Hey i need help finding the orthocenter for the following points, please show the steps you used. thanks
(-3, 0.5)
(3,3)
(2,-1)
Answers
9514 1404 393
Answer:
(81/43, -31/43) ≈ (1.884, -0.7209)
Step-by-step explanation:
The orthocenter is the intersection point of the altitudes of the triangle. Since this is not a right triangle, you need to find the equations of the altitude lines, then find where those intersect.
If we call the points A, B, C in the order given, the attached figure shows the altitude lines from B perpendicular to AC, and from C perpendicular to AB.
We can find the equation of the perpendicular line using the form ...
∆x·x +∆y·y = c
where ∆x and ∆y are the differences in coordinates between the points on the "side" and the value of constant 'c' is determined by evaluating the expression using x and y as the coordinates of the opposite vertex.
__
For line AC, we have ...
(∆x, ∆y) = C - A = (2, -1) -(-3, 0.5) = (2+3, -1-0.5) = (5, -1.5)
Then the equation of the first altitude line is ...
5x -1.5y = 6(3) -1.5(3) = 10.5
Multiplying by 2 puts this in standard form:
10x -3y = 21
__
For line AB, we have ...
(∆x, ∆y) = B - A = (3, 3) -(-3, 0.5) = (3+3, 3-0.5) = (6, 2.5)
Then the equation of the second altitude line is ...
6x +2.5y = 6(2) +2.5(-1) = 9.5
Multiplying by 2 puts this in standard form:
12x +5y = 19
__
Then the orthocenter is the solution to the system of equations ...
10x -3y -21 = 0
12x +5y -19 = 0
This set of equations can be solved a number of ways. One is the "cross-multiplication" method.
1/(10·5 -12·(-3)) = x/(-3·(-19) -5·(-21)) = y/(-21(12) -(-19)(10))
x = 162/86 = 81/43
y = -62/86 = -31/43
The orthocenter is (81/43, -31/43) ≈ (1.884, -0.7209).
_____
Additional comment
The "cross multiply method" is based on cross products when the equations are written in standard form:
ax +by +c = 0
dx +ey +g = 0
Usually, the coefficients are written in two rows of four, with the first column repeated at the end:
a b c a
d e g d
Then cross products are formed, where diagonal down is positive and diagonal up is subtracted from that:
p1 = (ae -db), p2 = (bg -ec), p3 = (cd -ga)
These give the solution to the equations when put in the form ...
1/p1 = x/p2 = y/p3 ⇒ x = p2/p1 and y = p3/p1
If you look at videos of this "cross multiply method," they will put the 1/p1 at the end of the string of equalities, and they will use the coefficients in the order starting with b instead of a. I find the arrangement here easier to remember, so less likely to get confused.
Further comment
You can, of course, find the slopes of the lines, and their opposite reciprocals, then use a point-slope form or some other method for writing the equations of the altitude lines. The solution to the equations can proceed by substitution or elimination. No doubt these methods are somewhat familiar. The method used here tries to get to the answer much more quickly with a lot less writing. (Unfortunately, it takes a bit of writing to explain it all.)
Related Questions
A new food delivery robot on a college campus is doing test runs at a constant speed of 4 miles per hour. The average test run is 3 miles from the campus cafeteria but varies a distance of 1.5 miles more or less than that. If x is the number of hours the robot is performing a test run, the equation that can be used to find the minimum and maximum time (in hours) for a test run is ____. For each test run, the minimum time is ____ hours and the maximum time is ____ hours.
Answers
Answer:
For each test run, the minimum time is 45 mins or 3/4 ____ hours
and the maximum time is 67.5 minutes or 1 hour 7.5 minutes.
Step-by-step explanation:
If x is the number of hours the robot is performing a test run, the equation that can be used to find the minimum and maximum time (in hours) for a test run is 15 mins for 1 mile
For 4 miles the time is 1 hour
For 1 miles the time will be 1/4 hour
For 3 miles the time will be 3/4 hours
and for 4.5 miles the time will be 4.5/4 hours
Now reversing
1 hour = 4miles
15 mins= 1 mile
45 mins = 3 miles
30 mins = 2 miles
67.5 min= 4.5 miles
For each test run, the minimum time is ____ hours
the minimum distance is 3 miles and time taken is 45 mins
and the maximum time is ____ hours.
