Parts of a Right Triangle
In a right triangle, the sides that meet to form the
right angle are called the legs. The side opposite the
right angle is called the hypotenuse. The hypotenuse is
always the longest of the three sides. It is important
that you can correctly identify the sides of a right tri-
angle, regardless of what position the triangle is in.
leg
leg
60
1. Name the legs.
Practice
AABC is a right triangle.
TRIANGLES AND THE PYTH
hypotenuse
leg
leg
hypotenuse
hypotenuse
2. Name the hypotenuse.
leg
leg
APQR is a right triangle.
3. Name the legs.
4. Name the hypotenuse.
AHLM is a right triangle.
5. Name the legs.
6. Name the hypotenuse.
Review of Squares
and Square Roots
Before you study the Pythagorean theorem, le
first review squares and square roots. Just like add
tion and subtraction are inverses, so are squares
and square roots. In other words, they "undo" e
other. To square a number, you multiply it by itsel
For example, 52 means two factors of 5, or 5 times
5, which is 25. Written algebraically, it looks
this: 52= 5 x 5 = 25. A common mistake is to s
that you multiply by 2, since 2 is the exponent (the
small raised number). But the exponent tells you
how many times to multiply the base (bot
number) by itself.
Twenty-five is a perfect square. It can be writer
as the product of two equal factors. It would be he

The image of triangle ABC under a dilation whose center is A and scale factor is 4 is as follows.

What is dilation?

Dilation involves changing the size of a shape.

Main body:

Assume point P is the center of origin, then we have the following coordinates of ABC

A= (1,0)

B= (-2,0)

C= (0,2)

The scale factor (k) is given as:

K = 4

So, the dilation rule is represented as:

(x, y) = 4 *(x,y)

Using the above dilation rule, the following coordinates represent the image of triangle ABC

A' =(4,0)

B' =(-8,0)

C' =(0,8)

See attachment for the image of the dilation.

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Answer:

Before 1913, federal government revenues came mainly from taxes on goods—tariffs on imported products and excise taxes on items like whiskey.

Step-by-step explanation:

What was the tax rate in 1913?

The act levied a tax of one percent on all incomes above the exemption with additional surtax rates imposed for progressively higher incomes. Those surtax rates started at one percent for those making $20,000 or more and topped out at a mere six percent on incomes in excess of $500,000.

The only option that is not an equivalent of the product of both polynomials is; C: 6x³ + 11x² − 13x + 2

Equations can be classified as either identities or conditioned equations. For each quantity of the variables, an identity is true. Only specific values of the variables make a conditional equation legitimate. The equals sign ("="), which separates two expressions, is a symbol for an equation.

Here,

We are given the expression;

(3x - 2)(2x² + 5x - 1)

The result of expanding the polynomial is;

6x³ + 11x² − 13x + 2

Looking at the given options as seen online, the only one that is not an equivalent of the given product is;

Option C) : 6x³ + 11x² − 13x + 2 is correct

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Answer:

9x-y=10

Step-by-step explanation:

The formula for standard form is Ax + By =C

So, y=9x-10

-1(-9x+y=-10)

9x-y= 10

a) The coordinates of the midpoints P and Q are given as follows:

b) The two segments are parallel because they have the same slope.

The midpoints between two points is given by the mean of the coordinates of these two points.

Hence the midpoint P of AB has the coordinates given as follows:

The midpoint Q of AC has the coordinates given as follows:

Parallel segments are segments that have the same slope, that is, the same result of the division of the change in y by the change of x of two points in the segment.

The slopes of PQ and BC are given as follows:

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The domain of the inverse function is {5,6}. And as a compound inequality the domain is 5<x<6.

A mathematical statement that expresses the inequality between two inequalities combinedly.

from the graph, we have two inequalities, x>5 and x<6

it can be written as 5<x<6

A function that can be obtained by reversing the primary function is termed as inverse of function. it is denoted by f⁻¹.

