Answers
Answer:
The last option is the correct answer: [tex]r=\sqrt{\frac{A}{\pi}}[/tex]
Step-by-step explanation:
The area of a circle is given by :
[tex]A=\pi r^{2}[/tex] ; where r is the radius.
Now, lets find the radius.
[tex]\frac{A}{\pi } =r^{2}[/tex]
[tex]\sqrt{\frac{A}{\pi}} =r[/tex]
or [tex]r=\sqrt{\frac{A}{\pi}}[/tex]
Therefore, the last option is the correct answer.
Related Questions
solve this problem for N
N - (2 x 7) = 7
A.20
B.21
C.22
Answers
n-14=7
n+(14-14)=7+14
n=21
B. 21
Batteries of each type have lifetimes before recharge that can be assumed independent and Normally distributed. The mean and standard deviation of the lifetimes of the Duxcell batteries are 10 and 2 minutes respectively, the mean and standard deviation for the Infinitycell batteries are 14 and 3 minutes respectively.
Part a) What is the probability that a Duxcell battery will last longer than an Infinitycell battery?,
Answers
X = N(10, 2^2)
This means that:
mean is 10
variance is 2^2
Y = N(14, 3^2).
Asked:P(X > Y)
= P(X - Y > 0).
mean of Z = X - Y
= 10 - 14
variance = 2^2 + 3^2
Z = N(-4, 2^2 + 3^2).
Answer:
0.1562
Step-by-step explanation:
Step 1
In this tep we define the normal distributions in this problem. . The lifetime of the Duracell battery is a normal random variable X with parameters (10,4). The lifetime of the Infinitycell battery is a normal random variable with parameters (14.9).
Step 2
Define the new normal distribution formed by the combination of these two distributions. We define a random variable Z such that
,[tex]Z=X-Y.[/tex]
This will be a random variable with mean of [tex]\mu_x-\mu_y=10-14=-4[/tex]. This random variable will have a variance of [tex]\mu_z^2 =\mu_x^2+\mu_y^2=2^2+3^2=13[/tex]. The mean for this distribution is [tex]\mu_z=\sqrt{13}[/tex] . The random variable Z is normally distributed with parameters (-4,13)
Step 3
The next step is to calculate the probability that [tex]Z=X-Y>0[/tex]. For that we perform the calculation shown below. Along the way , we will use the method of transforming any normal distribution to the standard normal distribution.This transformation is [tex]z=\frac{x-u}{\sigma}[/tex].
[tex]P(Z>0)=P(X-Y>0)=P(Z>0)\\P(Z>0)=1-P(Z<0)\\P(Z>0)=1-P(\tfrac{0-(-4)}{\sqrt{13}})=1-P(\tfrac{4}{\sqrt{13}})\\P(Z>0)=1-0.8438\\P(Z>0)=0.1562[/tex]
a rabbit traveled 60 meters in 4 seconds. what is the unit rate? enter your answer into the box.
Answers
4 seconds
To find the unit rate, divide the distance by the time.
60/4=15meters per second= 15m/s
(03.07)
Graph with a line going through points zero comma two point five and four comma two point five.
Select the equation of the line that passes through the point (3, -1) and is parallel to the line on the graph.
y = -1
y = 3
y = x -1
y = 3x - 1 @ankitshaw
Answers
The slope of a line is m=(y2-y1)/(x2-x1 )
For the given line, x1=0, y1=2.5, x2=4, y2=2.5
Substituting the values in the formula,
m=(2.5-2.5)/(4-0)
m=0/4
m=0
Another line passes through (3,-1).
The line passes through (3,-1) is parallel to the line that passes through
(0,2.5) and (4,2.5).
Therefore the slopes of these two lines are equal.
m=0 and the line passes through (3,-1)
The formula is y-y1=m(x-x1)
X1=3 and y1=-1 m=0
Substituting these values in the formula,
y+1= 0(x-3)
y+1=0
y=-1
Use the table to determine the location with the greatest amount of rainfall and the location with the least amount of rainfall.
a.Using complete sentences, identify the location with the greatest amount of rainfall and the location with the least amount of rainfall, and express the ratio of their respective rainfalls as a whole number.
b. Describe what this ratio tells you.
