1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
olga55 [171]
2 years ago
11

Rotating Lines

Mathematics
1 answer:
Juliette [100K]2 years ago
6 0
<h3>Answer: </h3>

Check out the reference chart below for a summary of each rotation equation. It is attached as an image.

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

Explanation:

In each rotation, I'm assuming the center of rotation is the origin.

For 180 degree rotations, the line stays the same. This is because a line is a combination of two rays that point in opposite directions (eg: one ray points directly northwest and the other ray points directly southeast).

So effectively it's like starting to face the northwest direction, rotating 180 degrees to face the southeast direction, and you're on the same line when all is said and done.

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

For 90 and 270 degree rotations, it's a bit trickier. I'll go over counterclockwise rotations of each.

Let's start with a 90 degree counterclockwise rotation.

The slope of y = mx+b is m

The negative reciprocal of this is -1/m. We flip the fraction and the sign from positive to negative.

When doing a 90 degree counterclockwise rotation, the point (0,b) moves to (-b,0). It doesn't matter if b is positive or negative or zero.

Notice the y intercept of the original equation (b) is now connected to the x intercept of the answer.

We'll use this to find the y intercept

y-y1 = m(x-x1)

y-0 = (-1/m)(x - (-b))

y = (-1/m)(x + b)

y = (-1/m)x - b/m

this is the equation for 90 degree counterclockwise rotations. It also applies to 270 degree clockwise rotations.

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

Now onto 270 degree counterclockwise rotations

Once again, the center of rotation is assumed to be the origin.

Like before, the old slope m becomes -1/m so that the lines are perpendicular.

So far, everything appears to be the same as the last section. However, the y intercept will be different.

The point (0,b) rotates to (b,0) when doing a 270 degree counterclockwise rotation.

We'll use the point slope formula to find the equation we're after.

y - y1 = m(x - x1)

y - 0 = (-1/m)(x - b)

y = (-1/m)(x - b)

y = (-1/m)x + b/m

this is the equation for 270 degree counterclockwise rotations. It also works for 90 degree clockwise rotations.

So the only real difference is the -b/m changed to +b/m.

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

The chart summarizing everything is posted below.

You might be interested in
Carlos bought a $235 water heater with his credit card. He used the water heater for five years before replacing it. He paid off
Mazyrski [523]

First we will find the interest on:


P = $235 principal


t = 2 years


r = 0.1415 annual rate


A = future value 


I = A - P the interest


A = P(1 + r)^t


A = 235(1 + 0.1415)^2


A = $306.21


I = A - P


I = $306.21 - $235


I = $71.21


the interest was $71.21.



Next lets find the lifetime cost value:


Lifetime cost value = 306.21 + 5*1.56*52 + 5*0.78*52 = $914.61 (considering that 1 year = 52 weeks)



Now lets find the percentage what percentage the interest is of the lifetime cost:


(71.21/914.61)*100 = 7.79%

<span>← Previous Problem<span>Next Problem →</span></span>
3 0
3 years ago
Samantha threw an apple out of a window from the tenth floor of her apartment building.The equation y=-16t^2+120 can be used to
Savatey [412]
If the apple is on the ground, then the height y is 0. To find t, just plug in 0 for y:
0=-16t^{2}+120
-120=-16t^{2}
t^{2} =7.5
t=2.74

So the answer is 2.74 seconds.

Note that taking the square root of 7.5, mathematically, gives you both 2.74 and -2.74. But because we know that time did not go backwards, only the positive value for t was considered.
5 0
3 years ago
Read 2 more answers
Tom's toy box is shown below. The toy box is shaped like a rectangular prism. It has 5 wooden faces and is open on the top. What
mihalych1998 [28]

Answer:

1100 square inches

Step-by-step explanation:

The faces are all rectangles (area = length x width)

Area of bottom:  20 x 10 = 200

Area of front: 20 x 15 = 300

Area of back (same as front) = 300

Area of right side: 10 x 15 = 150

Area of left side (same as right side) = 150

Total of all the sides: 1100

8 0
2 years ago
A teacher was interested in knowing the amount of physical activity that his students were engaged in daily. He randomly sampled
klasskru [66]

Answer:

The standard error of the mean is 4.5.

Step-by-step explanation:

As we don't know the standard deviation of the population, we can estimate the standard error of the mean from the standard deviation of the sample as:

\sigma_{\bar{x}}\approx\frac{s}{\sqrt{n}}

The sample is [30mins, 40 mins, 60 mins, 80 mins, 20 mins, 85 mins]. The size of the sample is n=6.

The mean of the sample is:

\bar{x}=\frac{1}n} \sum x_i =\frac{30+40+60+80+20+85}{6}=52.5

The standard deviation of the sample is calculated as:

s=\sqrt{\frac{1}{n-1}\sum (x_i-\bar x)^2} \\\\ s=\sqrt{\frac{1}{5}\cdot ((30-52.5)^2+(40-52.5)^2+(60-52.5)^2+(80-52.5)^2+(20-52.5)^2+(85-52.5)^2}\\\\s=\sqrt{\frac{1}{5} *3587.5}=\sqrt{717.5}=26.8

Then, we can calculate the standard error of the mean as:

\sigma_{\bar{x}}\approx\frac{s}{\sqrt{n}}=\frac{26.8}{6}= 4.5

6 0
3 years ago
What are the factors of 36
nasty-shy [4]
1,36. 2,18. 3,12. 4,9. 6,6. I thibk thats all
4 0
2 years ago
Other questions:
  • A teacher has 30 students in a classroom. She has 6 seats in the front row. How many ways can she arrange her students to sit in
    5·1 answer
  • 3 inches is what fraction of a foot
    13·2 answers
  • HELP! HOW DO I DO THIS???
    14·1 answer
  • Helppp and plz hurryyy
    9·1 answer
  • Consider functions f and g below.
    5·2 answers
  • At 3:30 in the afternoon in mid-September the Kimball Tower casts a shadow about 290 feet long when the sun's rays come down at
    5·1 answer
  • Please help!! I'm super confused
    10·1 answer
  • Someone help me !! Please
    12·2 answers
  • 1.The value of 8 dimes is
    14·2 answers
  • What is the slope of this line?<br> y-9=1/5(x-2)
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!