What is the missing reason in Step 8? Pythagorean theorem definition of cosine substitution properties of multiplication
8.a2 = b2 – 2bccos(A) + c2 8.?
Answers
Pythagorean theorem is used only for the right angled triangles.
For ΔABC,
Cosine rule is
[tex] a^2=b^2 + c^2-2bc$ cosA $ [/tex]
[tex] b^2=a^2 + c^2-2ac$ cosB $ [/tex]
[tex] c^2=a^2 + b^2-2ab$ cosC $ [/tex]
Here we have used one of these formulae
Hence definition of cosine or Cosine rule is the right answer
Option 2) is the right answer
The missing reason in Step 8 is substitution of a² = b² – 2bc·cosA + c²
What is the law of cosines?The law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
Given is a proof using triangles property and rule,
In △ ABC, BD ⊥ AC
Prove: the formula for the law of cosines, a² = b² + c² – 2bccos
Statements Reasons
1. In △ ABC, BD ⊥ AC 1. given
2. In △ ADB, c² = x² + h² 2. Pythagorean thm.
3. In △ BDC, a² = (b – x)² + h² 3. Pythagorean thm.
4. a² = b² – 2bx + x² + h² 4. prop. of multiplication
5. a² = b² – 2bx + c² 5. substitution
6. In △ ADB, cos(A) = x/6 6. def. cosine
7. ccos(A) = x 7. mult. prop. of equality
8. a² = b² – 2bccos(A) + c² 8. ?
9. a² = b² + c² – 2bccos(A) 9. commutative property
We are asked to find the reason used in step 8,
From step 5) we have, a² = b² – 2bx + c².......(i)
From step 7) we have, ccos(A) = x
So, by substituting ccosA for x in eq(i), we get the equation for step 8)
a² = b² – 2bc·cosA + c²
Hence, the missing reason in Step 8 is substitution of a² = b² – 2bc·cosA + c²
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Related Questions
Laura has $18.45 left after buying a shirt for $16.80 and a sweater for $24.75
How much money did Laura start with?
plz explain
Answers
18.45+16.80+24.75 which equal $60
If you want to check your answer just subtract the amount spent on the shirt and sweater by 60 and you should get 18.45
Laura started with $60.00 before purchasing a shirt for $16.80 and a sweater for $24.75, and then had $18.45 remaining.
Calculating Laura's Starting Amount
To find out how much money Laura started with, we need to add the amount she spent on the shirt and sweater to the amount she had left over. Laura spent $16.80 on a shirt and $24.75 on a sweater. After these purchases, she had $18.45 remaining.
We calculate the total amount spent by adding the costs of the shirt and sweater together:
Cost of the shirt: $16.80
Cost of the sweater: $24.75
Total spent: $16.80 + $24.75 = $41.55
We now add the amount left after the purchases to find the total amount Laura started with:
Total spent: $41.55
Amount left: $18.45
Total amount Laura started with: $41.55 + $18.45 = $60.00
What is the first error he made in computing the standard deviation?
A.Yuri failed to find the difference between each data point and the mean.
B.Yuri divided by n instead of n -1.
C.Yuri did not subtract 9 - 14 correctly.
D.Yuri failed to square -2 correctly.
Answers
Answer:
Step-by-step explanation:
Given is a sample with 4 items as
12,9,14,21
Mean = average of 4 = 14
s is the square root of sample variance
Sample variance is the sum of squares divided by n-1
But Yuri divided by 4 = n instead of 3 = n-1
That is the only mistake he did and all other steps are right.
Nadia needs half of a dollar for bus fare. Which amount is equivalent to her bus fare?
Answers
The square of the sum of two consecutive natural numbers is greater than the sum of the squares of these two numbers by 112. find the two numbers.
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Then, ( 2a + 1 )^2 = 112 + a^2 + a^2 + 2a +1;
4a^2 + 4a + 1 = 113 + 2a^2 + 2a;
Finally, 2a^2 + 2a - 112 = 0;
a^2 + a - 56 = 0;
We use Quadratic Formula for this Quadratic Equation;
The solutions are a1 = 7 and a2 = -8;
But a is a natural number; so, a = 7;
The natural consecutive numbers are 7 and 8.
The two consecutive numbers are 7 and 8.
Consecutive numbers are numbers that follow each other. For example, 1,2,3,4 are consecutive numbers.
