Point Q with coordinates (4, –7) is rotated 270° clockwise about (5, 1) . What are the coordinates of its image?
Answers
Answer:
The coordinates of the image are [tex]Q'(x,y) = (13,0)[/tex].
Step-by-step explanation:
The rotation of point Q around a given point of reference is defined by:
[tex]Q'(x,y) = O(x,y) + r_{OQ}\cdot (\cos (\theta_{OQ}\pm \theta_{R}),\sin (\theta_{OQ}\pm \theta_{R}))[/tex] (1)
Where:
[tex]O(x,y)[/tex] - Point of reference.
[tex]r_{OQ}[/tex] - Magnitude of line segment OP.
[tex]\theta_{OQ}[/tex] - Direction of the point Q with respect to the point of reference, measured in sexagesimal degrees.
[tex]\theta_{R}[/tex] - Rotation angle, measured in sexagesimal degrees.
[tex]Q'(x,y)[/tex] - Rotated point.
Please notice that positive sign means counterclockwise rotation, whereas the negative sign represents clockwise rotation.
The magnitude and direction of the line segment are defined by the following expressions:
Magnitude
[tex]r_{OQ} = \sqrt{(x_{Q}-x_{O})^{2}+(y_{Q}-y_{O})^{2}}[/tex] (2)
Direction
[tex]\theta_{OQ} = \tan^{-1}\left(\frac{y_{Q}-y_{O}}{x_{Q}-x_{O}} \right)[/tex] (4)
Where:
[tex]x_{O}[/tex], [tex]y_{O}[/tex] - Coordinates of point of reference.
[tex]x_{Q}[/tex], [tex]y_{Q}[/tex] - Coordinates of the point Q.
If we know that [tex]x_{O} = 5[/tex], [tex]y_{O} = 1[/tex], [tex]x_{Q} = 4[/tex], [tex]y_{Q} = -7[/tex], then the magnitude and direction of the segment OQ are, respectively:
[tex]r_{OQ} = \sqrt{(4-5)^{2}+(-7-1)^{2}}[/tex]
[tex]r_{OQ} \approx 8.062[/tex]
[tex]\theta_{OQ} = \tan^{-1}\left(\frac{-7-1}{4-5} \right)[/tex]
[tex]\theta_{OQ} = \tan^{-1}\left(\frac{-8}{-1} \right)[/tex] (Which means that line segment OQ belongs to third quadrant)
[tex]\theta_{OQ} \approx 262.875^{\circ}[/tex]
If we know that rotation is clockwise, [tex]x_{O} = 5[/tex], [tex]y_{O} = 1[/tex], [tex]r_{OQ} \approx 8.062[/tex] and [tex]\theta_{OQ} \approx 262.875^{\circ}[/tex], then the rotated point is:
[tex]Q'(x,y) = (5,1) + (8.062\cdot \cos (262.875^{\circ}-270^{\circ}),8.062\cdot \sin (262.875^{\circ}-270^{\circ}))[/tex]
[tex]Q'(x,y) = (5,1) + (8, -1)[/tex]
[tex]Q'(x,y) = (13, 0)[/tex]
The coordinates of the image are [tex]Q'(x,y) = (13,0)[/tex].
Related Questions
The local comic book store is having a closeout sale on their last 375 comic books and Pokémon cards. They are selling single issue comics for $5.79 and Pokémon boosters packs for $3.35. Their goal is to make $1345 to break even on their current final cost. Approximately how many of each do they need to sell?
Answers
Answer:
36 comic books and 339 Pokemon cards
Step-by-step explanation:
Let number of comic books be [tex]x[/tex] and number of Pokemon cards be [tex]y[/tex].
Total number of comic books and Pokemon cards is [tex]375[/tex]. So we get the equation
[tex]x+y=375\\\Rightarrow y=375-x[/tex]
Cost of one comic book is $5.79 and one Pokemon card is $3.35.
They need to make $1345 so we get the equation
[tex]5.79x+3.35y=1345\\\Rightarrow 5.79x+3.35(375-x)=1345\\\Rightarrow 5.79x-3.35x+3.35\times 375=1345\\\Rightarrow x=\dfrac{1345-3.35\times 375}{5.79-3.35}\\\Rightarrow x=36.37\approx 36[/tex]
[tex]y=375-x=375-36\\\Rightarrow y=339[/tex]
So, they need to sell approximately 36 comic books and 339 Pokemon cards.