Needles, California 0.1173 m
Mt. Waialeale, Hawai'i 11.4 m
Puako, Hawai’i 25.6 cm
Death Valley, California 57 mm
Mojave, California 160 mm
Hilo, Hawai'i 480 cm,
Answers
The other person answered Part A, so I'll answer Part B (at least I'll try my best to)
"With every 200 millimeters of rain in Waialeale, there will be 1 millimeter of rain in Death Valley." <- This is what I put for B
Sorry I'm late and if I'm wrong, good luck to everyone who sees this
Juan must choose a number between 61 and 107 that is a multiple of 3, 5, and 9. Write all the numbers that he could choose. If there is more than one number, separate them with commas.
Answers
62,63,64,65,66,67,68,69,70,....100,101,102,103,104,105,106
multiple of 5 are
65,70,75,80,85,90,95,100,105
multiples of 3 and 5
75,90,105
multiples of 3,5,9
90
Part 1.] Does the graph have even symmetry, odd symmetry, or neither?
Part 2.] Does the graph of y = sin x have even symmetry, odd symmetry, or neither?
Part 3.] Identify the transformations that describe the graph of an even function. Select all that apply.
reflection over the x-axis; reflection over the y-axis; reflection over the line y=x; 90° rotation around the origin; 180° rotation around the originAnswers
The graph has even symmetry. You can see that because it is symmetric with respect to the y-axis.
Functions that have even symmetry have the following property:
[tex]f(x)=f(-x)[/tex]
Part 2
To answer this we can simply check if the property we mentioned earlier holds for this function.
[tex]sin(\frac{\pi}{2})\ne sin(-\frac{\pi}{2})[/tex]
We can see that sine does not have even symmetry.
In fact, sine function has the following property:
[tex]sin(x)=-sin(x)[/tex]
This is called odd symetry.
Part 3
Take a look at the function that you attached in the picture. We know that function has even symmetry.
Reflection over x-axis and 180° rotation around the origin would give us -f(x). We would not end up with the same function, so these two are out.
90° rotation around the origin would mean we swapped x and y so that one is out too. Reflection over the line y=x is a property of functions that have an odd symmetry.
We are left with reflection around y-axis and, as mentioned before, this is the property of evenly symmetric functions.
if you reflect fgh across the y-axis, what will be the coordinates of the vertices of the image f'g'h'
Answers
The correct answer is b.
Hope this helps =)
Therefore option B is correct.
What is reflection of coordinates?When a point is reflected across the line y = x, the x-coordinates and y-coordinates change their place. Similarly, when a point is reflected across the line y = -x, the x-coordinates and y-coordinates change their place and are negated. Therefore, The reflection of the point (x, y) across the line y = x is (y, x).
We have,
Coordinate of the vertex of the ΔFGH is F( -2 , -1 ) , G( 2 , 2 ) and H( 4 , -3 )
According to the question,
We know that when a point ( x , y ) reflected over y-axis , the y-coordinate remains the same but the x-coordinate is transformed into its opposite .i.e., ( -x , y )
So, Image of the Vertex F( -2 , -1 ) = F'( -(-2) , -1 ) = F'( 2 , -1 )
Image of the Vertex G( 2 , 2 ) = G'( -2 , 2 )
Image of the Vertex H( 4 , -3 ) = H'( -4 , -3 )
Hence, the image of the vertexes is F'( 2 , -1 ) ,G'( -2 , 2 ) and H'( -4 , -3 )
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In a billiards game, Pete hits a ball that is 20in. from a wall. The ball travels 34 in. until it hits the wall and bounces to a position that is 16 in. from the wall.
What proportion can be used to find the distance, x, the ball traveled after it bounced off the wall to get to the ending position? CHECK ALL THAT APPLY!
Contact your teacher if you cannot see the diagram.
Question 1 options:
34/x = 20/16
16/20 = x/34
20/16 = x/34
20/16 = 34/x
Answers
we know that
(20/34)=16/x----------> 20*x=34*16
therefore
case 1)34/x = 20/16------> 34*16=x*20-------> this proportion is correct
case 2)16/20 = x/34------> 16*34=20*x-------> this proportion is correct
case 3)20/16 = x/34------> 20*34=16*x------> this proportion is not correct
case 4)20/16 = 34/x-------> 20*x=16*34-------> this proportion is correct
the answer is
the proportion can be used to find the distance x are
1)34/x = 20/16
2)16/20 = x/34
4)20/16 = 34/x
In the wrong hands, which of the following pieces of personal information would present the greatest risk for identity theft?
a.
address
b.
date of birth
c.
mother’s maiden name
d.
social security number
Answers
Answer:
D. social security number
Step-by-step explanation:
If a thief uses your your Social Security number for financial gain, you’re a victim of financial identity theft, criminal identity theft and also utility fraud.