The graph shown is got from an inverse function whose domain is {5,6} and ranges are {-2, -4}

The graph also refers the inequalities of x>5 And x<6

compound inequalities of domain 5<x<6

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Answer:

See below

Step-by-step explanation:

3x³ + 5x² - 17x - 15

1. Distribute

(x + 3)(3x² - 4x - 5)

x(3x² - 4x - 5) + 3(3x² - 4x - 5)

2. Distribute

x(3x² - 4x - 5) + 3 (3x² - 4x - 5)

3x³ - 4x² - 5x + 3 (3x² -4x - 5)

3. Distribute

3x³ - 4x² - 5x + 3 (3x² -4x - 5)

3x³ - 4x² -5x + 9x² - 12x - 15

4. Combine like terms

3x³ - 4x² -5x + 9x² - 12x - 15

3x³ + 5x² - 5x - 12x - 15

5. Combine like terms

3x³ + 5x² - 5x - 12x - 15

3x³ + 5x² - 17x - 15

3x³ + 5x² - 17x - 15

I hope my answer helped you! If you need more information or help, comment down below and I will be sure to respond if I am online. Have a wonderful rest of your day!

Answer: 28 inches

Hope this helps!!! :)

(i) At a total of 100 Songs the Charges for Company A and Company B will be Equal.

(ii) I would rather suggest to select Company A as the Total Charges to download 150 Songs is $127.5

What is Linear Equation in One Variable?

A linear equation is a one-variable equation of a straight line. The variable's only power is 1. Linear equations in one variable have the form ax + b = 0 and are solved using simple algebraic techniques.

Solution:

Let, Songs Downloaded be x

Total Charges of Company A = 15 + 0.75*x

Total Charges of Company B = 0.9*x

(i) Total Charges by Company A = Total Charges by Company B

15 + 0.75*x = 0.9*x

15 = 0.15*x

x = 100 songs

(ii) Charges by Company A to Download 150 Songs = 15 + 0.75 * 150

= 15 + 112.5 = $127.5

Charges by Company B to Download 150 Somgs = 0.9 * 150 = $135

I would rather suggest to select Company A as the Total Charges to download 150 Songs is $127.5

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Step-by-step explanation:

A circle with a center of (-1,2) and a radius

of

[tex]2 \sqrt{2} [/tex]

The triangles can be proven congruent using the ASA congruence postulate.

Thus, we can say that triangles ABK and CBK are congruent.

Triangles are congruent, according to the Angle Side Angle Postulate (ASA), if any two angles and their included side are equivalent in the triangle. The side between two angles is considered an included side.

Additionally, you can confirm that your two triangles are congruent by employing the Side Angle Side Postulate (SAS). In this case, we choose a side and its equivalent side on two triangles rather than two angles.

Triangles are said to be congruent according to the SAS Postulate if any pair of corresponding sides and their included angle are also congruent.

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Answer:

[tex]y=\dfrac{1}{4}x+\dfrac{7}{2}[/tex]

Step-by-step explanation:

Given equation:

[tex]y=-4x-5[/tex]

If two lines are perpendicular to each other, their slopes are negative reciprocals.

The slope of the given equation is -4.

Therefore, the slope of the perpendicular line is ¹/₄.

[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Point-slope form of a linear equation}\\\\$y-y_1=m(x-x_1)$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $(x_1,y_1)$ is a point on the line.\\\end{minipage}}[/tex]

Substitute the found slope and point (-2, 3) into the point-slope formula to create the equation of the perpendicular line.

[tex]\implies y-3=\dfrac{1}{4}(x-(-2))[/tex]

[tex]\implies y-3=\dfrac{1}{4}(x+2)[/tex]

[tex]\implies y-3=\dfrac{1}{4}x+\dfrac{1}{2}[/tex]

[tex]\implies y=\dfrac{1}{4}x+\dfrac{7}{2}[/tex]

In the octagon, the measurement of the largest exterior angle is 90°.

In a two-dimensional plane, a polygon is a closed object formed of line segments rather than curves. Polygon is a word that combines the words, poly, which means numerous and gon, means sides.

The given exterior angles of the octagon are,

x°, 2x°, 3x°, 4x°, 5x°, 7x°, 8x°, and 10x°.

The sum of the exterior angle of the octagon is 360°.

Implies that,

x + 2x + 3x + 4x + 5x + 7x + 8x + 10x = 360

40x = 360

x = 9

The largest exterior angle = 10 x  = 10 × 9 = 90°.

The required largest exterior angle is 90°.

