Prove that the complete integral of the equation (px + qy − z)2 = 1 + p2 + q2 is √
ax+by+cz= a2+b2+c2.
Answers
The complete integral of the equation (px + qy − z)2 = 1 + p2 + q2 is √
ax+by+cz= a2+b2+c2.
We are given the equation (px + qy − z)2 = 1 + p2 + q2 and we want to find its complete integral.
To start, we will rewrite the given equation as (px + qy − z)2 - (p2 + q2) = 1. Notice that this is the equation of a hyperboloid of two sheets centered at the origin. We can rewrite this equation in matrix form as:
[x, y, z] [p, q, -1] [p, q, -1]T [x, y, z]T = 1 + p2 + q2
where T denotes the transpose of a matrix.
Let A = [p, q, -1]. Then, the above equation can be written as:
A T x · A T x = 1 + ||A||2
where ||A||2 is the square of the Euclidean norm of A. Letting a = p/||A||, b = q/||A||, and c = -1/||A||, we have:
a2 + b2 + c2 = (p2 + q2 + 1)/||A||2 = 1/||A||2
Multiplying both sides of the equation by ||A||2, we obtain:
a2 ||A||2 + b2 ||A||2 + c2 ||A||2 = 1
Substituting x = (a ||A|| x, b ||A|| y, c ||A|| z) into the equation A T x · A T x = 1 + ||A||2, we obtain:
(ax)2 + (by)2 + (cz)2 = a2 ||A||2 + b2 ||A||2 + c2 ||A||2 = 1
Therefore, the complete integral of the equation (px + qy − z)2 = 1 + p2 + q2 is given by:
√(a2 x2 + b2 y2 + c2 z2) = √(ax + by + cz)2 = √(a2 + b2 + c2)(x2 + y2 + z2)
Thus, the complete integral is √(ax + by + cz) = √(a2 + b2 + c2).
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Related Questions
Find the area to this figure please help
Answers
Answer:
the area of the circle is 25π or about 78.54 yards
Step-by-step explanation:
Area of circle=πr^2, r being radius
r=5
5^2=25
25π≈78.54 yards
what expressions are equivalent to 3 (9p+2r)
Answers
Answer:
greetings
27p+6r, is an expression which is equivalent to 3 (9p+2r).
A baseball bat strikes a ball with an average force of 2.0 x 104 Newtons. if the bat stays in contact with the ball for a distance of 5.0 x 10-3 meter, what kinetic energy will the ball acquire from the bat?1.0 x 102 J2.0 x 102 J2.5 x 101 J4.0 x 102 J
Answers
A baseball bat strikes a ball with an average force of 2.0 x 104 Newtons. if the bat stays in contact with the ball for a distance of 5.0 x 10-3 meter , then the kinetic energy will be 2× 10^2 J
kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes.
In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is 1/2 mv²
Favg= -1/2 MV2
Favg = -1 2 ( 5.0 x 10-3)^2
Favg = - 1/2 ×25 × 10 ^-6
Favg = 2× 10^2 J
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A standing wave can be mathematically expressed as y(x,t) = Asin(kx)sin(wt)
A = max transverse displacement (amplitude), k = wave number, w = angular frequency, t = time.
At time t=0, what is the displacement of the string y(x,0)?
Express your answer in terms of A, k, and other introduced quantities.
Answers
The mathematical expression y(x,t) = Asin(kx)sin(wt) provides a way to describe the behavior of a standing wave in terms of its amplitude, frequency, and location along the string.
At time t=0,
the standing wave can be mathematically expressed as
y(x,0) = Asin(kx)sin(w*0) = Asin(kx)sin(0) = 0.
This means that the displacement of the string is zero at time t=0.
However, it is important to note that this does not mean that the string is not moving at all. Rather, it means that the string is in a state of equilibrium at time t=0, with the maximum transverse displacement being A.
As time progresses, the standing wave will oscillate between the maximum positive and negative transverse displacement values, creating a pattern of nodes (points of zero displacements) and antinodes (points of maximum displacement).
The wave number k and angular frequency w are both constants that are dependent on the physical properties of the string and the conditions under which the wave is being produced.
