the vertex of triangle DEF are D(1,3) ;E(-1,0) ;F(2,-1) find the coordinate of orthocenter for this triangle
please with clear explanation I will give brainliest
Answers
Answer:
[tex]\sf \left(\dfrac{1}{11},\dfrac{3}{11}\right)[/tex]
Step-by-step explanation:
The orthocenter is the point of intersection of the altitudes of a triangle.
(The altitude is the line that passes through the vertex and is perpendicular to the side opposite the vertex).
**Refer to attached diagram**
DC, EA and FB are the perpendicular lines drawn from the three vertices D, E and F.
[tex]\sf Let\:(x_1,y_1)=D\implies(x_1,y_1)=(1,3)[/tex]
[tex]\sf Let\:(x_2,y_2)=E\implies (x_2,y_2)= (-1,0)[/tex]
[tex]\sf Let\:(x_3,y_3)= F\implies (x_3,y_3)=(2,-1)[/tex]
G (x, y) is the intersection point of the three altitudes of the triangle, i.e. the orthocenter.
To find the point of intersection (orthocenter), we need to create linear equations for the altitudes, then equate them to find the point of intersection.
We only need to work with 2 altitudes.
Calculate the slopes of two sides of the triangle.
[tex]\sf \implies Slope\:of\:EF=\dfrac{y_3-y_2}{x_3-x_2}=\dfrac{-1-0}{2-(-1)}=-\dfrac{1}{3}[/tex]
[tex]\sf \implies Slope\:of\:DF=\dfrac{y_3-y_1}{x_3-x_1}=\dfrac{-1-3}{2-1}=-4[/tex]
As the slopes of the altitudes are perpendicular to the slopes of the sides:
[tex]\textsf{Slope of DC}=\sf \dfrac{-1}{\textsf{slope of EF}}=\dfrac{-1}{-\frac{1}{3}}=3[/tex]
[tex]\textsf{Slope of EA}=\dfrac{-1}{\textsf{slope of DF}}=\sf \dfrac{-1}{-4}=\dfrac{1}{4}[/tex]
Using point D (1, 3) and the slope of DC to create a linear equation for DC using the point-slope form of a linear equation:
[tex]\sf \implies y-y_1=m(x-x_1)[/tex]
[tex]\sf \implies y-3=3(x-1)[/tex]
[tex]\implies \sf y=3x[/tex]
Using point E (-1, 0) and the slope of EA to create a linear equation for EA using the point-slope form of a linear equation:
[tex]\sf \implies y-y_1=m(x-x_1)[/tex]
[tex]\sf \implies y-0=\dfrac{1}{4}(x-(-1))[/tex]
[tex]\sf \implies y=\dfrac{1}{4}x+\dfrac{1}{4}[/tex]
To find the x-value of the point of intersection (the orthocenter), equate the equations and solve for x:
[tex]\implies \sf 3x=\dfrac{1}{4}x+\dfrac{1}{4}[/tex]
[tex]\implies \sf \dfrac{11}{4}x=\dfrac{1}{4}[/tex]
[tex]\implies \sf 11x=1[/tex]
[tex]\implies \sf x=\dfrac{1}{11}[/tex]
To find the y-value of the point of intersection (the orthocenter), substitute the found value of x into the equation for line DC:
[tex]\implies \sf y=3\left(\dfrac{1}{11}\right)[/tex]
[tex]\implies \sf y=\dfrac{3}{11}[/tex]
Therefore, the orthocenter is at point:
[tex]\sf \left(\dfrac{1}{11},\dfrac{3}{11}\right)[/tex]
Related Questions
The work of a student to find the dimensions of a rectangle of area 8 + 16x and width 4 is shown below:
Step 1: 8 + 16x
Step 2: 4(4) + 4(12x)
Step 3: 4(4 + 12x)
Step 4: Dimensions of the rectangle are 4 and 4 + 12x
In which step did the student first make an error, and what is the correct step?