Please help! Will give brainiest! HIGH POINTS
Evaluate each of the following using ...
Answers
Answer:
F(0)= 0, 0
F(3)= -21, 3
F(2)= -12, 2
Step-by-step explanation:
explanation
Please I really need help
This is my last point
Show workings please
Answers
Step-by-step explanation:
By rotating a figure,
The length of its sides remain the same.
Therefore line AB = line A'B'.
=> 5x - 14 = 4x + 4
=> x = 18
Hence, line AB = 5(18) - 14 = 76.
Answer:
The length of AB = 76 units.
Step-by-step explanation:
We know that Rotations, just like reflections, and translations, are termed isometric.
In other words, these transformations do not alter the size of the figure.
If the size and shape of the figure are not altered, then the figures are termed as congruent.
Given
AB = 5x-14
A'B' = 4x+14
As A'B'C'D' is obtained by rotating ABCD,
so
AB = A'B'
substituting AB = 5x-14 and A'B' = 4x+14
5x-14 = 4x+14
5x - 4x = 14+14
x = 18
Thus,
AB = 5x-14
substitute x = 18
AB = 5(18)-14
AB = 90 - 14
AB = 76
Therefore, the length of AB = 76 units.
Look at the graph shown below:
Which equation best represents the line?
A: y = 1 over 3 x - 1
B: y = 3x - 1
C: y = -x + 1 over 3
D: y = 3x + 1
Answers
What is the area the following circle either enter an exact answer in terms of pie or use 3.14 for pie enter your answer as a decimal. R=5 
Answers
Answer:
it will be 78.5 units ²
what is the image point of (8,-7) after a translation right 1 unit and up 4 units?
Answers
Answer:
It is (9,-3) because x increased by 1 unit and y increased by 4 units.
Answer:
[tex](9,-3)[/tex]
Step-by-step explanation:
[tex](8,-7)[/tex]
If it moves right, add the value to the x-coordinate. If it moves left, subtract the value from the x-coordinate. In this case, because the point was translated 1 unit right, we add 1 to the x-coordinate 8.
[tex](9,-7)[/tex]
If it moves up, add the value to the y-coordinate. If it moves down, subtract the value from the y-coordinate. In this case, because the point was translated 4 units up, we add 4 to the y-coordinate.
[tex](9,-3)[/tex]
I hope this helps!
Find the equation of the straight line that is parallel to 2y - x = 7 and bisects the line joining the points (3,1) and (1,-5).
Answers
Answer:
yes that is completely correct
What is the equation of the circle shown below?
Answers
Step-by-step explanation:
Equation of circle: (x + 2)² + (y - 2)² = r²
Radius of the circle is 3 (from -2 to 1).
Hence we have
(x + 2)² + (y - 2)² = 3² or (x + 2)² + (y - 2)² = 9.
Some helpp pleaseeeee
Answers
Step-by-step explanation:
( 1.5 * 10^0 ) * ( 2.0 * 10 ^ - 5)
= 1.5 * 1 * 2.0 * 10 ^ - 5
= 3 * 10^-5
Hope it will help :)
Based on the graph shown, which features are correctly stated?
Answers
Answer:
A, B, & C.
that's features that correctly stated.
5. Company “A” charges $80 a month for
their cell phone plan and $5 for a 100MB
of Data. Company “B” charges $60 a
month for their cell phone plan and $10 for
a 100MB of Data. How many mega bytes
of data used would make the cost from
either company equal?
Answers
Answer:
400 megabytes of data used will make the cost from either company equal.
Step-by-step explanation:
Given that:
Per month charges of Company A = $80
Charges of 100 MB = $5 = [tex]\frac{5}{100}=\$0.05\ per\ MB[/tex]
Let,
x be the number of MBs consumed.
y be the total charges.
y = 0.05x+80
Per month charges of Company B = $60
Charges of 100 MB = $10 = $0.1 per MB
y = 0.1x+60
For equal charges;
0.05x+80 = 0.1x + 60
80-60 = 0.1x-0.05x
20 = 0.05x
[tex]\frac{0.05x}{0.05}=\frac{20}{0.05}\\x=400[/tex]
Hence,
400 megabytes of data used will make the cost from either company equal.
if 15.5 gallons are $24.16, how much is it for one gallon
Answers
Answer:
$1.55870968
Step-by-step explanation:
24.16dollars/15.5gallons = 1.55870968dollars/gallon
A Stock company was selling at 28$ a share a month later it was selling at 21$ a share what is the percent loss ?
