Given points A(2,1,3) and B(0,5,0), find a point C in the xy-plane that creates an isosceles right triangle such that:_______

a. AB and BC are the legs of the right triangle
b. AB is the hypotenuse of the right triangle (for part ii, just doing all the set-up work is sufficient; you do not need to solve and find the points)

Answers

Answer 1

Answer:

a.)Point C will be either (4.82, 7.41) or ( -4.82, 2.59)

b.) -x² + 2x + 6y - y² - 5= 0  , 8y - 4x = 11

Step-by-step explanation:

As given A(2,1,3) and B(0,5,0)

a.)

We have to find C  in the xy-plane

⇒ C is of the form (x, y, 0)    ( Because in xy plane , z coordinate is zero)

Now, given AB and BC are the legs of the right triangle .

AB = (0, 5, 0) - (2, 1, 3) = ( -2, 4, -3)

BC = (x, y, 0) - (0, 5, 0) = ( x, y-5, 0)

As AB and BC are legs of triangle

⇒AB is perpendicular to BC

⇒(AB).(BC) = 0

⇒( -2, 4, -3).( x, y-5, 0) = 0

⇒ -2x + 4(y-5) -0 = 0

⇒ -2x + 4y- 20 = 0

⇒ -x + 2y- 10 = 0

⇒ 2y = 10 + x

⇒ y = 5 + [tex]\frac{x}{2}[/tex]         ......(1)

As ABC is making an isosceles right triangle

It means two sides of the triangle are equal

Now,

|AB| = √(-2)² + (4)² + (-3)² = √4+16+9 = √29

|BC| = √(x)² + (y-5)² + 0² = √(x)² + (y-5)²

Now,

AS AB = BC

⇒|AB| = |BC|

⇒|AB|² = |BC|²

⇒29 = (x)² + (y-5)²

⇒29 = (x)² + (5 + [tex]\frac{x}{2}[/tex] -5)²

⇒29 = (x)² + ( [tex]\frac{x}{2}[/tex] )²

⇒29 = x²( 1 + [tex]\frac{1}{4}[/tex] ) = x²(  [tex]\frac{5}{4}[/tex] )

⇒ x² = [tex]\frac{116}{5}[/tex] = 23.2

⇒ x = 4.82 , -4.82

⇒y = 7.41 , 2.59

so, Point C will be either (4.82, 7.41) or ( -4.82, 2.59)

b.)

As given AB is the hypotenuse of the right triangle

S, AC and CB will perpendicular to each other

Now,

AC = (x, y, 0) - ( 2, 1, 3) = ( x-2, y-1, -3)

CB = (0, 5, 0) - (x, y, 0) = (-x, 5-y, 0)

As, AC and CB is perpendicular

⇒(AC).(CB) = 0

⇒( x-2, y-1, -3).(-x, 5-y, 0) = 0

⇒(x-2)(-x) + (y-1)(5-y) - 0 = 0

⇒-x² + 2x + 5y - y² - 5 + y = 0

⇒-x² + 2x + 6y - y² - 5= 0    ..........(2)

As ABC is making an isosceles right triangle

It means two sides of the triangle are equal

Now,

|AC| = √(x-2)² + (y-1)² + (0-3)² = √(x-2)² + (y-1)² + 9

|BC| = √(x)² + (y-5)² + 0² = √(x)² + (y-5)²

Now,

AS AC = BC

⇒|AC| = |BC|

⇒|AC|² = |BC|²

⇒(x-2)² + (y-1)² + 9  = (x)² + (y-5)²

⇒x² + 4 - 4x + y² + 1 - 2y + 9 = x² + y² + 25 - 10y

⇒ 4 - 4x + 1 - 2y + 9 = 25 - 10y

⇒14 - 4x - 2y  = 25 - 10y

⇒10y - 2y - 4x = 25 - 14

⇒8y - 4x = 11            ........(3)

Now, we have 2 equations

-x² + 2x + 6y - y² - 5= 0  , 8y - 4x = 11

Solve these equations to find x, y


Related Questions

List all factors of 72

Answers

Answer:

1,2,3,4,6,8,9,12,18,24,36,72.

Step-by-step explanation:

What you are looking for are numbers that multiply by each other to equal 72, like 1 and 72 or 8 and 9. Factor tree may help, search it up :)

Promise Easy Points
Calculate the average monthly expenses for these individuals?

Kelsey spent $350 in May, $432 in June and $420 in July. Her average monthly expenses are:____

Michael spent $700 in April and May and $800 in June and July. His average monthly expenses are:____

Answers

$420+$350+$432=kelsey's monthly expenses  

$700+$800=Micheal's Monthly expenses

Michael's average monthly expenses are $750.

What is the average?

The average value in a set of numbers is the middle value, calculated by dividing the total of all the values by the number of values. When we need to find the average of a set of data, we add up all the values and then divide this total by the number of values.

Given that, Kelsey spent $350 in May, $432 in June and $420 in July.