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What is the equation in slope-intercept form of a line that is perpendicular to y=2x+2 and passes through the point (4,3)?
(A) y=−12x+5
(B) y=−2x+2
(C) y=2x+3
(D) y=2x+5
Answers
Answer:
y = - \(\frac{1}{2}\) x + 5
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 2x + 2 ← is in slope- intercept form
with slope m = 2
Given a line with slope m then the slope of a line perpendicular to it is
\(m_{perpendicular}\) = - \(\frac{1}{m}\) = - \(\frac{1}{2}\) , thus
y = - \(\frac{1}{2}\) x + c ← is the partial equation
To find c substitute (4, 3) into the partial equation
3 = - 2 + c ⇒ c = 3 + 2 = 5
y = - \(\frac{1}{2}\) x + 5 ← equation of perpendicular line
Help me!! DUE IN 30 MINUTES!!!!!
Answers
Answer:
9 adults
Step-by-step explanation:
What is the exact value of sin 120° ? Use a double-angle identity.
Answers
Answer:
\(\frac{\sqrt{3}}{2}\)
Step-by-step explanation:
Sine is positive in the second quadrant, so:
\(\sin \frac{240^{\circ}}{2}=\sqrt{\frac{1-\cos 240^{\circ}}{2}} \\ \\ =\sqrt{\frac{1.5}{2}} \\ \\ =\sqrt{3/4} \\ \\ =\frac{\sqrt{3}}{2}\)
A six-sided dice is rolled. Four points is scored if the roll comes up with 5 or 6 , one point if it comes up 1,2,3 or 4 . If x is the point reward, what is the variance of X. Give you answer-in the form a bc
Answers
The variance of X is -34/27.
The random variable X has two possible outcomes:
1 with probability of 2/6 = 1/3 or 4 with probability of 4/6 = 2/3.So, the expected value of X is:
E(X) = 1(1/3) + 4(2/3) = 3(2/3) = 11/3.
The squared deviation from the mean of a random variable is referred to as variance in probability theory and statistics. The square of the standard deviation is another common way to express variation. Variance is a measure of dispersion, or how far apart from the mean a group of data are from one another.
Now we can compute the variance of X by using the following formula:
Var(X) = E(X²) - [E(X)]².
The expected value of X² is:E(X²) = 1²(1/3) + 4²(2/3) = 29/3.
So,Var(X) = E(X²) - [E(X)]²= 29/3 - (11/3)²= 29/3 - 121/9= (87 - 121) / 27= - 34 / 27.
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least to greatest 4/5,-1.4, 0.75, -21/3,21%
Answers
Answer:
-21/3 < -1/4 < 21% < 0.75 < 4/5
Step-by-step explanation:
how many different collections of 40 coins can be chosen if there are at least 40 of each kind of coin?
Answers
Using the combination formula, there are approximately 2.77 x 10^21 different collections of 40 coins that can be chosen if there are at least 40 of each kind of coin.
To solve this problem, we can use a combination formula. We have to choose 40 coins from a set of coins where there are at least 40 of each kind.
Let's assume that we have four types of coins - quarters, dimes, nickels, and pennies. Since we have at least 40 of each type, we have a total of 160 coins.
Now, we have to choose 40 coins from these 160 coins. This can be calculated using the combination formula:
nCr = n! / (r! * (n-r)!)
where n is the total number of coins (160) and r is the number of coins we want to choose (40).
Plugging in the values, we get:
160C40 = 160! / (40! * (160-40)!)
Simplifying this expression, we get:
160C40 = 2,771,436,503,423,282,717,080
So, there are approximately 2.77 x 10^21 different collections of 40 coins that can be chosen if there are at least 40 of each kind of coin.
In conclusion, using the combination formula, we can calculate the total number of collections that can be formed from a given set of coins. This formula is useful in solving a variety of problems related to probability and combinatorics.