Answers
Answer:
25% loss
Step-by-step explanation:
21 is 75% of 28, because 7*3 = 21 and 7*4 = 28
100%-75%=25%
Hope this helps!
Answer:
the answer is 50 %
Step-by-step explanation:
C.P x (100+ (P/L) ) over 100 = S.P
C.P x 50 over 100 = 8500
C.P = 8500 x 100 over 50 = 17000 .Rs.
hope it helps make me the brainliest now !!!!!!!
3)Hubba Bubba Bubble gum
was $0.25 thirty years ago.
Today it costs $1.39. What is
the percent increase? Round
to the nearest hundredth (if
necessary).
V
Answers
Answer:
456% increase
Step-by-step explanation:
Step one:
given data
Hubba Bubba Bubble gum was $0.25 30 years ago
Today it costs $1.39.
Required
The percentage increase
Step two:
Percentage increase= change in price/ old price *100
Percentage increase= 1.39-0.25/0.25 *100
Percentage increase=1.14/0.25 *100
Percentage increase=4.56*100
Percentage increase=456%
OMG PLEASE HELP QUICK 7TH GRADE LEVEL
Answers
Answer:
26 Feet
Step-by-step explanation:
6 feet 6 inches x 4
Answer:
That would be about 26 feet.
Step-by-step explanation:
Subtract (−x − 2) − (4x + 3).
Answers
Answer:
-5x - 5
Step-by-step explanation:
1. Distributive Property
-x - 2 - 4x - 3
2. Combine like terms
-5x - 5
Hope this helped!
- tan^2x + sec^2x = 1 proving trig equations
Answers
Answer:
Is C
Step-by-step explanation:
Answer:
see explanation
Step-by-step explanation:
Using the trigonometric identity
sec²x = 1 + tan²x
Consider the left side
- tan²x + sec²x
= - tan²x + 1 + tan²x ← collect like terms
= 1
= right side, thus proven
Word Problem (x and y intertcept)
A doll company can make a doll at a cost of P35 per doll. If the selling of the doll is x pesos and the number of dolls sold per month is 500 - x,
a. Express the monthly profit in pesos as a function of x.
b. If the selling price of the doll is P85, determine the monthly profit. Use the result in letter a.
Answers
Answer:
a. Monthly profit is
(x-35)(500-x)
b. given x = 85, the profit is 20,750 pesos
Step-by-step explanation:
a. The monthly profit is the selling price - cost price
The cost price for the number of dolls made will be ;
35(500-x)
The value of the amount we are selling all is;
x(500-x)
So the monthly profit would be;
x(500-x)-35(500-x)
= (x-35)(500-x)
b. Given the selling price we 85 , we want to calculate the monthly profit
From
what we have, x will be 85
Substituting in the equation we have for A above, we have that;
(85-35)(500-85)
= 50* 415 = 20,750 pesos
I = $60, P = $750, r = 4%, t = ?. Find the amount of time.
Answers
Answer:
you need to be more specific.
Use the word virtue in a complete and strong sentence about someone you know.
Answers
Answer:
Step-by-step explanation:
My friend is a virtue because he is very upright. The patience that he has is very virtuous, and during school, he shows high moral standards and you can see it by the grades he gets.
Hope this helped a little!
Answer the following Question with Appropriate Explanation
Answers
Answer:
option is B)
Step-by-step explanation:
2x is a perfect square. but x is not a perfect square, this brings us to the conclusion tht prob x is 2.
2x = 4
hence correct option is B) 4x is a perfect square, as 4x4=16, which is a perfect square...
X
3
7
+
Simplify:X-3 X+4 x²+x-12
Rewrite the expression using the LCD:
Ox(x+4)-3(x-3)+7
(x-3)(x+4)
O x(x+4)+3(x – R+7
(x-3)(x+4)
x(x+4)+3(x-3) -7
(x-3)(x+4)
Answers
Answer: it’s a
Step-by-step explanation:
Answer: A for the first part and B for the second part
Step-by-step explanation:
PLEASE help ASAP!
Thank you!
