a. The orthocenter for the coordinates (0,0), (6,3), and (3,24) is (24, 2).
b. The orthocenter for the coordinates (5,3), (8,6), and (4,10) is (8, 6).
What is an orthocenter?The orthocenter is the point where the altitudes of the triangle coincide. An obtuse triangle, will be outside the triangle. For a right triangle, it will be the right angle vertex. Of course, each altitude is a line through a vertex and perpendicular to the opposite side.
The equation can be written as,
y -y₃= -(x₂ -x₁)/(y₂ -y₁)(x -x₃)
This can be rearranged to something resembling standard form:
(x₂ -1x₁)(x -x₃) +(y₂-y₁)(y -y₃) = 0
(x₂ -x₁)x +(y₂ -y₁)y = (x₂ -x1)x₃ +(y₂ -y1)y₃ . . . . line through P3, ⊥ to P1P2
Part(a).
For the first altitude, we choose P1 = (0, 0), P2 = (7, 3), and P3 = (3, 51). This gives,
(x₂-x₁) = 7 -0 = 7
(y₂ -y₁) = 3 -0 = 3
altitude equation = 7x +3y = 7·3 +3·51 = 174
For the second altitude, we choose P1 = (0, 0), P2 = (3, 51), and P3 = (7, 3). This gives,
(x₂ -x₁) = 3 -0 = 3
(y₂ -y₁) = 51 -0 = 51
altitude equation = 3x +51y = 3·7 +51·3 = 174
Finding the point of intersection of these two equations can be done in a number of ways. Cramer's Rule provides a quick solution:
x = ((3)(-174) -(51)(-174))/(7·51 -3·3) = 8352/348 = 24
y = ((-174)(3) -(-174)(7))/348 = 696/348 = 2
The orthocenter for the triangle is (24, 2).
Part(b).
We can use the method just described, or we can verify the triangle is a right triangle, as it appears to be when the points are plotted.
The slope of P1P2 is (6 -3)/(8 -5) = 3/3/ = 1.
The slope of P2P3 is (10 -6)/(4 -8) = 4/-4 = -1
These slopes have a product of -1, so the segments are perpendicular. The vertex P2 has the right angle, hence the orthocenter.
The orthocenter for the triangle is (8, 6).
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Which is the equation of a parabola with a directrix at y = −3 and a focus at (5, 3)?
The equation of a parabola with a directrix at y = −3 and a focus at (5, 3) is y=1/12(x - 5)² .
What is Parabola?A parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.
Given that the directrix of parabola is at y=-3.
Focus at (5, 3)
parabola with equation (x - h)² = 4p(y - k)
p = 3 because the distance from the focus to the directrix is 6 and p = 6/2.
The vertex (5, 0)
Substitute h as 5 and k as 0.
(x - 5)² = 4(3)(y - 0)
(x - 5)² = 12y
Divide both sides by 12
y=1/12(x - 5)²
Hence, the equation of a parabola with a directrix at y = −3 and a focus at (5, 3) is y=1/12(x - 5)² .
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What is an equation of the line that passes through the point (-3,-1) and is perpendicular to the line 6x−2y=6?
Answer:
write the given equation in a standard form
6x - 2y = 6
-2y = -6x + 6
y = 3x - 3
m = 3
if the lines are perpendicular to each other the product of their gradient is equal to -1
m1 × m2 = -1
3 × m = -1
3m = -1
m = -1/3
y - y1 = m(x - x1 )
y - (-1) = -1/3 ( x - (-3) )
y + 1 = -1/3 (x + 3 )
y + 1 = -1/3x - 1
y = -1/3x - 1 + 1
y = -1/3x
Dascha is making a decorative table to
sell at a fair. The cost of her materials is
$80. If the table sells for $150, what is
the percent of markup?
The required percent markup is 87.5%.
Here,
The markup is the difference between the selling price and the cost price, expressed as a percentage of the cost price.
Markup = ((Selling price - Cost price) / Cost price) x 100%
In this case, the cost of materials is $80 and the selling price is $150, so the markup is:
Markup = ((150 - 80) / 80) x 100%
Markup = (70 / 80) x 100%
Markup = 0.875 x 100%
Markup = 87.5%
Therefore, the percent markup is 87.5%.
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A cistern can be completely drained by a pipe It in 6 hours. It can be filled by two pipes in 4 hours and 5 hours.
respectively. How many hours will the empty cistern take to fill if all the pipes are running?
