find the exact value of cos(sin^-1(-5/13))

Answer:

the answer should be 1

Step-by-step explanation:

4[(-3+z)2-2(-3+z)]+1

4(-6+2z+6-2z)+1

-24+8z+24-8z+1

the 24's and 8's cancel out leaving 1

To find g(h(-3+z)), we first find h(-3+z) by substituting -3+z into h(a), it becomes z^2-8z+15. Then, substitute this into g(a) to get g(h(-3+z)), which results in 4z^2-32z+61.

In the field of Mathematics, to solve the problem g(h(-3+z)), we need to substitute h into g. It means that every 'a' in g(a) will be replaced by what h(-3+z) is equal to. To find h(-3+z), we substitute -3+z into h(a) in place of a. Therefore, h(-3+z) equals to (-3+z)^2-2(-3+z). After simplifying, it becomes z^2-6z+9+6-2z=z^2-8z+15. Next, we substitute this into g(a) to get g(h(-3+z))=4[z^2-8z+15]+1=4z^2-32z+61. Hence, the solution to the problem g(h(-3+z)) is 4z^2-32z+61.

https://brainly.com/question/17348508

#SPJ2

Answer:

Step-by-step explanation:

y - 3 = -4(x - 5)

Add 3 to both sides to isolate y

y = -4 ( x - 5 ) + 3

Distribute

y = -4x +20 + 3

Combine like terms

y = -4x + 23

y = mx + b is standard form for which m is slope

m = -4

Please mark for brainliest!! :D Thanks!!

If you have any questions or need more information, please comment below and I'll respond asap!!

Answer:

-4

Step-by-step explanation:

Distribute -4 into the parenthesis and get -4x + 20. The number paired with x is the slope, usually represented by m in y = mx + b. So, the slope is -4. For the full slope intercept format, move the -3 on left side to the right by adding 3. The new equation is y = -4x + 23.

Answer:

An average of 661 people per day.

Step-by-step explanation:

1. 365-7= 358

2. 236,758/ 358= 661.3351

3. Round to 661

Step-by-step explanation:

All three sides are equal, so this is an equilateral triangle.

Answer:

This is an equilateral triangle

Step-by-step explanation:

All three sides are equal, making it equilateral

Answer:

The fifth term is 620

Step-by-step explanation:

y=40×(-2)^(n-1)​

The 5th terms means n=5

y = 40 * (-2) ^ (5-1)

  = 40 * (-2) ^4

  = 40 * (16)

  = 640

The answer is [tex]\( 24xy^2 \)[/tex].

To solve the expression [tex]\( 3x \cdot 8y^4 \)[/tex], we first multiply the coefficients and the variables separately. The coefficients are 3 and 8, and when multiplied, they give us 24. For the variables, we multiply x by [tex]\( y^4 \)[/tex], keeping in mind that when multiplying exponents with the same base, we add the exponents. Therefore, [tex]\( x \cdot y^4 \)[/tex] remains [tex]\( xy^4 \)[/tex].

Combining the coefficient and the variables, we get \( 24xy^4 \). However, we can simplify this further by recognizing that any variable raised to the power of 1 is simply the variable itself. Thus, [tex]\( xy^4 \)[/tex] is equivalent to [tex]\( xy^2 \)[/tex] because [tex]\( y^1 \)[/tex] is just [tex]\( y \)[/tex].

So the final simplified expression is [tex]\( 24xy^2 \)[/tex].

Answer:

Step-by-step explanation:89

Answer:

-9

Step-by-step explanation:

find the value of i2, which is -9, find the value of i7, which is 6

Then, add them together and multiply by the number of terms in the sequence divided by 2 (6/2=3)

Answer:

1. Number line 2

2. Number line 1

3. Number line 4

4. Number line 3

Step-by-step explanation:

1. x – 99 ≤ -104

Solving by adding +99 on both sides

x - 99 +99 ≤ -104 +99

x ≤ -5

Number line 2 represent x ≤ -5

2. x – 51 ≤ -43

Adding +51 on both sides

x -51 +51  ≤ -43 +51

x ≤ 8

Number line 1 represent x ≤ 8

3. 150 + x ≤ 144

Adding -150 on both sides

150 + x -150 ≤ 144 -150

x ≤ -6

Number line 4 represent x ≤ -6

4. 75 < 69 – x

Adding +x on both sides

75 + x < 69 -x +x

x < 69 -75

x < -6

Number line 3 represent x < -6

The correct matches are:

[tex]\(x - 99 \leq -104\)[/tex] with the number line showing [tex]\(x \leq -5\)[/tex].