Now, average monthly expenses are

(350+432+420)/3

= 1202/3

= 400.67

Michael spent $700 in April and May and $800 in June and July.

Here, (700×2+800×2)/4

= (1400+1600)/4

= 3000/4

= $750

Therefore, Michael's average monthly expenses are $750.

To learn more about an average visit:

https://brainly.com/question/11195029.

#SPJ2

if sarah has 800 marbles and her teacher gave her 3,00 more and she have 100 to her 2 friends how much will sarah have??

Answers

Answer:

3600

Step-by-step explanation:

If she has 800 her teacher gives her 3000= 3800 and she gave 100 to both of her friends - 200 then she will have 3600.

Hope this helps! Please consider marking brainliest! I hope you have an amazing day. Always remember, your smart and you got this! -Alycia


[tex] \sqrt{361} [/tex]


Answers

Answer:

19

Step-by-step explanation:

Answer:

19

Step by step explanation:

Photo attached please help

Answers

it’s a

hope this helped!:)
I’m pretty sure it’s a

A quality assurance check is 91% accurate for non-defective devices and 97% accurate for defective devices. Of the devices checked, 84% are not defective. What is the probability of an incorrect conclusion? Round your answer to the nearest tenth of a percent.

Answers

The probability of an incorrect conclusion is,

In 84 % of inspection,  [tex]7.56\%[/tex]  of them labeled as not defective but they are actually defective

Probability :

It is given that,

A quality assurance check is 91% accurate for non-defective devices and 97% accurate for defective devices.

Of the devices checked, 84% are not defective.

Actual not defective is,

            [tex]\frac{91*84}{100} = 76.44\%[/tex]

In  [tex]84\%[/tex] of inspection,[tex]76.44\%[/tex] are actually not defective

But [tex]84- 76.44 = 7.56\%[/tex]  of them labeled as not defective but they are actually defective

Learn more about Probability here:

https://brainly.com/question/24756209

HELP QUICK ILL GIVE TOU BRAINLIEST
find the surface area...

Answers

Answer: 483.09 in^2
Explanation: 12, 11, 10, 12

Estimate the value of 50.
O A. between 5 and 6
O B. between 6 and 7
C. between 7 and 8
<
D. between 8 and 9
PLEASE HELPP

Answers

Answer: c

Step-by-step explanation: because 7 squared is 49 and 8 squared is 64

nvm i read the question wrong

Step-by-step explanation:

the person above is correct

How do you graph y=−x−2.

Answers

Y=-x-2 that is how you graph it
Isn’t that how you graph it? I’m confused about the question sorry!

A professor in one city earns $61,000 a year. A bioengineer earns $95,750. A nurse earns $50,600. What is the difference in salary between the professor and nurse?

Answers

Answer:

The difference is 10,400$.

Step-by-step explanation:

$61,000-$50,600=$10,400

Use the exchange rate £1 = $1.38 to convert $276 into £.

Answers

Answer:

£200

Step-by-step explanation:

$276*(1/$138)=200 pounds

PLZ HELP ME What shape is this cross-section?

Answers

Answer:

i'm not sure if i'm right but think is cube

my friend needs help with this can someone help

Answers

Answer:Sorry

Step-by-step explanation:

Hi would help but i dont know and im not in collage sorry.

What is the area of the composite figure?

Answers

Hello! :)

[tex]\large\boxed{A = 32.5cm^{2}}[/tex]

We can use the following area to solve for the area of the trapezoid pictured:

[tex]A = \frac{1}{2}(b_{1} + b_{2})h[/tex]

Where:

h = height

b1 = bottom base

b2 = top base

Plug in the given values:

[tex]A = \frac{1}{2}(9 + 4)(5)[/tex]

Simplify:

[tex]A = \frac{1}{2}(65)\\\\A = 32.5 cm^{2}[/tex]

At a restaurant, the bill comes to $30. If you decide to leave a 10% tip, how much total do you need to pay? Give your answer to the nearest dollar.

Answers

Answer:

33dollors if I'm not mistakin

How heavy is a 225 meter cable if a 3 meter section weighs 1.3 kg?

Answers

97.5 kg

225/3=75
75 x 1.3 = 97.5

PLEASE HELP




Consider the expressions (3m+2n−6) and (−4−6m+7n).

Part A


What is the sum of the expressions for n=−2 and m=5?


The sum is

Answers

Answer:

5, -20

Step-by-step explanation:

Given the equation (3m+2n−6) and (−4−6m+7n).

Also, given that we're told that

n = -2 and

m = 5, if we take a rather direct approach by substituting like for like, we have

(3m+2n−6) = (3 * 5 + 2 * -2 - 6)

(3m+2n−6) = 15 - 4 - 6

(3m+2n−6) = 15 - 10

(3m+2n−6) = 5

(−4−6m+7n) = (-4 - 6 * 5 + 7 * 2)

(−4−6m+7n) = -4 - 30 + 14

(−4−6m+7n) = -34 + 14

(−4−6m+7n) = -20

Therefore, we proceed to conclude that the answers are 5 and -20 respectively

What’s the answer pls

Answers

Answer:

Step-by-step explanation:

There is no bird on the spinner. therefore the answer is impossible because there is no bird.