Answers
Answer:
I thinks its B.14 units
Step-by-step explanation:
HOPE THIS HELPS IF NOT THEN SORRY
Answer:
the answer is 10
Step-by-step explanation:
the formula for distance is square root ( of the entire equation),, (x1-x2) squared +(y1-y2) squared so (-5-3)+(4-(-2)) ( both answers squared) equals -8 squared plus 6 squared and you get 64+36 which is 100 and the square root of 100 is 10. Hope that wasn't confusing
Peter made 19 1/8 cups of chili for the football game. If 14 3/4 cups of the chili were eaten how many cups of chili did peter have left over
Answers
Answer:
Step-by-step explanation:
4(4y - 3x) Need help
Answers
Answer:
4(4y - 3x)
16y -12x
you can't simplify this any further
I hope this helps
Find cos 0 for 0 = 30°
The 0 has a weird line through it
Answers
Answer: [tex]\frac{\sqrt{3}}{2}[/tex]
======================================================
Explanation:
The symbol [tex]\theta[/tex] is the greek letter theta. It's often used in trigonometry for angles.
In this case, [tex]\theta = 30^{\circ}[/tex]
So,
[tex]\cos(\theta) = \cos(30^{\circ}) = \frac{\sqrt{3}}{2}[/tex]
You'll need to use a reference sheet or the unit circle to determine the value of cos(30). Or you could use a 30-60-90 triangle template.
When using the unit circle, look in the upper right quadrant (which is quadrant 1). The angle 30 degrees is in this quadrant. Locate the terminal point and note how [tex]\frac{\sqrt{3}}{2}[/tex] is the x coordinate of the terminal point. This is due to [tex]x = \cos(\theta)[/tex]
Side note: [tex]\frac{\sqrt{3}}{2} \approx 0.866025[/tex]
What does sas stand for
Answers
Answer: Statistical Analysis System
Step-by-step explanation:
Please help!!! I’ll mark brainliest!! (Look at picture)
Answers
Answer:
[tex]\huge\boxed{\sf (x*4) + (x*5) = 54}[/tex]
Step-by-step explanation:
[tex]\sf x(4+5) = 54\\\\Applying \ distributive \ property\\\\(x*4) + (x*5) = 54\\\\Distributive\ Property\ is:\\\\A(B+C) = (A*B)+(A*C)\\\\\rule[225]{225}{2}[/tex]
Hope this helped!
~AH1807Write the rule for the linear function. Remember a function rule is written using f(x)
Answers
Answer:
The rule for the linear function wil be:
f(x) = xStep-by-step explanation:
We know that linear function can be represented using the slope-intercept formula
y = mx+b
where m is the slope and b is the y-intercept
Given the function is linear
Taking two points
(0, 0)(1, 1)Finding the slope between (0, 0) and (1, 1)
Using the formula
Slope = m = [y₂ - y₁] / [x₂ - x₁]
= [1 - 0] / [1 - 0]
= 1 / 1
= 1
Thus, the slope of the line = m = 1
We know that the value of y-intercept can be determined by setting x = 0, and determining the corresponding value of y.
From the graph, it is clear
at x = 0, y = 0
Thus, the y-intercept b = 0
Now, substituting m = 1 and b = 0 in the slope-intercept form of linear function
y = mx+b
y = (1)x + 0 ∵ m = 1, b = 0
y = x
f(x) = x ∵
Therefore, the rule for the linear function wil be:
f(x) = xA rectangular soccer field may have a width between 50 and 100 yards and a length
between 100 and 130 yards. One particular soccer field has a perimeter of 320 yards. Its
length measures 40 yards more than its width. What are the dimensions of this field?
The length is___yards.
The width is____yards.
Answers
Answer:
[tex]perimeter = 2(l + w) \\ but \: length = 40 \times width \\ perimeter = 2(40w + w) \\ p = 2(41w) \\ 320 = 2(41w) \\ w = \frac{320}{(2 \times 41)} \\ w = 3.902 \: yards \\ length = 40 \times w \\ = 40 \times 3.902 \\ l = 156.08 \: yards[/tex]
9514 1404 393
Answer:
length: 100 yardswidth: 60 yardsStep-by-step explanation:
Let w represent the width of the field. Then the length is (w+40), and the perimeter is ...
P = 2(L +W)
320 = 2((w+40) +w) . . . . fill in values
160 = 2W +40 . . . . . . . . . divide by 2, collect terms
80 = w +20 . . . . . . . . . . . divide by 2
60 = w . . . . . . . . subtract 20
w+40 = 100
The length is 100 yards; the width is 60 yards.