Let:
x = first number
(x + 1) = second number
From the question, this expression can be derived:
[(x) + (x + 1)] ²
This can be simplified to (2x +1)²
The second expression is : x² + (x +1)²
(2x +1)² - [x² + (x +1)²] = 112
expanding the bracket gives:
(4x² + 4x + 1) - [x² + (x² + 2x +1) = 112
4x² + 4x + 1 - 2x² - 2x - 1 = 112
Add like terms
2x² + 2x = 112
Divide both sides by 2
x² + x = 56
x² + x - 56 = 0
Factorise the equation
The factors of -56x² that add up to x are : 8x and -7x
(x² - 7x) (8x - 56) = 0
x(x - 7) 8(x -7) = 0
x = 7
or
x = -8
Since x cannot be a negative number, x is 7 and (x + 1) = 8
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99 POINT QUESTION!!!!!
The prime factorization of a number is 2^3*3^2*5. Which is a true statement about the factors of the number?
Answers
so the number = 8 * 9 * 5 = 360
15 is a factor of 360, so the answer would be the first one: Fifteen is a factor of the number because both 3 and 5 are prime factors
3² = 9
8 x 9 x 5 = 360
15 is a factor of 360, so the answer would be 1. Fifteen is a factor of the number because both 3 and 5 are prime factors
How many odd integers from 1000 through 9999 have distinct digits?
Answers
step 1
the number of elements in a list from n to m is (m-n)+1
therefore
from 1000 to 9999 there are
(9999-1000)+1 = 9000 elements
step 2
Since even and odd numbers alternate, half of our list is composed of odd numbers. Therefore, there are 4500 of them
step 3
To construct a 4 digit odd number, we have 5 choices for the last digit.
(1-3-5-7-9)
then
we have
8 choices for the first (not 0 or same as last digit),
8 choices for the second
7 choices for the third.
there are (8)(8)(7)(5) = 2240
the answer is 2240 odd numbers with distinct digits in the list
Kim solved the equation below by graphing a system of equations. log2(3x-1)= log4(x+8) What is the approximate solution to the equation?
Answers
The approximate solution of the equation log₂(3·x - 1) = log₄(x + 8), obtained algebraically is about x = 1.353. The solution cain be verified using the attached graph created with MS Excel
The steps used to find the solution of the logarithm equation can be presented as follows;
The logarithm equation is; [tex]\log_2(3\cdot x - 1) = \log_4(x + 8)[/tex]
Converting the logarithm to the same base, we get;
[tex]\log_4(x + 8) = \frac{\log_2(x + 8)}{\log_2 4}[/tex]
[tex]\log_4(x + 8) = \frac{\log_2(x + 8)}{2}[/tex]
[tex]\log_2(3\cdot x - 1) = \frac{\log_2(x + 8)}{2}[/tex]
2 × log₂(3·x - 1) = log₂(x + 8)
log₂(3·x - 1)² = log₂(x + 8)
(3·x - 1)² = (x + 8)
(3·x - 1)² - (x + 8) = 0
9·x² - 7·x - 7 = 0
Solving using technology, we get; x ≈ 1.353 and x ≈ -0.575
Plugging in the value of x = 1.353, we get;
log₂(3 × 1.353 - 1) ≈ 1.613
log₄(1.353 + 8) ≈ 1.613
The value of the logarithm are much different when x ≈ -0.575, the expression log₂(3·x - 1) have no solution. The solution is therefore; x ≈ 1.353
Please find the attached the graph of the equation log₂(3·x - 1) = log₄(x + 8), showing the solution as x ≈ 1.353
Final answer:
To solve the equation log2(3x-1) = log4(x+8), we can rewrite the equation and simplify it to calculate the approximate solution as x = 4.5.
Explanation:
To solve the equation log2(3x-1) = log4(x+8), we can start by using the property of logarithms that states log(a) = log(b) if and only if a = b. Based on this property, we can rewrite the equation as 3x - 1 = x + 8. Simplifying this equation, we get 2x = 9. Finally, by dividing both sides by 2, we find that x = 4.5. Therefore, the approximate solution to the equation is x = 4.5.
Factor and find all the zeros of 2x^3+3x^2-8x+3 if -3 is a zero
Answers
.. 2x^2 -3x +1
That, in turn, factors as
.. (2x -1)(x -1)
So the complete factorization is
.. (2x -1)(x -1)(x +3)
and the zeros are confirmed by a graphing calculator to be
.. x = 1/2, 1, -3
Pablo folds a straw into a triangle with side lengths of 4x -2 inches and 4x -1 inched . The perimeter of the triangle is 12x -6 inched . If x =1.5 what was the length of the straw before it was folded ?
Answers
12(1.5)^2 -6 = length
12(2.25) -6 =
27 -6 =
21
Your answer is: 21 inches
Have an amazing day mate!