[tex]\(x - 51 \leq -43\)[/tex] with the number line showing [tex]\(x \leq 8\)[/tex].

[tex]\(150 + x \leq 14475\)[/tex] with the number line showing [tex]\(x \leq 14325\)[/tex].

[tex]\(69 - x < 14475\)[/tex]  with the number line showing [tex]\(x > -14406\)[/tex].

To solve the given inequalities and match them to their corresponding number lines, we will first solve each inequality algebraically and then represent the solution on a number line.

1. For the inequality [tex]\(x - 99 \leq -104\)[/tex], we add 99 to both sides to isolate \[tex](x\)[/tex]:

[tex]\[x \leq -104 + 99\][/tex]

[tex]\[x \leq -5\][/tex]

This means that all values of [tex]\(x\)[/tex] less than or equal to -5 satisfy the inequality.

2. For the inequality[tex]\(x - 51 \leq -43\)[/tex], we add 51 to both sides:

[tex]\[x \leq -43 + 51\][/tex]

[tex]\[x \leq 8\][/tex]

This means that all values of [tex]\(x\)[/tex] less than or equal to 8 satisfy the inequality.

3. For the inequality [tex]\(150 + x \leq 14475\)[/tex], we subtract 150 from both sides:

[tex]\[x \leq 14475 - 150\][/tex]

[tex]\[x \leq 14325\][/tex]

This means that all values of [tex]\(x\)[/tex] less than or equal to 14325 satisfy the inequality.

4. For the inequality [tex]\(69 - x < 14475\)[/tex], we subtract 69 from both sides and reverse the inequality sign because we are dividing by a negative number (-1):

[tex]\[-x < 14406\][/tex]

[tex]\[x > -14406\][/tex]

This means that all values of [tex]\(x\)[/tex] greater than -14406 satisfy the inequality.

Now, let's represent these solutions on number lines:

- For [tex]\(x \leq -5\)[/tex], the number line will have a closed circle at -5 and shading to the left of -5.

- For [tex]\(x \leq 8\)[/tex], the number line will have a closed circle at 8 and shading to the left of 8.

- For[tex]\(x \leq 14325\)[/tex], the number line will have a closed circle at 14325 and shading to the left of 14325.

- For [tex]\(x > -14406\)[/tex], the number line will have an open circle at -14406 and shading to the right of -14406.

Matching the inequalities to the number lines:

- The inequality [tex]\(x - 99 \leq -104\)[/tex] corresponds to the number line with a closed circle at -5 and shading to the left.

- The inequality [tex]\(x - 51 \leq -43\)[/tex] corresponds to the number line with a closed circle at 8 and shading to the left.

- The inequality [tex]\(150 + x \leq 14475\)[/tex] corresponds to the number line with a closed circle at 14325 and shading to the left.

- The inequality [tex]\(69 - x < 14475\)[/tex] corresponds to the number line with an open circle at -14406 and shading to the right.

Therefore, the correct matches are:

[tex]\(x - 99 \leq -104\)[/tex] with the number line showing [tex]\(x \leq -5\)[/tex].

[tex]\(x - 51 \leq -43\)[/tex] with the number line showing [tex]\(x \leq 8\)[/tex].

[tex]\(150 + x \leq 14475\)[/tex] with the number line showing [tex]\(x \leq 14325\)[/tex].

[tex]\(69 - x < 14475\)[/tex]  with the number line showing [tex]\(x > -14406\)[/tex].

