1. Create a circle. Show and explain the difference between the following:
a. secant line and tangent line
b. inscribed angle and central angle
2. Describe the steps necessary to construct an inscribed circle in a triangle and contrast this with the steps you take to construct a circumscribed circle.
3. Demonstrate on the whiteboard how to find the center and radius of a circle using an equation. Be sure to provide a unique example of your own that shows how to find both of these.
4. A classmate is having difficulty finding the measures of
5. The world’s largest round pizza, which also happened to be gluten-free, was set in December 2012 by five Italian chefs. The diameter of this pizza was 131 feet. If this pizza was cut into 50 slices, what would be the area of each slice and the length of the crust for the slice? Explain what concepts of a circle these calculations related to.
Answers
a.
A secant line is a line which intersects the circle at 2 different points.
A tangent line is a line which has only one point in common with the circle.
Check picture 1: The orange line s is a secant line, the blue line t is a tangent line.
b.
An inscribed angle is an angle formed by using 3 points of a circle.
The main property of an inscribed angle is that its measure is half of the measure of the arc it intercepts.
Check picture 2: If [tex]m(\angle KML)=\beta[/tex], then the measure of arc KL is [tex]2 \beta [/tex].
A central angle is an angle whose vertex is the center of the circle, and the 2 endpoints of the rays are points of the circle.
The main property is: the measure of the central angle is equal to the measure of the arc it intercepts.
Check picture 2
2.
To construct the inscribed circle of a triangle, we first draw the 3 interior angle bisectors of the triangle.
They meet at a common point called the incenter, which is the center of the inscribed circle.
We open the compass, from the incenter, so that it touches one of the sides at only one point. We then draw the circle. (picture 3)
To draw the circumscribed circle, we first find the midpoints of each side. We then draw perpendicular segments through these (the midpoints.) They meet at one common point, which is the circumcenter: the center of the circumscribed circle.
We open the compass from the circumcenter to one of the vertices of the triangle. We draw the circle, and see that it circumscribes the triangle.
(picture 4)
3.
Given an equation of a circle: [tex]x^2-2x+y^2+6y+6=0[/tex].
To determine the center and the radius of the equation we must write the above equation in the form :
[tex](x-a)^2+(y-b)^2=r^2[/tex].
Then, (a, b) is the center, and r is the radius of this circle. We do this process by completing the square.
Note that [tex]x^2-2x[/tex] becomes a perfect square by adding 1, and
[tex]y^2+6y[/tex] becomes a perfect square by adding 9.
Thus we have:
[tex]x^2-2x+y^2+6y+6=0\\\\(x^2-2x+1)+(y^2+6y+9)-4=0\\\\(x-1)^2+(y+3)^2=2^2[/tex]
Thus, the center is (1, -3), and the radius is 2.
4. Not complete
5.
The radius of the pizza is [tex]\displaystyle{ \frac{131}{2}ft=65.5ft [/tex].
The surface of a circle with radius r is given by the formula [tex]\displaystyle{ A=\pi r^2[/tex],
and the circumference is given by the formula C=2πr.
Thus, the area of the whole pizza is given by [tex]\displaystyle{ A=\pi r^2= \pi\cdot65.5^2=4290.25 \pi [/tex] (square ft).
Each of the 50 slices, has an area of [tex]\displaystyle{ \frac{4290.25 \pi}{50} =85.805 \pi [/tex] (square ft)
Notice that the perimeter (the crust) of a slice is made of 2 radii, and the arc-like part.
The arc is 1/50 of the circumference, so it is [tex] \displaystyle{\frac{2 \pi r}{50} = \frac{2\cdot65.5\cdot \pi }{50}= 2.62 \pi [/tex].
So the perimeter of one slice is 65.5+65.5+2.62π=131+2.62π
Final answer:
The differences between secant and tangent lines, and between inscribed and central angles, involve the number of intersection points with the circle and the position of the vertex, respectively. Constructing an inscribed circle involves finding the incenter, while a circumscribed circle requires finding the circumcenter. To find the center and radius of a circle using its equation, we use the standard form (x - h)² + (y - k)² = r².