If x iS the time to fill it, which of the following equations could be used to solve for t?
© 4x + 5x - 6x = 1
) 15x + 12x - 10x = 60
O 4x + 5x = 20
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Rain off and
Therefore , it will take 1.71 hours to fill the empty cistern if all the pipes are running.
What is equation?equation: a statement that two expressions having variable- and/or number-filled expressions are equivalent. Since equations are essentially questions, the search for a way to respond to them has spurred the development of mathematics.
Here,
Let's assume that the pipelines in this instance are "a" to drain the water in 6 hours, "b" to fill the water on its own in 4 hours, and "c" to fill the water in 5 hours.
Therefore, if "a" can empty it in 6 hours, it has only completed 1/6 of the task in that time. In 6 hours, "a" would have completed 6/6 or 1 whole, the entire task, and the cistern would have been fully filled, but in 1 hour, it has only accomplished 1/6 of that.
In an hour, "b" has completed barely a quarter of the task.
In that case, "c" has only completed 1/5 of the task in an hour.
Now, assuming it took "x" hours, if "a," "b," and "c" are running, and "a" is draining while b & c filling it up.
The cistern must be filled in "x" hours, thus after 1 hour, all three have completed only 1/x of the task.
well, let's add their rates, to see what we get then.
=> 1/4 + 1/5 -1/6= 1/x
=> 15 + 20 - 10 / 60 = 1/x
=>35/60 =1/x
=> x=60/35
=>1.71 hours
Therefore , it will take 1.71 hours to fill the empty cistern if all the pipes are running.
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Diego has 27 ounces of juice he pours equal amounts for his 3 friend and has 6 ounces left
Answer:
3x + 6 =27
Step-by-step explanation:
A. The solution is 7! First I Subtracted 6 from both sides of the equation, Then I Divided both sides of the equation by the same term. Then I got "7" from doing the following.
B. The solution is the x in the answer which is the amount of juice he poured.
"Diego has 27 ounces of juice. He pours equal amounts for each of his 3 friends and has 6 ounces left for himself"
If i² = -1, then (i)(1 + 2)(2-3i)² = ?
Answer:
[tex]36-15i[/tex]
Step-by-step explanation:
[tex](i)(1+2)(2-3i)^2 \\ \\ =3i(2-3i)^2 \\ \\ =3i(9i^2 + 4 -12i) \\ \\ =3i(-9+4-12i) \\ \\ =3i(-5-12i) \\ \\ =-15i-36i^2 \\ \\ =36-15i[/tex]
What is the y-intercept of the line y=-2/3 x ?
a. -2/3
c. 1
b. 0
d. 3/2
D
A
C
B
Answer:
a
Step-by-step explanation: becasue i said so
Answer:
d) B => b. 0
Step-by-step explanation:
Now we have to,
→ find the y-intercept of the line.
Formula we use,
→ y = mx + b
→ b = y-intercept
Then the y-intercept is,
→ y = mx + b
→ y = (-2/3)x + 0
→ [ b = 0 ]
Hence, the y-intercept is 0.
need assistance please
The range is $53000
About 30 graduates made $25,00050% earn above $33,00025% of the graduates earn above $25,000The range of the starting salariesThe box and whisker plot represents the given parameter
From the plot, we have the following representations:
Least salary = 19,000
Highest salary = 72,000
The range is calculated as
Range = Highest salary - Least salary
Substitute the known values in the above equation, so, we have the following representation
Range = 72000 - 19000
Evaluate
Range = 53000
Amount made by about 30 graduatesWe have
Total graduates = 120
30 out of 120 is 1/4
This means that
30 graduates = Lower quartile
From the plot, we have
Lower quartile = 25,000
This means that
30 graduates = 25,000
Graduates with salary of above $33000The salary of $33000 is at the median position
This means that
50% earn above $33,000
25% of the graduates25% represents the lower quartile
From the plot, we have
Lower quartile = 25,000
This means that
25% of the graduates earn above $25,000
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2. Minji bought 4 necklaces, each
costing $9.00. Then,
She started with $60.00. How
much does she have now?
Answer:$24
Step-by-step explanation: if she started with $60 and she bought 4 neckless with the cost of each $9 we get so $9x4 is 36
$60-$36=24
$24
Complete the linear equation in slope-intercept from with the slope -8 and the y-intercept (0,10)
y=______x+______
Answer:
y = -8x + 10.