Hope this will helpful.
Thank you.
Answer:
The correct option is D. social security number
Step-by-step explanation:
Address : Address cannot do much harm in the case of identity theft because other identity related information cannot be stolen by using the address.
So, this option is rejected.
Date of Birth : By using date of birth, no information about anyone's personal identity can be found. So this also cannot be related to identity theft.
So, this option is also rejected.
Mother's maiden name : This option is also not helpful in case of identity theft as this will not give any valuable information about someone's identity.
So, this option is also rejected.
Social security number : This number holds every personal information about a person so by using this complete identity of a person can be accessed and will present the greatest risk for identity theft.
Hence, The correct option is D. social security number
Which of the following is a trinomial with constant term?
Answers
The answer is F) x+4y+7
Simplify the expression 6√2/√3
B. 6√3
C. 2√3
D. 2√6
Answers
I am going to say your answer is D. 2 squar6
Round to the nearest hundredth 42.052
Answers
place value and digits:
4 = tens
2 = ones
0 = tenths
5 = hundredths
2 = thousandths
5 is in the hundredths place. To find if you round up or down, look at the digit to the right (thousandths in this case). It is a 2, which is less than 5, so you round down.
42.052 rounded to the nearest hundredth is 42.05
hope this helps
A solid has a circular base of radius 3. If every plane cross section perpendicular to the x-axis is an equilateral triangle, then its volume is:
A) 36
B) 12sqrt3
C)18sqrt3
D)24sqrt3
E)36sqrt3
The correct answer was E. Hoping someone could explain how that is.
Answers
Area of equilateral trianlge= (1/2) tiime a^2 square root 3
Equation for the base is = x^2+y^2 =9
at x ,
y= square root (9-x^2)
Side of triangle a= 2y
d times Volume = (1/2)a^2 square root 3 dx
d times V= (square root 3) (9-x^2) dx
V= square root 3 INT (9-x^2) dx
-3V= 2square root INT (9-x^2) dx
0
V= 2 square root 3 ( 9x-x^3/3)
V=2 square root 3 ( 27-9)
V= 36 square root 3
The answer is E.
Final answer:
The correct answer is E) 36sqrt(3), which is the volume of the described solid.
Explanation:
The student's question relates to finding the volume of a solid with unusual properties: it has a circular base and each plane cross-section perpendicular to the x-axis is an equilateral triangle. We will solve this problem using calculus, specifically by integrating the cross-sectional area along the x-axis.
The circular base of the solid defines the limits of integration, as it gives the diameter of the solid which, in turn, represents its height because the triangle is equilateral.
The radius given is 3, so the diameter (and hence the height of the solid) is 6.
The area of an equilateral triangle with side length s is (sqrt(3)/4) * s².
Since the side length of the triangles is equal to the diameter of the base, s is 6.
Therefore, the area of each cross-section is (sqrt(3)/4) * 6².
Now, to find the volume, we integrate the area of each cross-section along the height of the solid. Doing this gives us:
Volume = [tex]int_0^6 ((sqrt(3)/4) * 6^2) dx[/tex]= (sqrt(3)/4) * 36 * 6
= 6*6*6*sqrt(3) / 4
= 36sqrt(3)
Thus, the correct answer is E) 36sqrt(3), which is the volume of the described solid.
Maria wants to solve the following system using the elimination method:
4x + 9y = 59
x + y = 17
What number should the equation x + y = 17 be multiplied by to eliminate y?
4
−4
9
−9
Answers
The first equation has 9y.
To eliminate y, you need to add to it -9y.
The second equation has y. In order for the second equation to have -9y, you must multiply both sides of the second equation by -9.
Answer: D -9
Answer:
d
Step-by-step explanation:
i took the test
In "how to eat an ice cream cone" what does the author compare a melting ice cream cone to?
A runaway train
A machine gun
An atomic bomb
A hand grenade
Answers
I hope this helps:)
The author compare a melting ice cream cone to c. an atomic bomb.
In the essay ""How to Eat an Ice Cream Cone"" by Eugene Field, the author humorously compares the act of eating a melting ice cream cone to an atomic bomb.
The comparison is used to illustrate the chaotic and uncontrollable nature of trying to consume the ice cream before it melts, much like the destructive and rapid spread of an atomic bomb's explosion.
The author does not compare the melting ice cream cone to a runaway train, a machine gun, or a hand grenade, as these are not mentioned in the context of the essay.