Answer:

The vertex is: [tex](\frac{3}{2},\ \frac{7}{2})[/tex]

The axis of symmetry is:

[tex]x=\frac{3}{2}[/tex]

Step-by-step explanation:

For a quadratic equation of the form:

[tex]y=ax^2 + bx +c[/tex]

The vertex of the parabola will be the point: [tex](-\frac{b}{2a},\ f(-\frac{b}{2a}))[/tex]

In this case we have the following equation:

[tex]y= -2x^2+6x-1[/tex]

Note that:

[tex]a=-2\\b=6\\c=-1[/tex]

Then the x coordinate of the vertex is:

[tex]x=-\frac{b}{2a}[/tex]

[tex]x=-\frac{6}{2(-2)}[/tex]

[tex]x=\frac{3}{2}[/tex]

Then the y coordinate of the vertex is:

[tex]y= -2(\frac{3}{2})^2+6(\frac{3}{2})-1[/tex]

[tex]y=\frac{7}{2}[/tex]

The vertex is: [tex](\frac{3}{2},\ \frac{7}{2})[/tex]

For a quadratic function the axis of symmetry is always a vertical line that passes through the vertex of the function.

Then the axis of symmetry is:

[tex]x=\frac{3}{2}[/tex]

Answer:

y=2

x=5

Step-by-step explanation:

Answer:

[tex]24\sqrt{5}\ yards[/tex]

Step-by-step explanation:

Let A1 be the area of one square and A2 be the area of second square

So,

A1 = s^2

where s is side of square

[tex]s^2=125\\\sqrt{s^2}=\sqrt{125}\\s=\sqrt{25*5}\\ s= \sqrt{5^2 * 5}\\ s= 5\sqrt{5}[/tex]

So side of one square is [tex]5\sqrt{5}[/tex]

To calculate the length of fence we need to find the perimeter of the square

So,

P1 = 4 * s

[tex]=4*5\sqrt{5} \\=20\sqrt{5}[/tex]

For second square:

[tex]A_2=s^2\\5=s^2\\\sqrt{s^2}=5\\{s}=\sqrt{5}[/tex]

The perimeter will be:

[tex]P_2 = 4*s\\=4 * \sqrt{5} \\=4\sqrt{5}[/tex]

So the total fence will be: P1+P2

[tex]= 20\sqrt{5}+4\sqrt{5} \\= 24\sqrt{5}\ yards[/tex]

Answer:

11

Step-by-step explanation:

In this problem we have the following expression

[tex]a + b^2[/tex]

Note that it depends on two variables a and b

Evaluating the expression for a = 2 and b = 3 means that you must replace the variable "a" with the number 2, and you must replace the variable "b" with the number 3

So we have that:

[tex]a + b^2 = 2 + 3^2[/tex]

[tex]a + b^2 = 2 +9[/tex]

[tex]a + b^2 = 11[/tex]

Finally the answer is 11

Answer:

x =3

Step-by-step explanation:

5(x + 1) = 4x + 8

Distribute the 5

5x+5 = 4x+8

Subtract 4x from each side

5x-4x+5 = 4x-4x+8

x+5 =8

Subtract 5 from each side

x+5-5 = 8-5

x =3

Answer:

One globe costs $60.79

Step-by-step explanation:

The area of the globe is

A = 4.π.r^2

Since the radius is 11 inches

A = 4*(3.14)*(11 in)^2 = 1519.76 in^2

We apply a rule of three

1 in^2  --------------------------------- $0.04

1519.76 in^2 ------------------------- x

x = (1519.76 in^2/ 1 in^2)*$0.04

x = (1519.76)*$0.04

x = $60.79

x ≈ $60.8

Answer: Second Option

$60.79

Step-by-step explanation:

To solve this problem let's suppose that the balloons are spherical

Then the surface area of a sphere is given by the formula:

[tex]SA=4\pi r^2[/tex]

Where r is the radius of the sphere

In this case we know that the radius of each globes is 11 inches, then:

[tex]SA=4(3.14) (11)^2[/tex]

[tex]SA=4(3.14)(121)[/tex]

[tex]SA=1519.76\ in^2[/tex]

If the material to make globes costs $ 0.04 per square inch then the cost of a globes is:

[tex]C=1519.76*0.04[/tex]

[tex]C=\$60.79[/tex]

Answer:

Answer is [tex]\sqrt{13}-\sqrt{11}[/tex]

Step-by-step explanation:

We need to divide

[tex]\frac{2}{\sqrt{13}+\sqrt{11}}[/tex]

For solving this, we need to multiply and divide the given term with the conjugate of [tex]{\sqrt{13}+\sqrt{11}[/tex]

The conjugate is: [tex]{\sqrt{13}-\sqrt{11}[/tex]

Solving

[tex]=\frac{2}{\sqrt{13}+\sqrt{11}} *\frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}} \\=\frac{2(\sqrt{13}-\sqrt{11})}{(\sqrt{13}+\sqrt{11})(\sqrt{13}-\sqrt{11})}\\We\,\, know\,\, that\,\, (a+b)(a-b) = a^2-b^2\\=\frac{2(\sqrt{13}-\sqrt{11})}{(\sqrt{13})^2-(\sqrt{11})^2}\\=\frac{2(\sqrt{13}-\sqrt{11})}{13-11}\\=\frac{2(\sqrt{13}-\sqrt{11})}{2}\\=\sqrt{13}-\sqrt{11}[/tex]

So answer is [tex]\sqrt{13}-\sqrt{11}[/tex]

Answer:

The correct Answer is D[tex]\sqrt{13} - \sqrt{11}[/tex]

Step-by-step explanation:

Answer:h= [tex]\frac{(p/0.7-b)}{r}[/tex]

Step-by-step explanation:

1) p/0.7=rh+b

2) (p/0.7)-b=rh

3) [tex]\frac{(p/0.7-b)}{r}[/tex] =h

h= [tex]\frac{(p/0.7-b)}{r}[/tex]

The equivalent equation for isolated variable h = [tex]\frac{p-0.7b}{r}[/tex] obtained from the equation p = 0.7(rh + b).

Isolating a variable is the re-arrangement of the equation for the required variable. Even though rewriting the terms, doesn't affect the logic of the equation. It forms an equivalent equation.

The given equation is p = 0.7(rh + b)

Where,

p - home pay

h - working hours

r - the rate of dollars per hour

b - bonus received

Re-writing the equation for the variable h:

p = 0.7(rh + b)

⇒ p/0.7 = rh + b

⇒ p/0.7 - b = rh

⇒ h = p/0.7r -b/r

∴ h = [tex]\frac{p-0.7b}{r}[/tex]

Therefore, variable h is isolated and obtained an equivalent equation.

Learn more about isolating the variable here:

https://brainly.com/question/23692213

#SPJ2

Answer: Option 'D' is correct.

Step-by-step explanation:

Since we have given that

3 times the square root of 2 +2 times the square root of 3 =3sqrt2+2sqrt3

We need to simplify the above expression.

In addition and subtraction, there is a rule of adding or subtracting of like terms is possible only.

But, here, √2 ad √3 are unlike term.

so, we cannot simplified it.

Hence, Option 'D' is correct.

Step-by-step explanation:

Please mark brainliest and have a great day!

Answer:

Step-by-step explanation:

3√2 + 2√3 is as simple an expression as you'll get for this quantity.

Answer:

3. B. 128

4. B. 50.24

Step-by-step explanation:

3. The Books

The numbers of the problem make it quite easy.

We can easily figure out how many books will be stacked in each dimension of the trunk.

Length:

Books: 10 inches

Trunk: 40 iinches.

How may books can you fit lengthwise in the trunk?  40 / 10 = 4

We then to the same for the width:

Books: 6 inches

Trunk: 24 inches

How many books can you stack wide-wise in the trunk?  26/4 = 4

And then the height:

Books: 2.5 inches

Trunk: 20 inches

How many books stacked in height? 20 / 2.5 = 8

So, we can saw the trunk is 4 books long, by 4 books wide by 8 books thick... so, the number of books you can fit in that trunk is:

B = 4 * 4 * 8 = 128

4. Tomato juice

First, we have to find the volume of that big can...

Since it's a cylinder, its volume is calculated by:

V = π * r² * h and we have everything we need:

V = 3.14 * 4² * 12 = 192 * 3.14 = 602.88 cu in

We know a cup of tomato juice takes 12 cu inches...