Step-by-step explanation:
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is -8 and the y-intercept is (0, 10), so the equation is y = -8x + 10.
Use 3.14 for the value of pie , and round your answers to two decimal places, where applicable. If the radius of a circle is 3 cm, what is its circumference?
Answer:
18.85
Step-by-step explanation:
C=2πr
2(pi)3=18.85
A line passes through the points (4, 440) and (6, 560)
on a cooridinate grid. What is the equation of the line?
A. y = 60x + 200
B. y = 120x + 2
C. y = 200x + 60
D. y = 2x + 120
The equation of a line is given by y = mx + c
What exactly is a line equation?The equation of a line is given by:
y = mx + c
where m is the slope of the line and c is the y-intercept.
Example:
The slope of the line y = 2x + 120.
The slope of a line that passes through (4,440) and (6,560) is 0.676.
We have,
The equation of the line y = mx + c.
The line passes through (4,440) and (6,560).
This means,
We can find the slope using the coordinates (4,440) and (6,560).
Slope = m
m = (4,440) / (6,560)
m = 0.676
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Solve the equation n/(200 - n) = 3/5
The value of n after solving the given equation comes to 75.
What are equations?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions.
For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
A mathematical statement in which two expressions are rendered equal to one another is known as an algebraic equation.
A variable, coefficients, and constants make up an algebraic equation in most cases.
So, we have the equation:
n/(200 - n) = 3/5
Now, solve for x as follows:
n/(200 - n) = 3/5
5n = 600 - 3n
8n = 600
n = 600/8
n = 75
Therefore, the value of n after solving the given equation comes to 75.
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9 i don't know how to do
Answer:
9. see below for proof; θ ∈ {45°, 135°, 225°, 315°}
11. see below for proof; x ∈ {109.47°, 250.53°}
Step-by-step explanation:
You want to prove two trig identities, then use those to solve trig equations.
9. 2tan²(θ)sin²(θ) = 1It is often useful to use the identities ...
tan = sin/cossin² +cos² = 1(i) Doing that here, we get ...
[tex]2\tan^2\theta \sin^2 \theta=1\\\\2\dfrac{\sin^2\theta}{\cos^2\theta}\sin^2\theta=1\qquad\text{replace tan}\\\\\dfrac{2\sin^4\theta}{1-\sin^2\theta}=1\qquad\text{replace $\cos^2$}\\\\2\sin^4\theta=1-\sin^2\theta\qquad\text{multiply by $(1-\sin^2\theta)$}\\\\2\sin^4\theta+\sin^2\theta-1=0\qquad\text{put in standard form}[/tex]
(ii) Substituting x=sin²(θ), we have the quadratic ...
2x² +x -1 = 0
The quadratic formula gives the solutions as ...
x = (-1 ±√(1 -4(2)(-1)))/(2(2)) = (-1±3)/4 = -1, 1/2
The value of sin(θ) is the square root of these values. √-1 is imaginary, so does not give us any solutions. √(1/2) = (±√2)/2, corresponding to ...
sin(θ) = (±√2)/2
θ ∈ {45°, 135°, 225°, 315°}
11. sin(x)tan(x)/(1 -cos(x)) = 1 + 1/cos(x)(i) Substituting as suggested above, we have ...
[tex]\dfrac{\sin x\tan x}{1-\cos x}=1+\dfrac{1}{\cos x}\\\\\dfrac{\sin^2 x}{\cos x(1-\cos x)}=\qquad\text{substitute for $\tan x$}\\\\\dfrac{1-\cos^2 x}{\cos x(1-\cos x)}=\qquad\text{substitute for $\sin^2 x$}\\\\\dfrac{1+\cos x}{\cos x}=1+\dfrac{1}{\cos x}\qquad\text{cancel the common factor}\\\\1+\dfrac{1}{\cos x}=1+\dfrac{1}{\cos x}\qquad\text{Q.E.D.}[/tex]
(ii) Using this result, the equation can be written ...
(1 +1/cos(x)) +2 = 0
1/cos(x) = -3 . . . . . . . . . subtract 3
x = arccos(-1/3) . . . . . . . solve for x
x ∈ {109.47°, 250.53°} . . . . approximately
please help! Big Points here! I really need the help!