Answers
Answer:
Step 1: 8 + 16x Step 2: 4(4) + 4(12x) Step 3: 4(4 + 12x) Step 4: Dimensions of the rectangle are 4 and 4 + 12x In which step did the student first make an error and what is the correct step? Step 3; 4 + (4 + 12x) Step 3; 4 + (4 ⋅ 12x) Step 2; 4(2) + 4(4x) Step 2; 4(2) + 4x(2)
The square root of 83
Answers
Answer:
The answer would be 9.1104335791443.
WILL GIVE BRAINLIEST Drag the values into the boxes to classify each number as rational or irrational.
Answers
Answer:
See below ~
Step-by-step explanation:
5/4 = 1/25 ⇒ Rational√100 = 10 ⇒ Rational7.3 ⇒ Rational0.00247 ⇒ Rational√4/9 = 2/3 = 0.66666... ⇒ Rational√2/24 = 1/2√1/6 ⇒ Irrational0.01.... ⇒ RationalA chemist is mixing a 50% salt solution with an 18% salt solution to make 40 L of a new solution that will contain 30% salt. How much of each of the original solutions should the chemist use?
Answers
Answer:
See below
Step-by-step explanation:
Sum of ingredients = sum of final solution
.50 x where x = 50 percent amount then 40 - x = 18% amount
.50 x + .18(40 - x) = .30 ( 40 )
.50 x + 7.2 - .18x = 12
.32x = 4.8
x = 15 L of 50% 25 liters of 18%
If an item has a retail price = $475 and a discount of $45, what percentage discount is offered?
Answers
Answer:
A 9.47% discount was given
Step-by-step explanation:
The question is asking us to find how much percent of 475 is 45. Therefore we do the following:
[tex] \frac{45}{475} \times \frac{100}{1} = \frac{4500}{475} [/tex]
after dividing we get 9.473 which is approximately 9.47%
To check if what we got is correct we do the opposite and instead find 9.47% of 475.
[tex] \frac{475}{1} \times \frac{9.47}{100} = \frac{4498.25}{100} [/tex]
After dividing we find that is equal to 44.98 which is approximately 45.
1. How many triangles are congruent to AABC?
А.
2
В.
3
С.
4
D.
5
Answers
Answer:
Answer: 3, 3 of them are translated
Answer:
B. 3 triangles are congruent to ABC
Step-by-step explanation:
2 triangles are congruent if all corresponding side lengths are equal, or using SAS (Side-Angle-Side theorem) to confirm 2 corresponding sides & the included angle is equal those of another triangle.From the picture, triangle ABC is not a right triangle. The length of AB is graphed with a length of 4. From point C to segment AB is a height of 3. The triangle in quadrant I (top right) has both of these dimensions.
Similarly, the triangle on the x-axis (positive side) below it (between Q1 and Q4) is also congruent to ABC.
The triangle fully in quadrant III (bottom left) is congruent to ABC with a height of 3 and base length of 4.
The triangle in the bottom right corner (Q4) has a height of 4, therefore it is not congruent to ABC which has a height of 3.
The triangle on the x-axis (negative left side) is also not congruent to ABC. It may have a height of 3 and base length 4, but the angles are different which means the other side lengths don't correspond to ABC.
[+ I'm not sure if your teacher would consider this a real answer but I figured I'd mention that in proofs tri.ABC is also congruent to tri.ABC.]
HELPPPPP QUICKLYYY !!! The picture shows a cylinder of height x which has been inscribed in a sphere of radius r. Find the volume of a cylinder in terms of x.
Answers
See attached. We can draw a right triangles contained within both shapes.
The triangle has height x (same as the cylinder), base 2r (same as the diameter of the cylinder), and hypotenuse 6 (same as the diameter of the sphere).