Answer:

Impossible - There's no bird on the wheel.

Step-by-step explanation:

Which type of interest will earn you more money in a savings account?

Answers

Answer:

a compound interest that has the lowest interest rate .  

Step-by-step explanation:

three and two forths - one and two thirds =
3 2/4 - 1 2/3 =

Answers

Answer:

1.83333333333

Step-by-step explanation:

pls mark me brainliest

ok thanks :) :) :) :)


Solve for x:
3x – 2(6 - x) = 7x + 2(5 + x) – 6

Answers

Answer:

-4

Step-by-step explanation:

3x-2(6-x)=7x+2(5+x)-6

First, distribute.

3x+ −12+ 2x = 7x + 10 + 2x −6

Combine like terms on each side.

5x − 12= 9x + 4

To solve for x, pick a side to solve. I will solve the right x.

5x - 12 = 9x + 4

-5x         -5x

-12 = 4x + 4

Subtract 4 to isolate the variable.

-16 = 4x

Divide 4.

-4 = x

Helppppppppppppp meeeeee

Answers

Answer:

9

Step-by-step explanation:

because 9 in a third is 3

3 + 12 is 15

9x2 is 18

and 15 is 3 less than 18

Mr. Jones charged Roberta $105 for parts and $25 per hour for labor to repair her bicycle. He spent 2 hours repairing her bike. Which expression would be used to correctly calculate the amount Roberta owes him?

a.(2 × 25) + 105

b.(105 x 2) + (25 x 2)

c.2 x (25 + 105)

d.2 x 25 × 105

Answers

Answer:

a.(2 × 25) + 105

Step-by-step explanation:

Let the number of hours spent repairing the bike = x

Mr. Jones charged Roberta $105 for parts and $25 per hour for labor to repair her bicycle.

Therefore, our algebraic expression

= $25× x + $105

= 25x + 105

We were told in the question that He spent 2 hours repairing her bike.

Hence:

25 × 2 + 105

Therefore, the correct option is option a

a.(2 × 25) + 105

Isabella is making a display board for the school trip. The display board is 2.50m by 2m rectangle. She needs to cover the entire display board with card. How much card would she needs?

Answers

Answer:

5m^2

Step-by-step explanation:

Given data

Length of board= 2.50m

Width of board= 2m

The cover need should have the same area as the board

Area= Length*Width

Area= 2.5*2

Area=5m^2

Hence the area of the cover is 5m^2

An environmental study was conducted. It reported that houses with more inhabitants do not necessarily produce more or less trash. What can be determined?
A. There is no correlation between number of inhabitants and amount of trash produced.
B. There is a correlation between number of inhabitants and amount of trash produced. However, there is no causation. This is because there is probably an increase in the amount of trash produced with an increase in the number of inhabitants.
C. There is a correlation between number of inhabitants and amount of trash produced. There may or may not be causation. Further studies would have to be done to determine this.

Answers

Answer:

C.

Step-by-step explanation:

There's no problem for now but it still needs to be monitored in case something might happen.

how much is 1/2÷8×57÷13+15

Answers

Answer:

Use the BODMAS method and at the end work the fraction left by adding it to 15

My answer was 15.274

What is the equation in slope-intercept form for the line that passes through the points (–1, 3) and (–2, –3)?

Answers

Answer:

The answer is y = 6x + 9.

-

Step-by-step explanation:

☆Remember: [tex]m =  \frac{y_2 - y_1}{x_2 - x_1} [/tex]

☆Slope-intercept: [tex]y = mx + b[/tex]

---

☆We need to find the slope first. So just plug in according to the slope formula!

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

---

☆Now we need to find the y-intercept. We are going to plug in the slope and a set of coordinates to find the y-intercept.

[tex]y = mx + b \\ 3 = 6( -1 ) + b \\ 3 = - 6 + b \\ \frac{ + 6 = + 6 \: \: \: \: \: \: \: \: \: \: }{9 = b \: \: \: \: \: \: \: \: \: \: \: } [/tex]

---

☆Now all we need to do is plug in and we are done!

[tex]y = 6x + 9[/tex]

▪Happy To Help <3

Can you help me please with this

Answers

-39/100 is the answer according to me ..

Plz mark me as brainliest if u found it helpful

What is the measure of angle M?

Answers

Answer:

35°

Step-by-step explanation:

4x+11 = 5x-24

5x-4x=x

x=24+11

x=35

5/= 1.5 x 25

What is 1.5 × 25

A. 22.5

B. 37.5

c. 52.5

D. 85.5​

Answers

Answer:

37.5 is the answer

Step-by-step explanation:

all you do is just multiply the 2 numbers

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