The atomic bomb metaphor is used to emphasize the urgency and potential mess that can result from not eating the ice cream cone quickly enough, likening it to a catastrophic event that is difficult to contain once it starts."
a^9*8a^4 will mark as brainiest
Answers
Final result :
23a13
Step by step solution :
Step 1 :
Equation at the end of step 1 :
23a9 • a4
Step 2 :
Multiplying exponential expressions :
2.1 a9 multiplied by a4 = a(9 + 4) = a13
Final result :
23a13
Which is the most appropriate unit to measure the length of a slice of pizza?
Answers
which ordered pairs lie on the graph of the exponential function f(x)=-3^2x+5
select each correct answer
(3,724)
(-2,76)
(0,4)
(1,-4)
(-2,76)
Answers
It must be -3^(2x) + 5 In that case 0,4 works. That gives (-3)^0 = -1
-1 + 5 = 4
Unfortunately so does 1,-4 because -3^2 = - 9
- 9 + 5 = - 4
If (-2,76) is the answer, then the question should be written (-3)^(2x) - 5
Until the ambiguity is gone, we're not sure.
Answer:
1,-4
0,4
3,-724
Step-by-step explanation:
If a girl were going up a escalator how many steps would she have to take if it was not moving/10929120/99020363?utm_source=registration
Answers
72 steps!
It took me like an hour to figure it out but I got it!
Hope this helps!
@SamSamySamantha
y=2x-7
3x-4y=8 can someone solve for x and y ? What method is used ? elimination or subtraction ?
Answers
Valerie drives to and from work 5 days a week. The distance from her home to work is 6 miles. Over the course of 5 days, her car uses 14 of a tank of gas. Complete each sentence.
Answers
______________
The correct answer is
A round trip is 12 miles.
In a 5 day week she drives 60 miles (5 · 12)
Now we need to know how much gas she uses.
_________________
Hope this helps you
Jonathon's Yo-Yo has a string that is three and a half feet long. If Jonathon cuts six inches off of the string, how many feet will the string be
Answers
1/2 ft= 6 inches
=3 1/2 feet - 6 inches
substitute # of inches in for 1/2 foot
=3 ft 6 inches - 6 inches
=3 feet
ANSWER: 3 feet
Hope this helps! :)
Find the directional derivative of f(x, y, z) = xy2z3 at p(4, 1, 1) in the direction of q(0, −7, 9).
Answers
Answer:
Step-by-step explanation:
Since we have been given two points, P(4,1,1) and Q(0,-7,9), we can take the vector between the two toget the directional vector.
So, vector PQ = v = <-4, -8, 8>. Now, we find its UNIT VECTOR.
Since the UNIT VECTOR is just the vector divided by its magnitude, we get that the unit vector u = v / |v|
|v| = [tex]\sqrt{(-4)^2+(-8)^2+(8^2)}[/tex] = [tex]\sqrt{144}[/tex] = 12
Dividing each part of the original vector v by 12 gives us the unit vector.
Now, we take the partial derivatives, Fx, Fy, and Fz.
Fx = [tex]y^2z^3[/tex]
Fy = [tex]x(2y)z^3[/tex]
Fz = [tex]xy^2(3z^2)[/tex]
plugging in the original point P(4,1,1) gives us the following values for the gradient, which is just the vector of the partial derivatives.
ΔF = <1,8,12>
Now we just use the equation Duf = ΔF * u, where we take the DOT PRODUCT of the gradient at the given point with the direction unit vector.