Explanation:
Differences Between Secant Lines, Tangent Lines, Inscribed Angles, and Central Angles
A secant line is a line that intersects a circle at two points, while a tangent line touches the circle at exactly one point. An inscribed angle is an angle formed by two chords in a circle that meet at one point in the circle, whereas a central angle is an angle whose apex (vertex) is the center of the circle.
Constructing Inscribed and Circumscribed Circles in a Triangle
To construct an inscribed circle (incircle) of a triangle, draw the angle bisectors of the triangle's angles to find the incenter, and then use this point to draw the circle that touches all three sides of the triangle. For a circumscribed circle (circumcircle), draw the perpendicular bisectors of the triangle's sides and the point where they intersect will be the circumcenter; use this as the center to draw the circle that passes through all three vertices of the triangle.
Finding the Center and Radius of a Circle Using an Equation
Given a circle's equation in the standard form (x - h)² + (y - k)² = r², the center is at point (h, k), and the radius is the square root of r². For example, if we have the equation (x - 3)² + (y + 2)² = 16, the center is (3, -2) and the radius is 4 units.
Area and Length of the Crust of a Circular Pizza Slice
For the world's largest round pizza with a diameter of 131 feet, the radius would be 65.5 feet. The area of each slice is a fraction of the whole pizza's area, which can be found using the formula A = πR², and the length of the crust for the slice corresponds to the arc length of that slice.
Area of whole pizza = 3.14 * 65.5² ≈ 13478.2 ft²
Area of each slice = total area /50 = 13478.2/50 ≈ 269.5 ft²
Length of crust = 2πR * (slice area/total area)
= 2 * 3.14 * (1/50)
≈ 0.125 ft
Related Questions
Jessica can paint a room in 12 hours. If Mary helps, it takes them 10 hours. Without help how long would it take for mary to finish this job?
Answers
Find the domain and range of the following function ƒ(x) = 3|x + 7| - 2?
A.Domain: [-2,∞) Range: (-∞,∞)
B.Domain: (-2,∞) Range: (-8∞,∞)
C.Domain: (-∞,∞) Range: [-2,∞)
D.Domain: (-∞,∞) Range: (-2,∞)
Answers
Answer:
D.Domain: (-∞,∞) Range: (-2,∞)
is the right answer.
Step-by-step explanation:
Given is a function as
[tex]f(x) = 3|x+7|-2[/tex]
This is an absolute value function hence the minimum value of
|x+2|=0
Hence the range cannot be below -2
For maximum no limit
[tex]Range = [-2,\infty)[/tex]
Domain can be any real value
Hence option d is correct answer.
Answer:
A) domain: (-∞, ∞) range: (-∞, ∞)
Step-by-step explanation:
HELP PLEASE!
Which of the following possibilities will form a triangle?
Side = 10 cm, side = 5 cm, side = 6 cm
Side = 10 cm, side = 6 cm, side = 3 cm
Side = 11 cm, side = 5 cm, side = 6 cm
Side = 11 cm, side = 5 cm, side = 5 cm
Answers
Answer:
d
Step-by-step explanation:
Side = 11 cm, side = 5 cm, side = 5 cm
Find area of a rectangle
Answers
Area = l*w = 72
L = 72/w
Perimeter = 2l+ 2w
44 = 2(72/w) +2w
22 = 72/w +w
W^2-22w +72 =0
(w-18) (w-4) = 0
W = 18
L = 4
One inlet pipe can fill an empty pool in 4 hours, and a drain can empty the pool in 8 hours. how long will it take the pipe to fill the pool if the drain is left open?
Answers
Given that one inlet pipe can fill an empty pool in 4 hours. The outlet pipe can empty the pool in 8 hours. We know that rate x time = work done.
Rate = work done / time
Rate of inlet pipe = ¼ hours.
Rate of outlet pipe = 1/8 hours
Here 1 represent the work completed.