So, how many cups of juice fit in that can?

C = 602.88 cu in / 12 cu in/cup = 50.24 cups

Given - Length of the Rectangular Plot : 16 meters

Given - Width of the Rectangular Plot : 4 meters

We know that - Area of a Rectangle is given by : Length × Width

[tex]:\implies[/tex]  Area of the Rectangular plot = (16 × 4) m²

[tex]:\implies[/tex]  Area of the Rectangular plot = 64 m²

Given : Square plot and Rectangular plot have same Area

[tex]:\implies[/tex]  Area of Rectangular plot = Area of Square plot

[tex]:\implies[/tex]  Area of Square plot = 64 m²

We know that - Area of Square is given by : Side × Side

[tex]:\implies[/tex]  Side × Side = 64 m²

[tex]:\implies[/tex]  S² = 64 m²

[tex]:\implies[/tex]  S² = 8² m²

[tex]:\implies[/tex]  S = 8 m

Answer : One side of the Square Garden plot is 8 meter

Answer:

Side of the square plot = 8 m

Step-by-step explanation:

l = 16 m

b = 4 m

Area of the rectangular plot = l * b

                                                 = 16 * 4 = 64 m²

Area of square plot = Area of rectangular plot

          side  *  side    = 64 m²

                    side       =  √64 = 8 m

[tex]x(x-3)=x\\x^2-3x-x=0\\x^2-4x=0\\x(x-4)=0\\x=0\vee x=4[/tex]

[tex]\text{Hey there!}[/tex]

[tex]\text{In order for you can do the distributive property then work from there}[/tex]

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

[tex]\text{Subtract by the value of x on your sides!}[/tex]

[tex]\text{Your new equation becomes: x}^2\text{- 3x = x}[/tex]

[tex]\text{Like}\downarrow[/tex]

[tex]\text{x}^2\text{- 3x - x = x - x}[/tex]

[tex]\text{x - x = 0 }[/tex]

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

[tex]\text{Our equation becomes: x}^2\text{- 4x = 0}[/tex]

[tex]\text{Next, we have to FACTOR on the LEFT side of your equation}[/tex]

[tex]\text{x(x - 4) = 0}[/tex]

[tex]\text{Set the numbers to FACTOR out to 0}[/tex]

[tex]\text{Like: x = 0 or x - 4 = 0}\text{ (solve that and you SHOULD have the x-values)}[/tex]

[tex]\boxed{\boxed{\bf{Answer: x = 0\ or \ x =4}}}\checkmark[/tex]

[tex]\text{Good luck on your assignment and enjoy your day!}[/tex]

~[tex]\frak{LoveYourselfFirt:)}[/tex]

Answer:

A. 3,1,1/3,1/9,1/27 because you are multiplying by 1/3 every time to get new term.

C. 1,6,36,216,1296 because you are multiplying by 6 every time to get new term.

The other sequences you are not multiplying repeatedly to get new terms..

Step-by-step explanation:

The sequence is geometric sequence is 3, 1, 1/3, 1/9, and 1/27.

The sequence is a geometric sequence that is 1, 6, 36, 216, 1, 296.

A sequence is a list of elements that have been ordered in a sequential manner, such that members come either before or after.

If the common ratio between the two successive terms must be constant. Then the sequence is called a geometric sequence.

The sequences are given below;

3, 1, 1/3, 1/9, 1/27

The common ratio between the terms are;

[tex]\rm \dfrac{a_2}{a_1}=\dfrac{1}{3}\\\\\dfrac{a_3}{a_2}=\dfrac{\dfrac{1}{3}}{1}= \dfrac{1}{3} \times \dfrac{1}{1}=\dfrac{1}{3}\\\\\dfrac{a_4}{a_3}=\dfrac{\dfrac{1}{9}}{\dfrac{1}{3}}=\dfrac{3}{9}=\dfrac{1}{3}[/tex]

The sequence is a geometric sequence.

The sequences are given below;

1, 6, 36, 216, 1, 296.