The maximum distance is 3+ 1.5= 4.5 miles and time taken is 67.7 minutues or 1 hour 7.5 minutes.
Which ordered pair does NOT lie on the graph of h(x) = -2x + 7?
A(-1,9)
B(-2,3)
None of the other answers are correct C(3, 1)
D (4,-1)
Answers
The temperature of a pot of water is 50⁰ F. The temperature rises 15⁰ per minute. Write an equation below in y=mx + b form to express this situation.
Answers
Answer:
The equation to express this situation is y = 15x + 50.
Step-by-step explanation:
Initially, the temperature of a pot of water is 50⁰, i.e. at after x = 0 minutes the temperature was, y = 50⁰.
And the temperature rises 15⁰ per minute.
So, after x₁ = 5 minutes, the temperature was, y₁ = 125⁰.
And similarly after x₂ = 10 minutes, the temperature was, y₂ = 200⁰.
Compute the equation to express this situation as follows:
[tex](y-y_{1})=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\cdot (x-x_{1})\\\\(y-125)=\frac{200-125}{10-5}\times (x-5)\\\\y-125=15x-75\\\\y=15x-50[/tex]
Thus, the equation to express this situation is y = 15x + 50.
7a(3b - 2c + 4)
Help?
Answers
Answer:
21ab-14ac+28aStep-by-step explanation:
[tex]7a(3b - 2c + 4)\\\\\mathrm{Distribute\:parentheses}\\\\=7a\times \:3b+7a\left(-2c\right)+7a\times\:4\\\\\mathrm{Apply\:minus-plus\:rules}\\\\+\left(-a\right)=-a\\\\=7\times\:3ab-7\times\:2ac+7\times\:4a\\\\Simplify\\\\21ab-14ac+28a[/tex]
Choose the parabola that represents the trinomial below by finding the
correct vertex.
x2 + 7x + 12
Answers
HELP MEEE. :)
giving out brainliest
Answers
Answer:
(5, 7)
Step-by-step explanation:
before the translation, A is (2, 5)
by adding 3 to A and 2 to y, we get (5, 7)
Which angles are corresponding angles?
Answers
Answer:
I and H
Step-by-step explanation:
Nick bought 6 tickets to an air show if he spent 156 dollars how much did the tickets cost
Answers
Answer:
$26 dollars
Step-by-step explanation: when you divide 156 divided by 6 you get 26
Answer:
26 dollars per ticket
Step-by-step explanation:
156/6=26
26x6=156
A boat travels with velocity vector (25, 25 StartRoot 3 EndRoot). What is the directional bearing of the boat?
Answers
Answer:
The directional earing is N 30° E
Step-by-step explanation:
The given velocity vector is (25, 25·√3), therefor, we have;
R = 25·i + 25·√3·j
The angle of inclination of the direction of the boat with the horizontal = Arctan (y/x)
Where;
(x, y) = The coordinate of the components of the velocity of the boat
Therefore;
Arctan (y/x) = Arctan (25·√3/25) = Arctan (√3) = 60° North of East
The directional bearing which is the bearing relative to the northern direction is given as follows;
The directional earing = 90° - 60° = N 30° E.
Answer:
A on Egde
Step-by-step explanation:
i got it right
given the graph g(x), find g(-6)
Answers
Answer:
+2
Step-by-step explanation:
Draw a vertical line through x = -6. Determiene the corresponding y-value. It is +2. This is g(-6).
The value of the function g(x) when x = - 6, g(-6) is 2.
What is a function?A function can be defined as the outputs for a given set of inputs.
The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.
Given, A function g(x).
Now, The value of the function g(-6) is the same as the value of the y-coordinate when the x-coordinate is - 6.
From the graph, we observe that when x = - 6, y = 2.
So, The value of g(-6) = 2.
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Trick or treaters went trick or treating in their neighborhood,if there are 40 homes in the neighborhood and 3/4 of them were home to give out candy,how many homes did they visit
Answers
How many isosceles triangles with whole-number length sides have a perimeter of 20 units?
Answers
Answer:
There are 9 possible isosceles triangles
Step-by-step explanation:
Inequalities
Let's consider an isosceles triangle with equal side lengths x and unequal side y.