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You need to build a model that predicts the volume of sales (Y) as a function of advertising (X). You believe that sales increase as advertising increase, but at a decreasing rate. Which of the following would be the general form of such model? (note: X^2 means X Square)
A. Y ^ = b0 + b1 X1 + b2 X2^2
B. Y ^ = b0 + b1 X + b2 X / X^2
C. Y ^ = b0 + b1 X + b2 X^2
D. Y ^ = b0 + b1 X
E. Y ^ = b0 + b1 X1 + b2 X2
Answers
The general form of such a model that predicts the volume of sales (Y) as a function of advertising (X) in which sales increase as advertising increases, but at a decreasing rate is given by Y^ = b0 + b1X + b2X². Option C.
The general form of the model that fits the description of the sales model that is given in the problem is C. Y^ = b0 + b1X + b2X². Where Y^ represents the predicted or estimated value of Y. b0, b1, and b2 are the coefficients of the model, and they represent the intercept, the slope, and the curvature of the relationship between X and Y, respectively.
In this model, the variable X has a quadratic relationship with the variable Y because of the presence of the squared term X². This indicates that the effect of X on Y is not linear but curvilinear, which means that the effect of X on Y changes as X increases. Specifically, the effect of X on Y increases initially but then levels off or diminishes as X becomes larger. Answer option C.
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Trains Two trains, Train A and Train B, weigh a total of 339 tons. Train A is heavier than Train B.
The difference of their weights is 251 tons. What is the weight of each train?
Answers
Answer:
trains two trains?
Step-by-step explanation:
i do not understand
Answer:
Train A weights 295 tons and train B weights 44 tons
Step-by-step explanation:
339-251=88
88÷2=44
251+44=295
295+44=339
A regular pentagon has side lengths of 7 inches. It is dilated by a scale factor of 2. What is the resulting perimeter of the pentagon
Answers
A regular pentagon has side lengths of 7 inches. It is dilated by a scale factor of 2. What is the resulting perimeter of the pentagon?Main Answer:The resulting perimeter of the pentagon is
:Given, A regular pentagon has side lengths of 7 inches and it is dilated by a scale factor of 2.To find: The resulting perimeter of the pentagonSolution:The perimeter of the regular pentagon is given as:P = 5 × s where s is the length of the side of the pentagon.
So, the perimeter of the original pentagon is:P = 5 × 7P = 35 inchesThe scale factor of the dilation is 2.Therefore, the side of the new pentagon is 7 × 2 = 14 inches.So, the perimeter of the new pentagon is:P = 5 × 14P = 70 inchesTherefore, the resulting perimeter of the pentagon is 70 inches.
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HELP WILL GIVE BRAINLIEST
Answers
Answer:
C
Step-by-step explanation:
Verify, give me brainliest, and rate, and say thank you
Answer:
log ( \(\frac{x^2-2x(x-1)}{(x-1)^2}\) )
Step-by-step explanation:
to find (f ◦ g)(x) , substitute x = g(x) into f(x)
= log ( ( \(\frac{x}{(x-1)^}\) )² - 2(\(\frac{x}{x-1}\) ) )
= log ( \(\frac{x^2}{(x-1)^2}\) - \(\frac{2x}{x-1}\) ) ← express as a single fraction
= log ( \(\frac{x^2-2x(x-1)}{(x-1)^2}\) )
Can someone help me with this
Answers
Answer:
the 3rd answer is the one which can actually form a proper plot off of that sentence
Answer:
The 3rd choice
Cinderella we know is the name of a princess in a story not a plot.
"In the woods" doesn't completely describe a plot.
so the answer is "The big bad wolf wants to eat the pigs, so he blows down their houses."
Wally is rebuilding his fence. Each section of the fence will have 12 vertical boards that are each eight inches wide. He’s going to attach them to two horizontal pieces of wood and add a diagonal piece to brace the fence section. The horizontal pieces will be five feet apart.
What is the approximate measure of angle θ?