The common ratio between the terms are;

[tex]\rm \dfrac{a_2}{a_1}=\dfrac{6}{1}=6\\\\\dfrac{a_3}{a_2}=\dfrac{36}{6}=6\\\\ \dfrac{a_4}{a_3}=\dfrac{216}{36}=6\\[/tex]

The sequence is a geometric sequence.

More about the sequence link is given below.

brainly.com/question/21961097

#SPJ2

Answer:

[tex]\large\boxed{y=2\pm\sqrt{28+4x}}[/tex]

Step-by-step explanation:

[tex]2(x-2)^2=8(7+y)\\\\\text{exchange x to y, and vice versa:}\\\\2(y-2)^2=8(7+x)\\\\\text{solve for y:}\\\\2(y-2)^2=(8)(7)+(8)(x)\\\\2(y-2)^2=56+8x\qquad\text{divide both sides by 2}\\\\(y-2)^2=28+4x\iff y-2=\pm\sqrt{28+4x}\qquad\text{add 2 to both sides}\\\\y=2\pm\sqrt{28+4x}[/tex]

Answer:

y is inverse:  2 ±[tex]\sqrt{28+ 4x}[/tex] .

Step-by-step explanation:

Given: 2(x - 2)²=8(7+y)​.

To find: Find inverse.

Solution : We have given

2(x - 2)²=8(7+y)​.

Step 1: inter change the x and y.

2(y - 2)²=8(7+x)​.

Step 2:

Solve for y

On dividing both sides by 2

(y - 2)² = 4 (7+x)​.

Distributes 4 over ( 7 + x)

(y - 2)² =  28 + 4x

Taking square root both sides.

[tex]\sqrt{(y-2)^{2} } = ±\sqrt{28+ 4x}[/tex].

y - 2 =  ±[tex]\sqrt{28+ 4x}[/tex].

On adding both sides by 2

y =  + 2 ±[tex]\sqrt{28+ 4x}[/tex] .

Therefore, y is inverse : 2 ± [tex]\sqrt{28+ 4x}[/tex].

Answer:

AAS

Step-by-step explanation:

Given one pair of congruent angles and one pair of congruent sides.

When two lines intersect they form a pair of congruent angles (vertical angles are equal)

So the two triangles are congruent by AAS.

Answer: AAS

Step-by-step explanation:

By the figure we can say that the have one side equal. Now for the angles we have that in the part of the triangles are touching we have an oposite angles then these two angles have to be equals. Finally the two triangles have two equal angles and ones equal side. we can conclude that the correct method is AAS.

Answer:

The area is 0.015m²

Step-by-step explanation:

Step one

Given that the expression for area of the circle is A = π r²

Step two

Now our raduis r= 7cm - - - meter =7/100= 0.07

And Pi = 3.142

Step three

Substituting r in the formula for area we have

A= 3.142*(0.07)²

A=3.142*0.0049

A= 0.015m²

Answer:

Center of the circle= (-9,6)

Radius = 5 units

Step-by-step explanation:

Once the equation of the circle has been written in the format

(x-h)²+(y-k)²=r² , (h,k) is the center while r is the radius of the circle.

From the given equation, -h=9 therefore h= -9.

-k = -6 therefore k = 6. r² = 25 therefore r= √25=5

Center of the circle= (-9,6) radius = 5 units

Answer:

The graph in the attached figure

Step-by-step explanation:

we have

[tex]y=-\left|x\right|+2[/tex]

we know that

The graph is inverted V-shaped.

The x-intercepts are the points (-2,0) and (2,0)

The vertex is the point (0,2)

The domain is all real numbers

The range is all real numbers less than or equal to 2

using a graphing tool

The graph in the attached figure

[tex]\bf (\stackrel{x_1}{-3}~,~\stackrel{y_1}{-1})~\hspace{10em} slope = m\implies \cfrac{2}{5} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-(-1)=\cfrac{2}{5}[x-(-3)]\implies y+1=\cfrac{2}{5}(x+3) \\\\\\ y+1=\cfrac{2}{5}x+\cfrac{6}{5}\implies y=\cfrac{2}{5}x+\cfrac{6}{5}-1\implies y=\cfrac{2}{5}x+\cfrac{1}{5}[/tex]

Answer:

Option C is correct.