This gives us <1,8,12> * <-1/3, -2/3, 2/3> = 7/3
The directional derivative of [tex]\( f \)[/tex] at point [tex]\( P(4, 1, 1) \)[/tex] in the direction of [tex]\( Q(0, -7, 9) \)[/tex] is [tex]\( \frac{7}{\sqrt{6}} \).[/tex]
To find the directional derivative of the function [tex]\(f(x, y, z) = xy^2z^3\)[/tex] at the point [tex]\(P(4, 1, 1)\)[/tex] in the direction of [tex]\(Q(0, -7, 9)\)[/tex], we'll use the formula for directional derivative:
[tex]\[ D_{\mathbf{u}} f(x, y, z) = \nabla f \cdot \mathbf{u} \][/tex]
where [tex]\( \nabla f \)[/tex] is the gradient vector of [tex]\( f \)[/tex] and [tex]\( \mathbf{u} \)[/tex] is the unit vector in the direction of [tex]\( Q \).[/tex]
First, let's find the gradient vector [tex]\( \nabla f \):[/tex]
[tex]\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \][/tex]
[tex]\[ = \left( y^2z^3, 2xyz^3, 3xy^2z^2 \right) \][/tex]
Now, evaluate [tex]\( \nabla f \)[/tex] at the point [tex]\( P(4, 1, 1) \):[/tex]
[tex]\[ \nabla f(P) = \left( (1)^2(1)^3, 2(4)(1)(1)^3, 3(4)(1)^2(1)^2 \right) \][/tex]
[tex]\[ = (1, 8, 12) \][/tex]
Next, find the unit vector in the direction of [tex]\( Q \):[/tex]
[tex]\[ \mathbf{u} = \frac{\mathbf{Q} - \mathbf{P}}{\|\mathbf{Q} - \mathbf{P}\|} \][/tex]
[tex]\[ = \frac{(0 - 4, -7 - 1, 9 - 1)}{\sqrt{(0 - 4)^2 + (-7 - 1)^2 + (9 - 1)^2}} \][/tex]
[tex]\[ = \frac{(-4, -8, 8)}{\sqrt{16 + 64 + 64}} \][/tex]
[tex]\[ = \frac{(-4, -8, 8)}{4\sqrt{6}} \][/tex]
[tex]\[ = \left( -\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \][/tex]
Now, calculate the dot product [tex]\( \nabla f \cdot \mathbf{u} \):[/tex]
[tex]\[ D_{\mathbf{u}} f(P) = \nabla f(P) \cdot \mathbf{u} \][/tex]
[tex]\[ = (1, 8, 12) \cdot \left( -\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \][/tex]
[tex]\[ = -\frac{1}{\sqrt{6}} + \left(-\frac{16}{\sqrt{6}}\right) + \frac{24}{\sqrt{6}} \][/tex]
[tex]\[ = \frac{7}{\sqrt{6}} \][/tex]
Therefore, the directional derivative of [tex]\( f \)[/tex] at point [tex]\( P(4, 1, 1) \)[/tex] in the direction of [tex]\( Q(0, -7, 9) \)[/tex] is [tex]\( \frac{7}{\sqrt{6}} \).[/tex]
A number cube is rolled 450 times. The number 3 comes up 67 times.
What is the experimental probability of rolling a 3? Write your answer as a fraction in simplest form.
Ok so I did this if I am right?
67/3 = 22.3
450/3 = 150
22.3/150 = ?
or
22/150 =
This where I get lost. I need a fraction. @whpalmer4 or @Taz_kapadia or @tcarroll010
Answers
Which unit would you use to measure the weight of a flea?
mg
g
ounces
kg
Answers
Quadrilateral ABCD is inscribed in this circle.
What is the measure of angle C?
Enter your answer in the box.
Answers
For the quadrilateral ABCD and the inscribed circle, the measure of angle c will be 62⁰.
We already know that the opposite angles of an inscribed quadrilateral are supplementary in nature.
Therefore, In this problem
m<D+m<B=180⁰
m<C+m<A=180⁰
First, we have to find the value of x
m<D+m<B=180⁰
then substitute the values to solve for x, we get
(x+20)⁰ + (3x)⁰ = 180⁰
4x = 180⁰ - 20⁰
x = 160/4
x = 40⁰
second, to find the measure of angle A
m<A = (2x+38)⁰
put the value of x from first step,
angle A = 2(40)⁰+38⁰=118⁰
third, to find the measure of angle C
m<C+118⁰=180⁰
m<C = 180⁰-118⁰
angle C = 62⁰
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Alexandra paid \$7$7dollar sign, 7 to park her car for 333 hours at the parking garage. The garage charges an hourly parking rate. Write an equation that shows the relationship between the number of hours parked (p)(p)left parenthesis, p, right parenthesis and the cost in dollars (
c.(
c.left parenthesis, c, right parenthesis.
Answers
c= a*p
where c= cost, a= rates per hour, p= hours of parking
Alexandra pays 7$ for 3 hours parking. So, a would be:
c= a*p
7$= a * 3 hr
a= 7$/3hr
Then the final equation would be:
c= (7/3) *p
Answer:
c=7/3p
Step-by-step explanation:
Let's find the constant of proportionality.
in the proportional relationship between p, the number of hours parked, and c, the cost in dollars, one constant of proportionality is the cost per hour. It is the number we multiply by the number of hours to get the total cost.
p×?=c
px?=c
?= [tex]c/p[/tex]
= 7/3
the constant of proportionality is 7/3 this means we can multiply 7/3 by the numbers of hours to get the total cost
Now, let's write the equation: total cost
total cost=cost per hour×number of hours
[tex]c=7/3p[/tex]
Dante is making a necklace with 18 rows of tiny beads in which the number of beads per row is given by the series 3 + 10 + 17 + 24 + ...
a. If you were to write this series in summation notation, give
i.the lower limit of the sum
ii.the upper limit of the sum
iii.the explicit formula of the sum
b. Find the total number of beads in the necklace. Explain your method for finding the total number of beads.