Let x be the time taken to fill the pool by the inlet pipe if the drain is left.
Work done by inlet pipe – work done by outlet pipe = 1 work completed.
x/4 – x/8 = 1.
To solve the rational equation, first we clear the fractions by multiplying it with LCD (4,8)=8 to each item.
8(x/4) – 8(x/8) = 8(1)
2x – x = 8
X = 8
Time taken for the pipe to fill the pool if the drain is left open = 8 hours
3x+1<2x-3 what is x
Answers
3x less 2x-4
x less -4
how to solve log base 5 of x + log base 5 of (4x-1)
Answers
As the bases are the same the number doesn't matter
Log (a) + log(b) =log (ab)
Log(4x^2-x)
(x )(4x-1)
x=0.25 or x=0
Log(4×0.25^2-0.25) = log(0) = Infinity
Same with 0. So no answer
To solve for log base 5 of x + log base 5 of (4x-1), combine the two logarithms using the property that the sum of logarithms is equal to the logarithm of the product: log base 5 of x + log base 5 of (4x-1) = log base 5 of (x(4x-1)).
To solve the logarithmic expression log5(x) + log5(4x-1), we can combine the two logarithms using the property that the sum of logarithms is equal to the logarithm of the product. This gives us:
log5(x) + log5(4x-1) = log5(x(4x-1))
So the solution to the expression is log5(x(4x-1)).
If a barber is capable of cutting the hair of 35 people per day and he works 7 days per week, how many haircuts could he give during the months of april, may, and june?
Answers
April=1,225
May=1,225
All June April and May is 3,675 haircuts
What is the probability that the next flood will occur four years from now?
Answers
How many feet does it take to go from 0 to 60?
Answers
A sample of water is heated from 10.0 (degrees Celsius) to 15.0 (degrees Celsius) by the addition of 125 J of heat. What is the mass of the water?
(I understand what the question is asking me and I know that I have to use the formula q=m•c•DT but I don’t understand how to plug in a specific heat capacity if they don’t give it to me. Please help!!)
Answers
△RST is mapped to △R′S′T′ using the rule (x, y)→(x, −y) followed by (x, y)→(3x, 3y) .
Which statement correctly describes the relationship between △RST and △R′S′T′ ?
△RST is not congruent to △R′S′T′ because the rules do not represent a sequence of rigid motions.
△RST is congruent to △R′S′T′ because the rules represent a reflection followed by a rotation, which is a sequence of rigid motions.
△RST is congruent to △R′S′T′ because the rules represent a reflection followed by a translation, which is a sequence of rigid motions.
△RST is congruent to △R′S′T′ because the rules represent a rotation followed by a translation, which is a sequence of rigid motions.
Answers
(y->-y) followed by a dilation (x->3x, y->3y). Reflections and rotations ARE congruent because rigid motions, but dilations/contractions change size, are NOT rigid, and therefore NOT congruent.
What is the value of x? x+13=9
Enter your answer in the box in simplest form.
x =
Answers
x=6
that answer is 6
A rectangle with an area of 120 in 2 has a length 8 inches longer than two times its width. What is the width of the rectangle?
Answers
width = x
length = 2x+8
area = l x w
x(2x+8)=120
2x^2+8x−120=0
x^2+4−60=0
(x+10)(x−6)=0
x=−10 and x=6
width has to be a positive number
Width = 6
inches.
Convert the base-6 number 12345 to decimal
Answers
A decimal is a way of expressing numbers that are not whole numbers, but rather include a fractional or decimal part. The decimal equivalent of the base-6 number 12345 is 1415.
To convert a base-6 number to decimal, we multiply each digit by its corresponding weight and add the products together. The weights are 6 to the power of the digit's position, starting from the rightmost digit.
For example, to convert the base-6 number 12345 to decimal, we would do the following:
(1 * 6^4) + (2 * 6^3) + (3 * 6^2) + (4 * 6^1) + (5 * 6^0)
= 1296 + 72 + 18 + 24 + 5
= 1415
Therefore, the decimal equivalent of the base-6 number 12345 is 1415.