Shoutout to: @Apologiabiology
Answer:
21
Step-by-step explanation:
Anybody know the answer to this?
Answers
If the radius of the circle is 1, the side length of the square is √2 and its area is (√2)^2 = 2.
The ratio of square area to circle area is 2/π ≈ 63.7%.
is -4 real, complex, pure imaginary, or nonreal complex. (More than one of these descriptions may apply.)
Answers
Hope this helps!
Find the slope and the y-intercept of the equation y − 3(x − 1) = 0.
slope = -3 and y-intercept = -3
slope = 3 and y-intercept = 3
slope = -3 and y-intercept = 3
slope = 3 and y-intercept = -3
Answers
The slope and the y-intercept of the equation y − 3(x − 1) = 0 are: slope = 3 and y-intercept = -3. The correct option is the fourth option (slope = 3 and y-intercept = -3)
To determine the slope and y-intercept of the equation, we will rewrite the equation in the form of the slope-intercept form of the equation of a straight line.
The slope-intercept form equation of a straight line is
y = mx + c
Where m is the gradient
and c is the y-intercept
From the question, the given equation is y − 3(x − 1) = 0
This can be rewritten in form of the equation of a straight line as
y -3x +3 = 0
Then,
y = 3x - 3
Now, compare this to y = mx + c (the slope-intercept form of the equation of a straight line)
Then,
m = 3 and c = -3
That is,
slope = 3 and y-intercept = -3
Hence, the slope and the y-intercept of the equation y − 3(x − 1) = 0 are: slope = 3 and y-intercept = -3. The correct option is the fourth option (slope = 3 and y-intercept = -3)
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The sum of two complementary angles is 90°. For one pair of complementary angles, the measure of the first angle is 15 less than twice the measure of the second angle. Write a system of equations that can be used to determine the measure of the first angle, a, and the measure of the second angle, b.
•a+b=90
a-2b=15
•a+b=90
2b-15=a
•a+b=90
2b+15=a
•a+b=90
2a-15=b
Answers
a---------------> first angle
b---------------> second angle
we know that
a+b=90
and
the measure of the first angle is 15 less than twice the measure of the second angle
a=(2b)-15
A system of equations that can be used to determine the measure of the first angle, a, and the measure of the second angle, b is
a+b=90
a=(2b)-15
A pair of shoes that normally costs $75 is on sale for $55. What is the percent decrease in the price, to the nearest whole percent?
Answers
which or the following will happen if you miss a monthly credit card payment?
A)You will be charged a late fee.
B)You loose reward points
C)Your APR will increase the next month
D)Both A & B
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Which of these is a correct statement regarding payday loans?
A.) Theyŕe easier to get than both car loans and credit cards.
B.) Theyŕe easier to get than credit cards but harder to get than car loans.
C.) Theyŕe easier to get than car loans but harder to get than credit cards.
D.) Theyre harder to get than both car loans and credit cards.
Answers
Answer:
A.) They are easier to get than both car loans and credit cards.
Step-by-step explanation:
Payday loans are also called cash advance loans or check advance loans.
These loans are comparatively very easy loans for a person to get. These are short term loans that come with high interest rates. The rates are based on the borrower's income and credit profile.
In this loan the principal amount is typically a small portion of the borrower's next paycheck.
A cube has a length of 9.3 cm. what is the surface area of the cube?
Answers
Explanation:
The surface area of the cube is given by
Surface Area = 6*(edge)²
Since the edge of the cube in this problem is its length, the egde is 9.3 cm. Hence, the surface area of the cube is calculated as
Surface Area = 6*(edge)²
= 6*(9.3 cm)²
Surface Area = 518.94 square centimeters
Why are angles measured in degrees minutes and seconds?
Answers
The use of the number sixty as a basis for the measurement of angles, coordinates and time measurements is linked to old astronomy and trigonometry. It was common to measure the elevation angle of a star and trigonometry uses right triangles.
help me please asap!!!!!!!
Answers
1. It is where the bisectors meet. This fact creates the next statement.
2. It is the same distance from the incenter to the sides of triangle. Three lines each drawn from the incenter perpendicularly to the sides will create 3 equal lines. These lines will each be equal. Put another way AI = BI = CI which are radii for the incircle.
Any two of AI, BI, and CI and make them equal to each other.
3x + 7 = 5x - 11 Now you can just use algebra to solve. Add 11 to both sides
3x + 7 + 11 = 5x
3x + 18 = 5x. Now subtract 3x from both sides.