Since it's a right triangle, the Pythagorean theorem holds and we have
x² + (2r)² = 6²
which we solve for r to get
x² + 4r² = 36
4r² = 36 - x²
r² = 9 - x²/4
r = √(9 - x²/4)
Then the volume of the cylinder, whose height is x and whose base has radius r, is
π r² x = π (9 - x²/4) x = (36x - x³) π/4
helpppppppp asappppppppppp pleaseeee
Answers
Length of sides:
A = 4 ftB = 8 ftC = 6 ftD = 10 ftSurface Area:
(perimeter of base)(length) + (base)(height)
(8+10+6)(4)+(8)(6)
(24)(4)+48
96+48
144 ft²
How many possible outcomes if I choose an odd number between 0 and 10 and a letter of an alphabet?
Answers
Answer:
104, with the assumption that "an alphabet" is the English alphabet
Step-by-step explanation:
26*4
Libby built a fence that was 56 \text { feet}56 feet56, start text, space, f, e, e, t, end text long over 444 days. She built the same length of fence each day.
How many inches of fence did Libby build each day?
Answers
Answer:
168 inches
Step-by-step explanation:
To find the fence made each day by Libby, we will divide 56 by 4.
convert the length of fence into inches.
1 foot = 12 inches.
A star with an initial radius of r=1. 0108 m and a period of 30. 0 days collapses suddenly to a radius of r=1. 0104 m
Answers
The initial angular velocity of the star will be [tex]w=2.4\times 10^{-6} \ \frac{rad}{sec}[/tex].
What is angular velocity?Angular velocity refers to how fast an object rotates or revolves relative to another point. It is a rate of change of angle.
Here for the 30 days period the angular velocity will be calculated as:
[tex]W=\dfrac{2\pi}{T}=\dfrac{2\times \pi}{30\times 24\times 3600}=2.4\times 10^{-6} \ \frac{rad}{sec}[/tex]
So the initial angular velocity of the star will be [tex]2.4\times 10^{-6} \frac{rad}{sec}[/tex]
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9) How much ice cream can the sugar cone hold if
you only fill the inside? Round your answer to the
nearest cubic millimeter.
60 mm
136 mm
140 mm
Answers
Answer:
13188mm^2
Step-by-step explanation:
D=60mm
H=136mm
L=140mm
then v=1/3xd^2x h= 1/3x3.14x60^2x136
SA=tc pl/2
=3.14x60x140= 13188mm^2
hope this helps!
The smallest angle of rotational symmetry for a regular polygon is 30°. how many sides does the regular polygon have? 9 12 20 30
Answers
The regular polygon must have 12 sides so the smallest angle of rotational symmetry is 30°
How many sides the polygon have?We know that the smallest angle of rotational symmetry is 30°, then the exterior angles must measure that.
Remember that the sum of the exterior angles of a regular polygon must be equal to 360°, then we have:
360° = n*30°
Where n is the number of sides
Solving for n we get:
n = 360°/30° = 12
We conclude that the regular polygon has 12 sides.
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Answer: B. 12
Step-by-step explanation: On Edge!!!
Many large cities have considered or passed proposals that ban the use of plastic grocery bags in stores. A concerned student would like to measure if residents of her hometown would consider a similar ban on plastic bags. Standing in a busy shopping area, the student approaches adults and asks, "Items made of plastics are a dangerous, growing source of pollution to our world’s oceans and threaten many animal species. Given this, would you support a ban on plastic shopping bags in our city stores?” How might the results of this survey be biased in estimating the proportion of all town residents who would support a ban on plastic bags?
The student may overestimate support for a plastic bag ban because only adults in the shopping area were chosen to participate.
The student may overestimate support for a plastic bag ban because the wording of the question may influence adults to support a ban.
The student may underestimate support for a plastic bag ban because the wording of the question may influence adults to not support a ban.
The student may underestimate support for a plastic bag ban because some adults may not want to answer questions about the environment.
Answers
Answer: The student may overestimate support for a plastic bag ban because the wording of the question may influence adults to support a ban.