For such more question on decimal
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BD =4x + 5 and BF =9x
How to slove this problem
Answers
The centroid theorem, the value of x is: 2.5.
According to the centroid theorem, the centroid of a triangle is two-thirds of the distance from each midpoint to each vertex.
Given:
BD = 4x + 5
BF = 9x
Thus:
BD = 2/3(BF)
Replace 4x + 5 = 2/3(9x) with
4x + 5 = 6x
4x - 6x = -5
-2x = -5
x = 5/2
x = 2.5
As a result of the centroid theorem, the value of x is: 2.5.
Complete question:
Point D is the centroid of △ABC. Use the given information to find the value of x.
BD=4x+5 and BF=9x
x=
A collection of dimes and nickels is worth $5.60. The ratio of dimes to nickels is 3:2. How many dimes and how many nickels are there?
Answers
$5.60 is just 560 pennies
3 dimes is just 30 pennies
2 nickels is just 10 pennies
so, the ratio from dimes to nickels is a ratio of 30:10 pennies, or 3:1 then, for a total amount of 560 pennies.
[tex]\bf \cfrac{dimes}{nickels}\qquad 3:2\qquad\cfrac{\stackrel{pennies}{30}}{\stackrel{pennies}{10}}\implies \cfrac{30}{10}\implies \cfrac{3}{1}\implies 3:1~ratio \\\\\\ \cfrac{\stackrel{dimes}{pennies}}{\stackrel{nickels}{pennies}}\qquad 3:1\qquad \cfrac{3}{1}\implies \cfrac{3\cdot \frac{560}{3+1}}{1\cdot \frac{560}{3+1}}\implies \cfrac{3\cdot 140}{1\cdot 140}\implies \cfrac{420}{140}\\\\ -------------------------------[/tex]
[tex]\bf \textit{now, how many dimes in 420 pennies?}\quad \cfrac{420}{10}\implies 42~dimes \\\\\\ \textit{and how many nickels in 140 pennies?}\quad \cfrac{140}{5}\implies 28~nickels\\\\ -------------------------------\\\\ 42~dimes + 28~nickels\implies \$5.60[/tex]
There are 42 dimes and 14 nickels there if the collection of dimes and nickels is worth $5.60
First, we need to convert dimes and nickel to pennies to have:
1 dime = 10 pennies
1 nickel = 5 pennies
Since the ratio of dimes to nickels is 3:2, this shows that we have 30 dimes and 10 pennies
Total pennies = 30 + 10 = 40 pennies
Since the total collection of dimes and nickels is worth $5.60, this means that the total collections in pennies are 560 pennies ($1 is 100 pennies)
Number of pennies for the ratio of 30 = 30/40 * 560
Number of pennies for the ratio of 30 = 420
Number of dimes = 420/10 = 42 dimes
Number of pennies for the ratio of 10 = 10/40 * 560
Number of pennies for the ratio of 10 = 140
Number of nickel = 140/10 = 14 nickels
Hence there are 42 dimes and 14 nickels there if the collection of dimes and nickels is worth $5.60
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An experimenter is randomly sampling 3 objects in order from among 48 objects. what is the total number of samples in the sample space?
Answers
Thanks (:
There are 15 boys and 20 girls in ms.smith's algebra class. what is the ratio of boys to girls
Answers
Write y-6x-9=0 in function notation
Answers
y - 6x - 9 = 0
y = 6x + 9
now, to put it in function notation, sub in f(x) for y
f(x) = 6x + 9 <== function notation
hope this helps:D
plz give brainiest
Deangelo has $7 worth of dimes and quarters in a jar. He has 7 more quarters than dimes. How many of each coin does he have? Enter your answers in the boxes. _dimes and _ quarters
Answers
q = d + 7
0.10d + 0.25(d + 7) = 7
0.10d + 0.25d + 1.75 = 7
0.35d = 7 - 1.75
0.35d = 5.25
d = 5.25 / 0.35
d = 15 <=== there are 15 dimes
q = d + 7
q = 15 + 7
q = 22 <== there are 22 quarters
Answer:
There are 15 dimes and 22 quarters in the jar.