18 = 5x - 3x
18 = 2x
x = 9
You should really check this. Do this by equating CI to either AI or BI. I leave it to you to figure out which one I choose.
Check
=====
This will be short so you can fill in the steps for yourself.
3x + 7 = 52 - 2x
5x = 45
x = 45/5
x = 9
Find an angle between 0° and 360° that is coterminal with 538°.
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Write the equation of the circle with center (−1, −3) and (−7, −5) a point on the circle.
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well, the radius will be the distance from the center to any point on the circle, it just so happen that we know -7, -5 is on it, thus
[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ -1 &,& -3~) % (c,d) &&(~ -7 &,& -5~) \end{array} \\\\\\ d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ r=\sqrt{[-7-(-1)]^2+[-5-(-3)]^2}\implies r=\sqrt{(-7+1)^2+(-5+3)^2} \\\\\\ r=\sqrt{36+4}\implies r=\sqrt{40}\\\\ -------------------------------[/tex]
[tex]\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{-1}{ h},\stackrel{-3}{ k})\qquad \qquad radius=\stackrel{\sqrt{40}}{ r} \\\\\\\ [x-(-1)]^2+[y-(-3)]^2=(\sqrt{40})^2\implies (x+1)^2+(y+3)^2=40[/tex]
Which number completes the square? x2 + 18x + _____?
Answers
Thus x^2 + 2(9)(x) + (9)^2 = x^2 + 18x + 81
At how many points does the graph of the function below intersect the x-axis?
y = 25x2 - 10x + 1
A. 0
B. 1
C. 2
Answers
Answer:
It is one.
Step-by-step explanation:
If you didn't get 1 re read the problem
Maren is buying carpet for her rectangular living room. The room is 4.8 yards wide and 5.2 yards long. How much carpet does she need to buy?
Answers
Answer
A = 24.96 yd^2
Step-by-step explanation:
We need to find the area of the room
Area is length times width
A = lw
We know the length is 5.2 yds and the width is 4.8
A = 5.2 * 4.8
A = 24.96 yd^2
The length of a rectangular garden is 8 feet longer than the width. The garden is surrounded by a 4-foot sidewalk. The sidewalk has an area of 320 square feet. Find the dimensions of the garden.
Answers
The width of the garden is 12 feet and the length of the garden is 20 feet.
Explanation:Let's denote the width of the garden as 'w'. The length will then be 'w + 8'. Measurements are in feet. The garden is surrounded by a 4-foot sidewalk. Therefore, the total length of the garden including the sidewalk is 'w + 8 + 4 + 4 = w + 16', and the total width is 'w + 4 + 4 = w + 8'.
The area of the sidewalk surrounding the garden is the area of the large rectangle minus the area of the garden. So, we set up the following equation:
(w + 16)(w + 8) - (w)(w + 8) = 320.
After simplifying, we can get a quadratic equation: w² + 8w - 320 = 0 and then solve this equation, which gives us w = 12 or w = -28. We ignore the negative solution as the width cannot be negative. So, the dimensions of the garden are:
Width: 12 feet
Length: 20 feet.
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The dimensions of a certain ibm machine are 20" x 17" x 48". if the size of the machine is increased proportionately until the sum of its dimensions equals 200", what will be the increase in the shortest side
Answers
Then new dimensions are
20x, 48x and 17x
The sum of the dimensions are ;
20x + 48x + 17x = 200
85x = 200
x = 200/85
Therefore each dimension was increased by a factor of x,
The new dimension of the shortest side is 17” * 200/85 = 40
So the increase in the shortest side ;
40-17 = 23”
Is the system of equations is consistent, consistent and coincident, or inconsistent?
y=−12x+3y=4x+2
Select the correct answer from the drop-down menu.
Drop-down Menu: consistent, consistent and coincident, or inconsistent
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You can eliminate an inconsistent system. That would happen when the lines are parallel.
! Hope this helps!
Each leg of a 45°-45°-90° triangle measures 14 cm. What is the length of the hypotenuse? 7 cm cm 14 cm cm
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This one helps too.
Ur welcome.
Kyle made a loaf of banana bread. He gave equal portions of 1\2 of the loaf to 4 friends what diagram could kyle use to find the fraction of the whole loaf of banana bread that each friend get
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Simplify and leave in radical form.
8√x^2 y^6
Answers
8sqrt(x•x•y•y•y•y•y•y)
Since there are 2 xs, we can pull them out of the radical.
8xsqrt(y•y•y•y•y•y)
Since there are 6 ys, we can pull them all out and there are 3 left now.
8xy^3
There is nothing left inside the radical so we can get rid of it!
Best wishes:)