Step-by-step explanation:
What is the surface area of the rectangular prism?
a. 144 m 2
b. 288 m 2
c. 128 m 2
d. 256 m 2
Answers
A = 2 * l * w + 2 * l * h + 2 * w * h
Answer: B. 288m 2
Step-by-step explanation:
A = 2 (wl+hl+hw)
2 · (4 · 16 + 4 · 16 +4 · 4) = 288m 2
f(x)=4x-1. g(x)=-2|x+3|. what is the y value when f(x)=g(x)
Answers
..............................................................
PLS HELP PLS ANSWER AS SOON AS POSSIBLE
Answers
Answer:
y>1/3x-3
Step-by-step explanation:
The slope for this equation is 1/3x-3. Because the line is dashed, we know that we're dealing with either < or >. The shading goes upwards, so it's greater than. That means, we would get y>1/3x-3.
Hope this helps!
Math in My World
Example 1
Blair is arranging 18 cupcakes on a shelf. Find the
factors of 18. Use arrays to find the factors.
Answers
Answer:
445455
Step-by-step explanation:
What is the height of a cone if the radius is 3 and the volume is 40π
Answers
Then the height of the cone is 11.537
I need help with all of these questions you see on the top. there the red squares. please someone smart. this is one of the problems I need to finish this tonight.
Answers
Answer:
49°
Step-by-step explanation:
The marked angles are "vertical" angles. They have the same measure.
∠TGF = ∠HGM = 49°
_____
You don't need to know the value of p to answer this question. It is 7.
Answer:
∠TGF = 49°
Step-by-step explanation:
From the diagram, we get that ∠TGF is equal to ∠MGH because they are vertical angles.
Hence,
⇒ ∠TGF = ∠MGH
⇒ ∠TGF = 49°
jake has an account that is compounded every month at a rate of 3.5%.if he invests $20,000,what will the balance be after 5 years
Answers
You can’t have 3.5 as your rate so you divide that by 100 and you will get 0.035 and then you multiply 20,000 x 0.035 x 5 and you will get 3,500
Jake’s balance will be $3,500 after 5 years
does anyone know how to do bearings
answer as many as you like
Answers
Nyla compared 790 and 709.her work is shown at the right. Is Nyla comparison correct ? If not, correct her mistake
Answers
Answer:
its correct so you don't need to do anything
Step-by-step explanation:
The solution is, result of subtraction = 81.
What is Subtraction?Subtraction is the process of taking away a number from another. It is a primary arithmetic operation that is denoted by a subtraction symbol (-) and is the method of calculating the difference between two numbers.
here, we have,
given that,
Nyla compared 790 and 709.
so, we have,
790 - 709
=81
Hence, The solution is, result of subtraction = 81.
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Does anybody know this? I followed the example and did the formula and it told me it was 11.99 but it was wrong. Find the length of the radius if the area of the sector is 361.6 and the central angle is 288 use 3.15 round your answer to the neatest whole number
Answers
Hi Student!
The first step that we need to take to solving this problem (and almost any problem), is to concentrate the important information for the problem that is provided. Looking at the problem we can see that we are trying to determine what the length of the radius is. In order to solve this problem we are provided with the area of a sector of 361.6 meters squared and a central angle of 288 degrees.
Now that we have the important information taken care of, we can move onto the part of actually solving the problem. We must first convert our degrees to radians because the equation that we will be using can only use radians for the angle and not degrees. We can simply convert our angle to radians by multiplying it by π/180.
[tex]288\ degrees\ *\ \frac{\pi }{180\ degrees}[/tex]
[tex]288\ degrees\ *\ \frac{3.14}{180\ degrees}[/tex]
[tex]\frac{288\ degrees\ *\ 3.14}{180\ degrees}[/tex]
[tex]5.024\ radians[/tex]
After completing this step, we have converted our degrees to radians and now we can move onto the step where we plug in our information into an equation and solve for our unknown.