Answer:
13.3
Step-by-step explanation:
7/200=x/380
cross multiply
7(380)=200x
2660=200x
/200 /200
13.3=x
hopes this helps please mark brainliest
Grade 7 Envision math volume 1
Answer:
The table of contents for Envision Math Volume 1 includes the following topics:
1. Whole Number Operations
2. Decimals
3. Fractions and Mixed Numbers
4. Algebraic Thinking
5. Geometry
6. Measurement
7. Data Analysis and Probability
8. Integers
9. Ratios and Proportional Relationships
10. Expressions and Equations
11. Geometric Measurement and Transformations
12. Statistics and Probability
This figure is made up of two rectangular prisms.
What is the volume of the figure?
Enter your answer in the box.
cm³
6 cm
2 cm
22 cm
↓
5 cm
13 cm
2
The aggregate volume of the two rectangular prisms is 330 cubic cm.
What is meant by the volume of rectangular prisms?The base's area multiplied by the height gives the rectangular prism its volume. So the formula for a rectangular prism's volume is as follows. Using cubic units, the volume of a rectangular prism = length × breadth × height
The volume is going to be the addition of two volumes.
We are going to divide the shape into two.
Volume = length × breadth × height
Section A and section B
Volume = length × breadth × height
Volume of section A is = 2 × 10 × 9
Volume of section A = 180 cm³
Volume = length × breadth × height
Volume of section B is = 3 × 5 × 10
Volume of section B = 150cm³
Total volume = volume of section A + volume of section B
Total volume = 180cm³ + 150cm³
Total volume = 330cm³
The complete question is:
The figure below is made of 2 rectangular prisms.
What is the volume of this figure?
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keeps a farmer keeps cows in an enclosed rectangular pasture with the length of 2.2 * 10 ^ 2 ft and a width of 3.1 * 10 ^ 3 ft what is the area of the rectangular pasture
Answer:
6.82 × 10⁵ ft²-------------------------------
Use dimensions of the rectangle to find its area:
A = lwA = 2.2 × 10² × 3.1 × 10³ = (2.2×3.1) × 10²⁺³ = 6.82 × 10⁵ ft²Which matrix equation can be used to solve the system? {(2x+3y=4),(x+y=7):}
Use matrices created using the coefficients of the unknowns and their related determinants to apply Cramer's rule.
How should a matrix equation be written?Three easy actions must be taken in order to describe this system as a matrix:
First, create a matrix with all the coefficients. An acronym for this is a coefficient matrix.
Multiply this matrix by the system setup's variables in another matrix.
In another matrix, place the solutions on the other side of the equal sign.
A = [[2, 3], [1, 1]]
A_{x} = [[3, 4], [1, 7]] d
A{y} = [[2, 4], [1, 7]]
det (A) = 2-3=-1
det (Ax) = 14 - 4 =10
Delta_{y} = [[2, 4], [1, 7]] det (A7), 22-2012
x = (det (Ax))/(det (A)) = 10
det (Ay) = 14 - 4 =10
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A shopkeeper buys 50 pencils for 80 rupees and sells them at 40 pencils for 90find his gain or loss percent
please help me! Help me solve this problem. The first three terms of an arithmetic sequence are 9, 16, and 23. Which of the following would be this sequences 20th term?
Answer:
142
Step-by-step explanation:
Definition: an = a1 + f × (n-1)
Sequence: 9, 16, 23, 30, 37, 44, 51, 58, 65 ...
Common Difference: +7
The sum of all numbers up through the 20th: 1510
20th value: 142
Hope this helps! :)
A worker is filling out his timesheet. He worked 8 hours on Monday, 7 hours and 30 minutes on Tuesday, 8¾ hours on Wednesday, 4 hours on Thursday, and 8¼ hours on Friday. If he earns $14.35 per hour, what will be his gross pay for this week? (Enter numeric value only. Round your answer to the nearest penny.)
The gross pay of worker this week is $432.325
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.
Number of money made in one hour = $ 14.35
Number of hours, worked on Monday = 8 hours
Number of hours, worked on Tuesday = 1/2 hours
Number of hours, worked on Wednesday = 8¾ hours
Number of hours, worked on Thursday = 4 hours
Number of hours, worked on Friday = 8¼ hours
Total number of worked = 4 + 8 + 8¼ + 8¾ + 1/2
= 29.5
Total gross pay = 14.35 × 29.5
= $432.325
Thus, The answer is $432.325
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find the ninth power of 10. Write it as repeated multiplication, with an exponent, and as a whole number.
