 3}{2 + 1} = \frac{4}{3}
  \]
2. The slope-intercept form of a line equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  We already found the slope \( m = \frac{4}{3} \), so we just need to find \( b \).
  Using the first point (-1, -3), plug the values into the slope-intercept form:
  \[
  -3 = \frac{4}{3}(-1) + b
  \]
  Calculate \( b \):
  \[
  -3 = -\frac{4}{3} + b
  \]
  Add \( \frac{4}{3} \) to both sides:
  \[
  b = -3 + \frac{4}{3}
  \]
  \[
  b = -\frac{9}{3} + \frac{4}{3}
  \]
  \[
  b = -\frac{5}{3}
  \]
3. Now we have \( y = \frac{4}{3}x - \frac{5}{3} \) in slope-intercept form.
4. Next, we will convert this to standard form, which is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) is positive.
  Multiply both sides of the slope-intercept equation by 3, the common denominator, to eliminate fractions:
  \[
  3y = 4x - 5
  \]
5. Rewriting in standard form, we move the \( x \)-term to the left side:
  \[
  -4x + 3y = -5
  \]
6. In standard form, \( A \) should be positive. If we multiply the entire equation by -1, we will make \( A \) positive:
  \[
  4x - 3y = 5
  \]
7. This is already simplified since the greatest common divisor (GCD) of 4, -3, and 5 is 1. Thus, the coefficients are already in their simplest integer values.
The final equation for the line in standard form is:
\[ 4x - 3y = 5 \]