The perimeter of that triangle is:
P = x + x + y =2x + y
The perimeter has a fixed value of 20 units, thus:
2x + y = 20
Solving for y:
y = 20 - 2x
The solutions of this equation have two restrictions:
* Both x and y must be positive
* Both x and y are integer (whole) numbers.
The first restriction leads to inequality:
20 - 2x > 0
Solving for x:
x < 10
Since x must also be positive, only the following numbers are valid solutions for x: {1,2,3,4,5,6,7,8,9}
There are 9 solutions for x, giving this set of 9 solutions for y: {18,16,14,12,10,8,6,4,2}
Thus, there are 9 possible isosceles triangles
Raj made a profit of $890 selling T-shirts. He wants to share the profit equally among three charities. How much should he give to each charity?
Answers
Answer:
890/3 thats the anwser
Step-by-step explanation:
Please Answer ASAP
Evaluate the expression for m = 8, n = 11, and p = –3.
what is mp − m − n =
Answers
Answer:
-43
Step-by-step explanation:
m=8, n=11, and p=-3
mp=8x-3=-24
-24-m=-24-8=-32
-32-n=-32-11=-43
Help I have 7 minutes
Answers
Answer:
c. is your answer dear.
Step-by-step explanation:
By SSS theorem which is Side Side Side.
Hope you got it.
What are the points of intersection between the graphs of the functions
f(x) = x²(x + 1) and g(x) = x + 1?
Answers
Answer:
(x1,y1) = (1, 2)
(x2,y2) = (-1,0)
Step-by-step explanation:
Given: f(x) = x²(x + 1) and g(x) = x + 1
To find the intersection, then set functions equal to each other to solve for x first.
x²(x + 1) = x + 1; divide by (x+1) from both sides
[tex]x^{2}[/tex] = 1; take the square root to both sides
x = ±1;
The graphs intersect at two points (x1,y1) and (x2,y2)
Use x=1
f(1) = x²(x + 1) = [tex]1^{2}[/tex](1+1) = 2
g(1) = x + 1 = 1 + 1 = 2
(x1,y1) = (1, 2)
Use x= -1
f(-1) = x²(x + 1) = [tex](-1)^{2}[/tex](-1+1) = 0
g(-1) = x + 1 = -1 + 1 = 0
(x2,y2) = (-1,0)
The points of intersection of two functions are required.
The points of intersection are [tex](-1,0)[/tex] and [tex](1,2)[/tex].
The given functions are
[tex]f(x)=x^2(x+1)[/tex]
[tex]g(x)=x+1[/tex]
Equating them with each other
[tex]x^2(x+1)=x+1\\\Rightarrow x^2=\dfrac{x+1}{x+1}\\\Rightarrow x^2=1\\\Rightarrow x=\pm 1[/tex]
Substitute the [tex]x[/tex] value in one of the functions.
[tex]x=1[/tex]
[tex]f(1)=1(1+1)=2[/tex]
[tex](1,2)[/tex]
[tex]x=-1[/tex]
[tex]f(-1)=(-1)^2(-1+1)=0[/tex]
[tex](-1,0)[/tex]
The points of intersection are [tex](-1,0)[/tex] and [tex](1,2)[/tex].
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1 litre of a liquid P has a mass of p grams. 1 Litre of a liquid Q has a mass of q grams. A liquid R is made by mixing a volume of liquid P with a volume of liquid Q in the ratio 3:7. Find an expression, in terms of p and q, for the mass of 50 litres of liquid R
Answers
Answer:
mass of 50 L of R = 15p + 35q
Step-by-step explanation:
1L of P = p g
1L of Q = q g
R = 3P + 7Q
total = 10
P = 3/10 R
Q = 7/10 R
R = 50 L
50 L × 0.3 = 15 L the amount of P in L
15 L of P = 15p g
35 L 0f Q = 35 q g
then
mass of 50 L of R = 15 p + 35 q grams
I need help plz like ASAP
I will give you a Brainist if you answer first
Answers
Answer:
Step-by-step explanation:
The proportion of x/y is 8/1. And to solve your question the value of y when x is 10 is 1.25 .