The value of the expression 10⁹ by the repeated multiplication will be 1,000,000,000.
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
The expression is given below.
⇒ 10⁹
The value of the expression by the repeated multiplication will be given as,
⇒ 10⁹
⇒ 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
⇒ 1,000,000,000
The value of the expression 10⁹ by the repeated multiplication will be 1,000,000,000.
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6 multiply by 3 subtract 2 divide by 4 add 7
Answer:
11
Step-by-step explanation:
6x3
=18
-2
=16
divided by 4
=4
+7
=11
Answer:13.5
Step-by-step explanation:
6x3-2/4+7
First multiply 6 and 3. Then divide 2 and 4.
18-0.5+7
Then subtract 0.5 from 18 and add 7
18-0.5= 16.5
16.5+7= 13.5
The following are the dimensions of each room in a house:
Living room = 20 ft x 15 ft Kitchen = 12 ft x 10 ft
Bedroom 1 = 12 ft x 12 ft Bedroom 2 = 12 ft x 10 ft
Bathroom = 8ft x 12 ft
What is the total square footage of the house?
The total square footage of the house is 3776 square feet.
What is dimension?
A database of research grants called Dimensions connects funding to publications, clinical trials, and patents as well as other outcomes. Dimensions is a part of the technology firm Digital Science, which has its global headquarters in London, United Kingdom.
Given:
The dimensions of each room are,
Living room = 20 ft x 15 ft
Kitchen = 12 ft x 10 ft
Bedroom 1 = 12 ft x 12 ft
Bedroom 2 = 12 ft x 10 ft
Bathroom = 8ft x 12 ft
So, the total square footage is,
(20 + 12 + 12 + 12 + 8) ft x (15 + 10 + 12 + 10 + 12) ft = 64 ft x 59 ft = 3776 square feet.
Hence, the total square footage of the house is 3776 square feet.
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The solutions of the quadratic equation
x²-x-12=0
Answer:
[tex]x=-3, 4[/tex]
Step-by-step explanation:
[tex]x^2-x-12=0 \\ \\ (x-4)(x+3)=0 \\ \\ x=-3, 4[/tex]
Richard found 93 - 67 by adding up on the open number line. Is he correct? Explain. Then write an addition equation to show how you could check his work
The addition equation is 93 - 67 = 26 in the number line.
What is number line?
A number line is an image of numbers plotted on a straight line, either horizontally or vertically. We can compare numbers and execute simple arithmetic operations on them easily by writing the numbers down on a number line. The starting point of a number line is often regarded as zero (0). These numbers to the left of zero are all negative while the numbers just on right of zero are all positive. As a result, we can argue that on a number line, the value of numbers grows as we approach the right. The numbers on the right are therefore larger than the ones on the left, according to this statement.
The entire question is depicted in the picture that is attached.
Richard added 26 to 67 by first marking point 67 on the line, then adding 2 tens and 2 threes until he reached 93 as opposed to subtracting 67 to 93 by subtracting 6 tens and 7 ones starting at point 93, which results in 26.
Take note that 26 is equal to 10 plus 10 plus three additional times.
To verify his work, we add the following: 67 + 26 = 93, which is also written as 93 - 67 = 26.
Hence the addition equation is 93 - 67 = 26.
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Given a rectangle with an area of 40 x 6y9z and a side of 10 x ³y ³, find
the length of the other side.
The length of the other side is 4x⁻².6y⁻².9z .
How to calculate area of rectangle ?
The area of a rectangle is equal to its length times its width. Follow the steps mentioned below to find the area of a rectangle:
Step 1: Note the dimensions of length and breadth from the given data.
Step 2: Find the product of length and breadth values.
Step 3: Give the answer in square units.
Given :
Area = 40x.6y.9z
length of one side = 10 x ³y ³
∵Area of rectangle = length ˣ breadth
∴ 40x.6y.9z = 10 x ³y ³ × length of other side
length of other side = [tex]\frac{40x.6y.9z}{10 x^{3} y^{3} } = 4x^{-2} . 6y^{-2} . 9z[/tex]
Hence , the length of the other side is 4x⁻².6y⁻².9z .
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What is the relationship between 5 and 20
20 is is equal to 5 multiplied by 4.
5 is equal to 20 divided by 4
Annual cost of insurance: $4,160. Employer pays 80 percent. 52 pay periods. Find the deduction per pay period.