Step-by-step explanation:
Deangelo has $7 worth of dimes and quarters in a jar.
Let he has d number of dimes and q number of quarters in a jar.
0.10d + 0.25q = 7 -------(1)
He has 7 more quarters than dimes.
q = d + 7 -------(2)
We replace the value of y from equation 2 in the equation 1
0.10d + 0.25(d + 7) = 7
0.10d + 0.25d + 1.75 = 7
0.35d = 7 - 1.75
0.35d = 5.25
d = [tex]\frac{5.25}{0.35}=15[/tex] dimes
Now we put this value in the equation number 2
q = 15 + 7 = 22
Therefore, number of dimes and quarters are 15 and 22 respectively.
Which statement is always true for any parallelogram?
a. all four angles are right angles
b. the diagonals bisect each other
c. all four sides are congruent
d. the diagonals are congruent
Answers
the diagonals bisect each other- correct
all four sides are congruent - wrong that would be a Rhombus
the diagonals are congruent - wrong, this is the case in a rectangle, but in a parallelogram, because of the angle one side has to be shorter than the other
so b. the diagonals bisect each other is always true and therefore the solution
The statement that is always true for any parallelogram is that the diagonals bisect each other.
Explanation:The statement that is always true for any parallelogram is b. "the diagonals bisect each other". To bisect means to divide into two equal parts. In a parallelogram, the diagonals (the line segments connecting opposite corners) will always divide each other into two equal parts. This can be proven by using properties of parallel lines and alternate interior angles.
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Which of the following statements must be true about this diagram? Check all that apply.
A. m1 + m2 = m4
B. m1 + m2 = m3
C. m4 is greater than m2
D. m4 is greater than m1
E. m3 is greater than m2
F. m2 + m3 = m4
Answers
how to determine if a number is prime or composite
Answers
Answer:
Look at how many factors it has.
If it has 2, it's prime.
If it has 3 or more, it's composite.
Step-by-step explanation:
I hope it helps!
Which of the following expressions is equal to 5 - 2m?
7 - 2(m + 1)
7 - (2m - 5)
7 - 2(m - 2)
Answers
The correlation coefficient (r) between the number of miles driven x and the number of gallons of gasoline used y is 0.925
What percent of the variation in the number of gallons of gasoline can be explained by differences in the number of miles driven?
7.5%
14.4%
85.6%
92.5%
Answers
Answer:
OptionC
Step-by-step explanation:
Given that the correlation coefficient (r) between the number of miles driven x and the number of gallons of gasoline used y is 0.925
Coefficient of determination = square of r
= 0.8556
=85.6%
Hence the percent of the variation in the number of gallons of gasoline can be explained by differences in the number of miles driven is 85.6%
Option C
The sum of two numbers is 28 and their difference is 4. What are the numbers?
Answers
x-y=4----(b)
(a)-(b)
2y=24
y=24/2=12----(c)
use (c) in (b)
x=4+12=16
All the best
Is the mass of one pound of lead greater than less than or equal to one pound of feathers?
Answers
Complete the pattern 5, 10, 30, 60, 180
Answers
5*2=10
10*3=30
30*2=60
60*3=180
180*2=360
360*3=1,080
Suppose a college student pays $500 for tuition fees. However, she also has to pay $350 for her textbooks (ouch!). What percent of her total education costs does she pay for her books?
Answer: Her books represent
percent (as a decimal) of her total costs.
Write your answer to the nearest tenth of a percent, but in decimal form. For example, if you saved 34.67%, you would round to 34.7% and enter .347 as your answer.
Answers
Textbook costs = $350
Total cost = 500 + 350 = $850
Percent of costs due to textbooks = 100*(350/850) = 41.1765%
Her books represent 41.2% of her total costs.
Answer: 0.412