Can someone help me this is due today
Answers
Answer:
13 miles
Step-by-step explanation:
It tells us that her house is 5 miles from the Post Office. So we know that each unit on the grid is 1 mile. Lucy walks from her house to the post office to the bank. The shortest path is the hypotenuse of the right triangle. We have one side length. Now, we count the post office to bank's length. It is 12 miles. The Pythagorean theroem is a^2+b^2=c^2 so 5^2+12^2=c^2 Solving this, we get c=13. The shortest path is 13 miles.
What is the interest you will pay if you borrowed $900 at 10% interest for 2 years?
Answers
Answer:
$180
Step-by-step explanation:
Simple interest formula:
I = R x T x P divided by 100
=(900 x 2 x 10) / 100
=18000 / 100 = 180
=$180
There are 12 pieces of candy in Miguel's Halloween bag. 3 are Reese's, 4 are M&M's, 3 are lollipops and 2 are sour gummy worms. What is the ratio of Reese's to the total number of pieces of candy in the bag and is this a part to part, part to whole or whole to part relationship?
Pick the correct answer
1. 3:12 part to part
2. 12:3 part to part
3. 3:12, part to whole
4. 12:3 part to whole
don’t use this question for points plz!
Answers
Answer:
3. 3:12 part to whole
Step-by-step explanation:
because there are 3 Reese's & 12 pieces of candy in total so it would be part of the candy to the entire bag of candy.
Find the GCF of 24, 48, and 480.
Answers
Answer:
The common factors for 24 , 48 , 480 are
1 , 2 , 3 , 4 , 6 , 8 , 12 , 24
1 , 2 , 3 , 4 , 6 , 8 , 12 , 24
1 , 2 , 3 , 4 , 6 , 8 , 12 , 24
The GCF (HCF) of the numerical factors 1,2,3,4,6,8,12,24
Step-by-step explanation:
Answer:
24
Step-by-step explanation:
Prime factorization.
24= 2*2*2*3
48= 2*2*2*2*3
480= 2*3*2*5*2*2*2
The numbers that appear the most often for all 3 numbers is: 2, 3, 4. I got 4 because of 2*2.
Multiply 2*3*4 which equals 24. The GCF of 24, 48, and 480 is 24.
plz help asap i need the right answer who ever is right get brainiest
Answers
X=18
Step-by-step explanation:
Solve -14=p - 11. Which statement shows the correct justification for the step used to solve the equation? Add 14 to each side. Add 11 to each side Subtract 14 from each side Subtract 11 from each side
Answers
Answer: 47
Step-by-step explanation: I’m not in high school so how would I know I just want to get brainly points
ABC is a central angke in a cirvle with center B. The length of the radius is 13 km and m
Answers
Answer:
[tex]\frac{1170pi}{360m}[/tex]km
Step-by-step explanation:
To find the length of the minor arc, you find the circumference of the circle multiplied by the arc angle, which is [tex]\frac{45}{m}[/tex] over 360 degrees.
Circumference of the circle with center B = 13×2×pi
= 26pi
Length of minor arc = 26pi × [tex]\frac{45}{m}[/tex] ×[tex]\frac{1}{360}[/tex]
= [tex]\frac{1170pi}{360m}[/tex]km
Which equation matches the story: Elena spends $17 and buys a $3 book and a bookmark for each of her 5 cousins. How much does each bookmark cost?
3(x+5)=17
3x+5=17
5(x+3)=17
5x+3=17
Answers
Answer:
$2.80
Step-by-step explanation:
17-3=14
14 divided by 5=
2.8
Add 0 since it's about money.
$2.80
What is the value of 2/11 divided by 3/11? *
5 points
6/121
2/3
3/2
121/6
Answers
Answer:
the answer is 2/3
Step-by-step explanation:
(2/11)/(3/11)=2/3
Answer:
Exact Form:
2 /3
Decimal Form:
0. 6 repeating
Step-by-step explanation:
Find the y-intercept of the line on the graph.
Enter the correct answer
Answers
85 ÷5 jejeisisossjjxjsjsjskkskxkxkxo
Answers
Answer:
17
Step-by-step explanation:
Which graph shows the solution to the inequality −2(2x+3)≤26?
Answers
Answer:
x≥−8
Step-by-step explanation:
Step 1: Simplify both sides of the inequality.
−4x−6≤26
Step 2: Add 6 to both sides.
−4x−6+6≤26+6
−4x≤32
Step 3: Divide both sides by -4.
−4x/−4 ≤ 32/−4
x≥−8