Show all your work.
Answers
Answer:
a)
i.the lower limit of the sum
: n=1
ii.the upper limit of the sum
: n=18
iii.the explicit formula of the sum
: [tex]a_n=7n-4[/tex]
b) 1,125 beads
Step-by-step explanation:
a) The number of beads per row of Dante's necklace is given by the series:
3 + 10 + 17 + 24 + ...
The first term of this series is [tex]a_1=3[/tex].
There is a constant difference of [tex]d=10-3=7[/tex].
To write this series in summation notation, we need to determine the explicit formula which is given by:
[tex]a_n=a_1+d(n-1)[/tex]
We plug in the values to get:
[tex]a_n=3+7(n-1)[/tex]
[tex]\implies a_n=3+7n-7[/tex]
[tex]\implies a_n=7n-4[/tex]
The summation notation is given by:
[tex]\sum^{18}_{n=1}(7n-4)[/tex]
b) The total number of beads in the necklace is given by the sum of the first 18 terms of the sequence.
This is given by [tex]S_n=\frac{n}{2}(2a_1+d(n-1))[/tex]
We substitute the values to obtain:
[tex]S_{18}=\frac{18}{2}(2\cdot3+7(18-1))[/tex]
[tex]S_{18}=9(6+119)[/tex]
[tex]S_{18}=9(125)[/tex]
[tex]S_{18}=1125[/tex]
Therefore the total number of beads is 1125
The arithmetic series 3, 10, 17, 24,... in which each term increases by 7 can be described in summation notation with a lower limit of 1, an upper limit of 18, and an explicit formula of 7n - 4. The total number of beads in Dante's 18-row necklace, computed using the arithmetic series sum formula, is 1125 beads.
Explanation:The series described here is an arithmetic series, where each term increases by the same amount, 7 in this case.
a. To express this series in summation notation, you must know:
The lower limit of the sum: In the series, the first term is 3, so the lower limit, often denoted as n, is 1.The upper limit of the sum: As there are 18 rows, n, or the number of terms in our sum, will be 18.The explicit formula for the sum: For an arithmetic series, the sum is given by the formula S = n/2 * (a + l) where n is the number of terms, a is the first term and l is the last term. However, we need the general term to express the series in summation notation, which is given by a + (n-1)d in an arithmetic progression. Here, a=3 and d, the difference is 7. So the general term is 3 + (n-1)*7 = 7n - 4.b. To find the total number of beads in the necklace, you use the formula for the sum of an arithmetic series. For the last term, l, we use the formula a+(n-1)d = 3+(18-1)*7 = 122. Hence, the total number of beads is S = 18/2 * (3 + 122) = 1125.
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If two angles of one triangle are congruent to two angles in another triangle, then what must be true of the third angles of the triangles?
Answers
Final answer:
If two angles of one triangle are congruent to two angles in another triangle, the third angles must also be congruent since the sum of angles in any triangle must equal 180 degrees. This principle is supported by geometric theorems about the properties of triangles and the requirements for their congruence and similarity.
Explanation:
If two angles of one triangle are congruent to two angles in another triangle, then the third angles in both triangles must also be congruent. This is because the sum of the angles in any triangle must equal 180 degrees or two right angles. Since two angles are congruent in both triangles, whatever is left from the 180 degrees after subtracting these two angles must be what is left for the third angle, making them congruent by necessity.
There are several related theorems in geometry that confirm this. Theorem 11, also known as the Side-Angle-Side (SAS) Postulate, states that if one side and the two adjacent angles of two triangles are congruent, then the triangles are congruent. While this theorem refers mainly to the congruency of triangles, it supports our understanding that if two angles are congruent, the third must be as well to satisfy the congruence of all three angles.
Furthermore, in the context of similar triangles, when two triangles have congruent angles, they are similar, meaning all their angles are congruent, and their sides are in proportion. However, if we only consider the angles, as in the original question, knowing that two angles are congruent across two triangles is enough to deduct that the third angles will be congruent, regardless of side lengths or overall similarity.