Answers
Answer:
20-degree angle
Step-by-step explanation:
hope this helps
Answer:
Approx. 40 degrees
Step-by-step explanation:
1. Solve for the unknown in each triangle. Round each answer to the nearest tenth.
Answers
The values of the missing sides are;
a. x = 35. 6 degrees
b. x = 15
c. x = 22. 7 ft
d. x = 31. 7 degrees
How to determine the valuesTo determine the values, we have;
a. Using the tangent identity;
tan x = 5/7
Divide the values
tan x = 0. 7143
x = 35. 6 degrees
b. Using the Pythagorean theorem
x² = 9² + 12²
find the square
x² = 225
x = 15
c. Using the sine identity
sin 29= 11/x
cross multiply the values
x = 11/0. 4848
x = 22. 7 ft
d. sin x = 3.1/5.9
sin x = 0. 5254
x = 31. 7 degrees
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urgent help matlab.Thanks in advanc Write a M function file 'tconvert.m', which can convert coordinates (x, y) into Polar from Cartesian coordinates.
Answers
A M function file 'tconvert.m', which can convert coordinates (x, y) into Polar from Cartesian coordinates. is:
"```matlab
function [theta, r] = tconvert(x, y)
```"
To create the M function file 'tconvert.m' that converts Cartesian coordinates (x, y) into Polar coordinates, follow these steps:
1. Open the MATLAB Editor or any text editor and create a new file named 'tconvert.m'.
2. In the file, start with the function declaration line: `function [theta, r] = tconvert(x, y)`.
3. Inside the function, write the conversion code using MATLAB's built-in functions:
- Calculate the angle theta using `atan2(y, x)`, which returns the angle in radians.
- Calculate the radius r using `sqrt(x^2 + y^2)`, which gives the distance from the origin.
4. End the function with the `end` keyword.
5. Save the file in a directory accessible by MATLAB.
The function 'tconvert' takes the Cartesian coordinates (x, y) as input and returns the corresponding Polar coordinates (theta, r). The angle theta represents the direction in radians, and the radius r represents the distance from the origin.
The function can be called from the MATLAB command window or from another script or function file.
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Complete question:
Write a M function file 'tconvert.m', which can convert from Cartesian coordinates (x,y) into Polar coordinates.
Terry had his car repaired at Ace Auto. He was charged $50 per hour for labor plus $150 for parts. His total bill for the repair before tax was $375. How many hours of labor was Terry charged for?
a 2.5
b 4.5
c 7.5
d 10.5
Answers
Answer:
4.5
Step-by-step explanation:
Lucky you, had the same test today.
1. IF HEIGHTS OF FEMALE COLLEGE STUDENTS ARE NORMALLY DISTRIBUTED WITH A MEAN OF 64 INCHES AND A STANDARD DEVIATION OF 10 INCHES, WHAT IS THE PROBABILITY OF RANDOMLY SELECTING A FEMALE COLLEGE STUDENT WITH A HEIGHT LESS THAN 63 INCHES.
2. WHAT IS THE PROBABILITY OF RANDOMLY SELECTING A FEMALE COLLEGE STUDENT WITH A HEIGHT BETWEEN 61 AND 69 INCHES?
3. WHAT IS THE PROBABILITY OF FINDING AN AVERAGE OF GREATER THAN 65 INCHES WITH A SAMPLE OF 35 FEMALE COLLEGE STUDENTS?
Answers
a)0.1587 (or 15.87%)
b)0.6827 (or 68.27%)
c)0.0228 (or 2.28%)
1. Using the Normal Distribution formula with a mean (μ) of 64 inches and standard deviation (σ) of 10 inches, the probability of randomly selecting a female college student with a height less than 63 inches is 0.1587 (or 15.87%).
2. Using the Normal Distribution formula with a mean (μ) of 64 inches and standard deviation (σ) of 10 inches, the probability of randomly selecting a female college student with a height between 61 and 69 inches is 0.6827 (or 68.27%).
3. Using the Central Limit Theorem, the probability of finding an average of greater than 65 inches with a sample of 35 female college students is 0.0228 (or 2.28%).
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Mortimer has 3/4 of his birthday cake left on his party how many 3/16 size portions can you share with his family
Answers
Given:
Fraction of cake left = ¾
Let's find the number of 3/16 size portions of the remaining cake he can share with his family.
To find the amount of 3/16 size portions he can share, divide the remainder by 3/16.