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Answer:

350

Step-by-step explanation:

Let "a" represent the number of adult tickets sold and "s" represent the number of student tickets sold. We can set up the following equation to represent the information given:

a + s = 627

s = a - 73

Substituting the second equation into the first equation gives us:

a + (a - 73) = 627

2a - 73 = 627

2a = 700

a = 350

Thus, there were 350 adult tickets sold.

[tex]x=-2 \implies y=(1/3)^{-2}=3^{2}=9\\\\x=-1 \implies y=(1/3)^{-1}=3\\\\x=0 \implies y=(1/3)^0=1\\ \\ x=1 \implies y=1/3\\\\x=2 \implies y=(1/3)^2 =1/9[/tex]

Using these points, the correct graph is shown in option C.

It takes 11.55 years for money to double if invested at 6% compound interest.

What is compound interest?

Calculating the answer exactly:  P⁰°⁰⁶ⁿ=A .

We don’t know the initial value of the principal but we do know that the accumulated value is double (twice) the principal.

P⁰°⁰⁶ⁿ = 2P

We divide both sides by  P

e⁰°⁰⁶ⁿ = 2

Using natural logarithm:

0.06 n = ln(2)

n = ln(2)/0.06

n = 11.55 years

Hence, It takes 11.55 years for money to double if invested at 6% compound interest.

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Answer:

500,000 can be written in expanded form in two different ways:

500,000 = 500,000

500,000 = 5 x 100,000 + 0 x 10,000 + 0 x 1,000 + 0 x 100 + 0 x 10 + 0 x 1

In both cases, the expanded form shows the value of each digit in the number. The first way simply lists the number itself, while the second way breaks down the number into its component parts and shows the value of each digit in relation to the place value it holds in the number.

Answer:

18.6

Step-by-step explanation:

She should divide both sides by -5 while multiplying the other side by -5. She also did not replace the > symbol with a symbol.

An inequality in mathematics is a relationship that makes a non-equal comparison between two numbers or other mathematical expressions. It is most commonly used to compare the sizes of two numbers on a number line.

From given question :

-5x-3.5>6.5

-5x > 10

X > -50

Since, we have to solve for x

so, we will isolate x on anyone side

step-1:

Add both sides by 3.5

- 5x - 3.5 + 3.5 > 6.5 + 3.5

- 5x > 10

step-2:

Divide both sides by -5

we know that whenever we divide both sides by negative sign , it's inequality gets revered

so, we get

- 5x / -5 < - 50 / -5

we get

x < 10

so,

She should divide both sides by -5 while she multiplied by -5 on other side

and she also did not change the > symbol to a < symbol

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Answer:

A system of two linear inequalities in two variables is a set of two linear inequalities that involve two variables, such as x and y. A system of two linear equations in two variables is a set of two linear equations that involve two variables, also typically x and y.

There are several key differences between a system of two linear inequalities in two variables and a system of two linear equations in two variables:

Inequalities use symbols such as "less than" or "greater than" to indicate the relationship between the variables, whereas equations use the equal sign (=).

The solution to a system of linear inequalities is a region in the coordinate plane, whereas the solution to a system of linear equations is a single point.

The solution to a system of linear inequalities is found by graphing the inequalities on the coordinate plane and finding the region where they intersect, whereas the solution to a system of linear equations is found by solving the equations for the values of the variables.

The solution to a system of linear inequalities may include an infinite number of points, whereas the solution to a system of linear equations is always a finite number of points.

Overall, the main difference between a system of two linear inequalities in two variables and a system of two linear equations in two variables is the way in which the solutions are represented and found.

The graph's representation of the inequality system is

y≤3x-2 and y≥-4x+2. So, option a is correct.

One of the basic ideas of social justice theories is the existence of unequal status, rights, and opportunities. In public conversation, it might be confusing because various people often interpret it differently. There are some frequent variants, though. In mathematics, an unjust comparison between two numbers or other mathematical expressions is referred to as an inequality. A relationship between two expressions or values that is not equal to each other is known as inequality in mathematics. So inequality emerges from an imbalance. As an illustration, suppose you wish to get a 250 new bicycle. Different types of inequality are indicated by a variety of notations, including:

A less than sign, such as a, denotes that an is greater than b.