[tex]Sector\ Area = radius^2\ *\ \frac{angle}{2}[/tex]
[tex]361.6\ m^2 = radius^2\ *\ \frac{5.024}{2}[/tex]
[tex]361.6\ m^2 = radius^2\ *\ 2.512[/tex]
[tex]\frac{361.6\ m^2}{2.512} = \frac{radius^2\ *\ 2.512}{2.512}[/tex]
[tex]143.949\ m^2 = radius^2[/tex]
[tex]\sqrt{143.949\ m^2} = \sqrt{radius^2}[/tex]
[tex]12\ m = radius[/tex]
Therefore, after solving our equation for our unknown, we can see that our radius is really close to 12 meters (so we can just round up as the problem stated).
the volume of a cube with edge lengths of 2/3 units
What is the edge length of a cube with volume 27/64 cubic units?
Pleas helppppppppppppp
Answers
Answer:
[tex]a=\sqrt[3]{\frac{27}{64}} = 0.75[/tex]
edge length = 0.75
Equation: [tex]a=\sqrt[3]{V}[/tex]
A. 225
B. 25
C. 150
D. 50
Answers
Step-by-step explanation:
V=1/3hπr².
(h=3cm, r=5²cm
V= 1/3 x 3 x π x 5²
V= 25π
A letter is chosen at random from the word CALIFORNIA. Find the probability of all the letters in the word
Answers
Answer:
C - 1/10A - 2/10L - 1/10I - 2/10F - 1/10O - 1/10R - 1/10N - 1/10Do not put A and I more than once, we already measured it before.
10 letters in all
Find how many there are per letter
C - 1/10
A - 2/10
L - 1/10
I - 2/10
F - 1/10
O - 1/10
R - 1/10
N - 1/10
Consider z1 = 4 – 4 startroot 3 endrooti and z2 = –16i. What is z1 – z2? 4 – 12startroot 3 endroot i 4 12startroot 3 endroot i 4 – 4( startroot 3 endroot – 4)i 4 4 (startroot 3 endroot – 4)i
Answers
The answer is: [tex]z_1-z_2[/tex]= 4-4([tex]\sqrt{3i}[/tex]-4i)
The correct option is: (D)
What are Complex numbers?A complex number is the sum of a real number and an imaginary number. A complex number is of the form a + ib and is usually represented by z. Here both a and b are real numbers. The value 'a' is called the real part which is denoted by Re(z), and 'b' is called the imaginary part Im(z).
Since we have given : [tex]z_1[/tex]==4-4[tex]\sqrt{3i}[/tex] and [tex]z_2[/tex]= -16i
The difference of z1 - z2 would be:
So,
[tex]z_1-z_2[/tex] = 4-4[tex]\sqrt{3i}[/tex] - (-16i)
[tex]z_1-z_2[/tex] =4-4[tex]\sqrt{3i}[/tex]+16i
[tex]z_1-z_2[/tex] =4-4([tex]\sqrt{3i}[/tex] -4i)
Hence, [tex]z_1-z_2[/tex]= 4-4([tex]\sqrt{3i}[/tex]-4i)
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Answer:
4 – 4( StartRoot 3 EndRoot – 4)i
it's C.
Step-by-step explanation:
took the L for yall bc my shtoopid ahh thought it was D, which is 4 + 4
A middle school took all of its 6th grade students on a field trip to see a symphony at a theater that has 3080 seats. The students filled 1078 of the seats in the theater. What percentage of the seats in the theater were filled by the 6th graders on the trip?
Answers
Answe
about 30-45 Percent.
Step-by-step explanation:
A 10 inch long piece of string is approximately 25.4 centimeters long.
What is the length, in centimeters, of a string that is 45 inches long, rounded to the nearest tenth?
Answers
If 10 inches is equal to 25.4 centimeters. Then the 45 inches will be equal to 114.3 centimeters.
What are ratio and proportion?A ratio is an ordered set of integers a and b expressed as a/b, with b never equaling 0. A percentage is a mathematical expression in which two things are equal.
A 10 inches long piece of string is approximately 25.4 centimeters long.
If the length of the string is 45 inches. Then the length in cm will be
Let x be the length. Then we have
x/45 = 25.4/10
x = 114.3 cm
More about the ratio and the proportion link is given below.
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