Thus, we have;
\(\frac{3}{4}\div\frac{3}{16}\)To divide the fractions, change the division symbol to multiplication and flip the fraction on the right.
We have:
\(\begin{gathered} \frac{3}{4}\ast\frac{16}{3} \\ \\ =\frac{3\ast16}{4\ast3} \\ \\ =\frac{48}{12} \\ \\ =4 \end{gathered}\)Therefore, he can share 4 portions of 3/16
ANSWER:
4
The large sphere has a diameter of 20 feet. a large sphere has a diameter of 20 feet. a smaller sphere with a radius of 4 feet is cut out of the center of the larger sphere. which expression represents the volume, in cubic units, of the shaded part of the sphere? four-thirdsÏ€(103) four-thirdsÏ€(43) four-thirdsÏ€(103) â€" four-thirdsÏ€(43) four-thirdsÏ€(203) four-thirdsÏ€(43) four-thirdsÏ€(203) â€" four-thirdsÏ€(43)
Answers
Volume is a three-dimensional scalar quantity. The correct option is B, (4/3)π(10)³ - (4/3)π(4)³.
We have been given that a large sphere has a diameter of 20 feet. A smaller sphere with a radius of 4 feet is cut out of the center of the larger sphere. We are asked to find the volume outside smaller sphere and inside larger sphere.
A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.
The volume, in cubic units, of the shaded part of the sphere is the difference between the volume of the larger sphere and the smaller sphere. Therefore, the volume can be written as,
The volume of the sphere = (4/3)π(10)³ - (4/3)π(4)³
Hence, the correct option is B, (4/3)π(10)³ - (4/3)π(4)³.
Complete Question:
The large sphere has a diameter of 20 feet. A large sphere has a diameter of 20 feet. A smaller sphere with a radius of 4 feet is cut out of the center of the larger sphere. Which expression represents the volume, in cubic units, of the shaded part of the sphere? Four-thirdsπ(103) + Four-thirdsπ(43) Four-thirdsπ(103) – Four-thirdsπ(43) Four-thirdsπ(203) + Four-thirdsπ(43) Four-thirdsπ(203) – Four-thirdsπ(43)
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a rational number between
1
3
and
5
3
Answers
Answer:
3
Step-by-step explanation:
i don't really understand the question laid out
Help quick or else trouble
Answers
Answer:
1/21
Step-by-step explanation:
2/7 ÷ 6 =
2/7 x 1/6 = 2/42
Simplify by dividing the numerator and denominator by 2
1/21
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
Simplify this expression.
2(10) + 2(x – 4)
explain your steps. will give brainliest for right answers.
Answers
Answer:
12 + 2x OR 2x + 12
Step-by-step explanation:
2(10) + 2(x - 4)
Distribute;
20 + 2x - 8
Collect like terms;
12 + 2x OR 2x + 12
4. For which of the following probability assignments are events A and B independent?
P(AN B“) – 0.3. P(AN B) = 0.12, and P(Aºn B) = 0.4.
B
P(An BC) = 0.3. P(An B) = 0.3, and P(Aºn B) = 0.3.
C) P(An Bº) = 0.1. P(ANB) = 0.1, and P(Aº n B) = 0.4.
P(An Bº) = 0.3. P (AnB) = 0.0, and P(
AB) = 0.2.
E) P(An Bº) = 0.5, P (ANB) = 0.1 and P (Aºn B) = 0.4
Answers
Answer:
A
Step-by-step explanation:
formatting is so bad lol
but yeah its the first one because for independent events, the probably of a and b is the product of both probabilities, just like how 0.3*0.4=0.12
Question 4 of 10
Which of the following could be the ratio between the lengths of the two legs
of a 30-60-90 triangle?
Check all that apply.
□A. √2:√2
B. 15
□ C. √√√√5
□ D. 12
DE √3:3
OF. √2:√5
←PREVIOUS
SUBMIT
Answers
The ratios that could be the lengths of the two legs in a 30-60-90 triangle are √3:3 (option E) and 12√3 (option D).