Step-by-step explanation:

Slope of line JK

J (1,-6) K (0,4)

slope of JK = y₂ - y₁ / x₂ - x₁

Slope of JK = 4-1/0-(-6) = 3/6 = 1/2

so slope of JK = 1/2

Slope of line LM

L (-5,-3) and M(0,3)

slope of LM = y₂ - y₁ / x₂ - x₁

slope of LM = 3-(-5)/0-(-3) = 3+5/3 = 8/3

Slope of line NO

N(-6,-5) and 0 (0,0)

slope of NO = y₂ - y₁ / x₂ - x₁

slope of NO = 0-(-6)/0-(-5) = 6/5

So, slope of line NO = 6/5

Slope of PQ

P(-5,4) Q(0,-2)

slope of PQ = y₂ - y₁ / x₂ - x₁

slope of PQ = -2-4/0-(-5) = -2/5

so slope of line PQ = -2/5

the two lines are perpendicular if slope of one line is m then slope of other line is -1/m

The slope of given line =m= -5/6

The slope of line perpendicular to the given line = -1/m = 6/5

So, line NO is perpendicular to the given line as its slope is 6/5

Option C is correct.

Answer:

The coordinates of the point in question is (1, 3).

Step-by-step explanation:

Point (-1, 7) is above and to the left of the point (4, -3). The point in question is to the right and below the point (-1, 7).

What will be the horizontal distance between the point (-1, 7) and the point in question?

The horizontal distance between the point (-1, 7) and (4, -3) is 5. Let the horizontal distance between the point (-1, 7) and the point in question be [tex]a[/tex]. Let the horizontal distance between the point in question and point (4, -3) be [tex]b[/tex].  

[tex]\displaystyle \frac{a}{b} = \frac{2}{3}[/tex].

[tex]\displaystyle a = \frac{2}{3} \; b[/tex].

[tex]\displaystyle b = \frac{3}{2}\; a[/tex].

However,

[tex]a + b = 5[/tex].

[tex]\displaystyle a + \frac{3}{2}\; a = 5[/tex].

[tex]\displaystyle \frac{5}{2}\; x= 5[/tex].

[tex]a = 2[/tex].

In other words, the point in question is 2 units to the right of the point (-1, 7). The x-coordinate of this point shall be [tex]-1 + 2 = 1[/tex].

The vertical distance between the point (-1, 7) and the point (4, -3) is 10. Similarly, the point in question is [tex](2/5) \times 10 = 4[/tex] units below the point (-1, 7). The y-coordinate of this point will be [tex]7 - 4 = 3[/tex].

Thus, the point in question is (1, 3).

Answer:

To solve our given problem we will use section formula :]

Section Formula states that, when a point divides a line segment internally in the ratio m:n, So the coordinates are :]

[tex]\tiny: \implies (x,y) = \bigg \lgroup x = \frac{m. {x}_{2} +n. {x}_{1} }{m + n} ,y= \frac{m. {y}_{2} +n. {y}_{1} }{m + n} \bigg \rgroup \\ \\ \\ [/tex]

Let

(-1 , 7) = (x₁ , y₁)

(4 , -3) = (x₂ , y₂)

m = 2

n = 3

[tex]\tiny: \implies (x,y) = \bigg \lgroup x = \frac{2 \times 4 +3 \times - 1 }{2 + 3} ,y= \frac{2 \times - 3 +3 \times 7}{2 + 3} \bigg \rgroup \\ \\ \\ [/tex]

[tex]\tiny: \implies (x,y) = \bigg \lgroup x = \frac{8 - 3 }{2 + 3} ,y= \frac{ - 6 +21}{2 + 3} \bigg \rgroup \\ \\ \\ [/tex]

[tex]\tiny: \implies (x,y) = \bigg \lgroup x = \frac{5 }{5} ,y= \frac{15}{5} \bigg \rgroup \\ \\ \\ [/tex]

[tex]\tiny: \implies (x,y) = \bigg \lgroup x = 1,y= 3 \bigg \rgroup \\ \\ [/tex]