Therefore, The  system of inequalities that is represented by the graph is

y≤3x-2 and y≥-4x+2. So, option a is correct.

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The graph that shows the straight line → x - 5y = 27 is plotted and attached.

The general equation of a straight line is -

y = mx + c

where -

[m] is slope of the line.

[c] is the y - intercept of the line.

Given is the equation of the line as -

x - 5y = 27

The given equation of the line is -

x - 5y = 27

Rearranging the equation, we get -

5y = x - 27

y = (1/5)x - (27/5)

y = 0.2x - 5.4

Now, refer to the graph attached. The slope [m] of the equation will be -

m = 0.2 and the [y] - intercept will be y = - 5.4.

Therefore, the graph that shows the linear relation → y = 0.2x - 5.4 is plotted and attached.

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Answer:

$0.48/pound

Step-by-step explanation:

$5.76 divided by 12

Answer:

The dividend in the synthetic division below is x+2:

     4 | -6 -5 4 -7

     -   --------------

     -   -1 -1 1 -5

     0 | -5 -4 3 -2

     -   --------------

     2 | x+2

In synthetic division, the numbers in the top row represent the coefficients of the polynomial being divided, and the number in the bottom row represents the value of the dividend. In this case, the dividend is x+2 because it is the only number in the bottom row that is not a result of the division operation

The mean and the standard deviation of the distributions are

The distribution is given as:

A uniform continuous model

The mean and the standard deviation of this type of model are

Mean = (a + b)/2

Standard deviation = (b−a)/√12

Using the above formulas, we have

a. U(0, 15)

Mean = (0 + 15)/2 = 7.5

Standard deviation = (15−0)/√12 = 4.3301

b. U(90, 230)

Mean = (90 + 230)/2 = 160

Standard deviation = (230 − 90)/√12 = 40.4145

c. U(4, 92)

Mean = (4 + 92)/2 = 48

Standard deviation = (92 - 4)/√12 = 25.4034

Hence, the standard deviation of U(4, 92) is  25.4034

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From the given parameters, the area of the triangle is 288m².

How to calculate the area of a triangle?

A triangle is a polygon with three angles, three vertices and three sides whose sum of angles is 180⁰

Let the vertices of the triangle be labelled ABC

/AB/=16m

/BC/=45m

<ABC=127⁰

The given information formed an obtuse angled triangle.

Area of the triangle (A) = [tex]\frac{1}{2}[/tex]ACsinB

Substituting the various values in the formula we have;

A=[tex]\frac{1}{2} *16*45*sin127^{0}[/tex]

simplifying the equation to get;

A=8*45*0.7986

A=287.5088

A=288m²

Therefore, the area of the triangle to the nearest whole number is 288m²

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The Area of the shaded part is 3.44 cm²

The formula used to find the Area of a circle is given by

Where r is the radius of the circle

The formula used to find the Area of a Square is given by

Here we know

The radius of the circle = 2m

As we know Area of a Circle = 2πr

=> Area of given circle = 2(3.14)2 = 12.56 cm²

Area of circle = 12.56 cm²

As we can see the circle is inscribed in a square

Then the Length of the side square = diameter of the circle

=> Length of side = 2 × Radius of circle = 2 × 2 cm = 4 cm

Length of side = 4 cm

As we know Area of the square = side²

=> Area of given square = 4 × 4 = 16 cm²

Now using the above value we will friend the area of the shaded part

Area of shaded part = Area of the square - Area of the circle

= 16 cm² - 12.56 cm² = 3.44 cm²

Therefore,

The Area of the shaded part is 3.44 cm²

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The equivalent equations of y - 4 = 2/3(x - 1) are y - 4 = 2/3x - 2/3 and y  = 2/3x + 10/3

From the question, we have the following parameters that can be used in our computation:

y - 4 = 2/3(x - 1)

We start by expanding the bracket

So, we have the following representation

y - 4 = 2/3x - 2/3

Next, we add 4 to both sides of the equation

This gives

y - 4 + 4 = 2/3x - 2/3 + 4

Evaluate the like terms

So, we have the following representation

y  = 2/3x + 10/3

Hence, the equivalent equation is y  = 2/3x + 10/3

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