In a 30-60-90 triangle, the angles are in the ratio of 1:2:3. The sides of this triangle are in a specific ratio that is consistent for all triangles with these angles. Let's analyze the given options to determine which ones could be the ratio between the lengths of the two legs.
A. √2:√2
The ratio √2:√2 simplifies to 1:1, which is not the correct ratio for a 30-60-90 triangle. Therefore, option A is not applicable.
B. 15
This is a specific value and not a ratio. Therefore, option B is not applicable.
C. √√√√5
The expression √√√√5 is not a well-defined mathematical operation. Therefore, option C is not applicable.
D. 12√3
This is the correct ratio for a 30-60-90 triangle. The ratio of the longer leg to the shorter leg is √3:1, which simplifies to √3:3. Therefore, option D is applicable.
E. √3:3
This is the correct ratio for a 30-60-90 triangle. The ratio of the longer leg to the shorter leg is √3:1, which is equivalent to √3:3. Therefore, option E is applicable.
F. √2:√5
This ratio does not match the ratio of the sides in a 30-60-90 triangle. Therefore, option F is not applicable. So, the correct option is D. 1 √2.
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if the correlation between two variables is close to 1.0, we cannot conclude that the explanatory variable causes changes in the response variable when the study is ______.
Answers
Answer: Observational
Step-by-step explanation:
If it is a experiment then you can conclude that a variable caused a response
In statistics, correlation refers to the degree to which two variables are related or associated with each other. When the correlation between two variables is close to 1.0, it indicates a strong positive relationship between them. However, it is important to note that this strong correlation does not necessarily imply a cause-and-effect relationship between the two variables, especially in a study where certain conditions may not be met.
A study is a systematic investigation aimed at acquiring knowledge and understanding of a particular phenomenon or subject matter. In order to establish a causal relationship between two variables, certain conditions need to be met, such as the establishment of a temporal sequence, a clear association between the variables, and the elimination of any confounding variables that may influence the outcome. These conditions are essential for any study seeking to establish a cause-and-effect relationship between variables.
In the absence of these conditions, even if the correlation between two variables is close to 1.0, it cannot be concluded that the explanatory variable causes changes in the response variable. This is because correlation does not imply causation. Correlation simply indicates the strength of the association between two variables, but it does not provide information about the direction or the underlying mechanisms of the relationship.
For instance, consider a study examining the relationship between smoking and lung cancer. The study finds a strong positive correlation between smoking and lung cancer, indicating that smokers are more likely to develop lung cancer than non-smokers. However, this correlation alone does not necessarily prove that smoking causes lung cancer. There may be other factors such as air pollution, family history, or genetics that could influence the development of lung cancer.
In summary, the correlation between two variables is a measure of the strength of the association between them. A high correlation does not necessarily imply a cause-and-effect relationship between the variables, especially in a study where certain conditions may not be met. It is important to consider these conditions and rule out other factors that could influence the outcome before drawing any conclusions about the causal relationship between two variables.
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What is the value of x in the triangle?
Answers
Answer:
45 degree=√2/2
√2/2=12√2
we use criss-cross method
√2x/√ 2=12√2/√ 2
×=24
Step-by-step explanation:
if it confused you this is (/)is dividing
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U7L2 Cool Down
The measure of the arc from B to A not passing through C is 26 degrees.
1. What is the measure of angle BOA ?
2. What is the measure of angle BDA?
3. What is the measure of angle BCA ?
degrees
degrees
degrees
Answers
Using the inscribed angle theorems, the measure of the indicated angles are:
1. m∠BOA = 26°
2. m∠BDA = 13°
3. m∠BCA = 13°
What is the Inscribed Angle Theorems?Based on the inscribed angle theorem, the following relationships are established:
Inscribed angle = 2(measure of intersected arc)Central angle = measure of intersected arcGiven:
Intercepted arc BA = 26°
1. ∠BOA is central angle
Thus:
m∠BOA = 26° (inscribed angle theorems)
2. ∠BDA is inscribed angle.
m∠BDA = 1/2(30) = 13° (inscribed angle theorems)
3. m∠BCA = m∠BDA = 13° (inscribed angle theorems)
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