Find the orthocenter& circumcenter of a triangle when their vertices are A(1, 2), B(2, 6), C(3,-4).
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Note:
Ortho centre :a point of intersection of altitudes of a triangle meets the opposite angle.
Given:
For Orthocentre:.
A(1, 2), B(2, 6), C(3,-4). are vertices of a triangle:
Slope of AB[m1]=[tex] \frac{6-2}{2-1} [/tex]=4
Since it is perpendicular to CX.
slope of CX=m2
we have for slope of perpendicular
m1m2=-1
m2=-¼
It passes through the point C(3,-4)
equation of line CX becomes;
(y-y1)=m(x-x1)
y+4=-¼(x-3)
4y+16=-x+3
x+4y+16-3=0
x+4y+13=0........[1]
again:
Slope of AC[m1]=[tex] \frac{-4-2}{3-1} [/tex]=-3
Since it is perpendicular to BY
slope of BY=m2
we have for slope of perpendicular
m1m2=-1
m2=⅓
It passes through the point B(2,6)
equation of line BY becomes;
(y-y1)=m(x-x1)
y-6=⅓(x-2)
3y-18=x-2
x-3y+18-2=0
x-3y+16=0.........[2]
Subtracting equation 1&2.
x+4y+13=0
x-3y+16=0
-__________
7y-3=0
y=[tex] \frac{3}{7} [/tex]
again
Substituting value of y in equation 1.
x+4*[tex] \frac{3}{7} [/tex]+13=0
x=-13-[tex] \frac{12}{7} [/tex]
x=[tex] \frac{-103}{7} [/tex]=-14[tex] \frac{5}{7} [/tex]
So
orthocenter is (-14[tex] \frac{5}{7} [/tex],[tex] \frac{3}{7}[/tex])
And for circumcenter.Circumcentre: a point of intersection of perpendicular bisector of the triangle.
Now
X,Y and Z are the midpoint of AB,AC and BC respectively.
X(a,b)=([tex] \frac{2+1}{2} [/tex],[tex] \frac{2+6}{2} [/tex])=
([tex] \frac{3}{2} [/tex],4)
Slope of AB=4
Slope of OX=-¼
Equation of line OX passes through ([tex] \frac{3}{2} [/tex],4)is
y-4=-¼(x-[tex] \frac{3}{2} [/tex])
4y-16=-x+[tex] \frac{3}{2} [/tex]
8y-16*2=-2x+3
2x+8y=3+32
2x+8y=35
x+4y=[tex] \frac{35}{2} [/tex]........[1]
again
Y(c,d)=([tex] \frac{3+1}{2} [/tex],[tex] \frac{-4+2}{2} [/tex]=(2,-1)
Slope of AC:-3
Slope of OY=⅓
Equation of line OY passes through (2,-1) is
y+1=⅓(x-2)
3y+3=x-2
x-3y=3+2
x-3y=5......[2]
Multiplying equation 2 by 3 and
Subtracting equation 1&2.
x+4y=35/2
x-3y=5
-_______
7y=[tex] \frac{25}{2} [/tex]
y=[tex] \frac{25}{14} [/tex]
Substituting value of y in equation 2.
x-3*[tex] \frac{25}{14} [/tex]=5
x=5+[tex] \frac{75}{14} [/tex]
x=[tex] \frac{145}{14} [/tex]
x=10[tex] \frac{5}{14} [/tex]
circumcenter of a triangle: (10[tex] \frac{5}{14} [/tex],1[tex] \frac{11}{14} [/tex])
Related Questions
in a school, students either study french, german or spanish. They study one language each. 11 percent of students study french 27percent students study spanish. what fraction of the students study German
Answers
Answer:
62%
Step-by-step explanation:
First, you got the formulate: 1-27%-11%
Second, calculate the difference: 0.73-0.11
Third, the sum is 0.62
Then you multiply 0.62 ×100\100
Write a single fraction: 0.62×100\100
Calculate the product or quotient: 62\100
100% to a percentage: 62%
That is your answer 62%
Find the missing length and area of each figure
Answers
LM = √(8² + 6²) = √(64 + 36) = √100 = 10 yd
Area = 2 × (1/2) × 6 × 8 = 48 yd²
PLEASE HELP W THE QUESTIONS IN IMAGE ILL GIVE BRAINLIEST
Answers
Answer:
53
Step-by-step explanation:
Nassim spends $15.00 on colored pencils, $10.00 on a sketchbook, $3.50 on erasers, and $4.95 on a colored pencil blender. He hands the clerk $35.00.
What could he receive in change?
Answers
Answer: 1.55
Step-by-step explanation:
55 cents or 0.55
step-by-step explanation:
15 + 10 + 3.50 + 4.95
33.45
35 - 34.45
0.55
PLEASE SOMEONE HELPPPPP
Simplify (36)18
O a
Ob
Oc
Od
33
312
324
3108
Answers
Answer:
d. 3^108
Step-by-step explanation:
Just multiply 6 and 18.
slope intercept form of y + 6 = (x - 4)
Answers
remove the parenthesis
y + 6 = x - 4
y + 6 + 4 = x
add 4 to both sides
y + 10 = x
simplify & switch sides to x being the solution
x = y + 10
and here’s your answer
The sales tax in one town is 8%.So, the total cost of an item can be written as c=0.08cents .what is the total cost of an that sells for $12 ?its due tmrr asap
Answers
Answer: 12.96
Step-by-step explanation: you should multiply 12 * 0.08 = 0.96
then just add 0.96 + 12 = 12.96
help please due monday
Answers
Answer:
I think it is D, but I am not sure.
Step-by-step explanation:
Have a nice day!
What is the slope of 5x 2y =- 3?
Answers
The slope of 5x 2y =- 3 is -5/2.
To find the slope of the equation 5x 2y =- 3, we need to rearrange it into the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).
We start by rearranging the equation 5x 2y = -3 so that all the terms with y are on one side and all the terms with x are on the other side.
5x -2y = -3
Now, we divide both sides by -2 to isolate y.
-5x/2 = y + 3/2
Finally, we subtract 3/2 from both sides to get the equation in the slope-intercept form.
-5x/2 = y - 3/2
Therefore, the slope of the equation 5x 2y = -3 is -5/2.
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Why are sine and cosine ratios always less than 1?
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Answer:
Step-by-step explanation:
The simple explanation is that the hypotenuse and sides of a right triangle are always shorter than one another. Therefore, any side's to any hypotenuse's ratio will never be greater than 1. The figure is no longer a triangle when the angle is 0 or 90 degrees because one of the sides vanishes and the other side lines up with the hypotenuse. For this reason, sin and cosine have values of 0 or 1 in these extreme situations.
Someone please help me !!???? ASAP show steps please
Answers
Answer:
To find the answer, use the Pythagorean theorem. It would be the last choice.
Step-by-step explanation:
The formula for the Pythagorean theorem is:
a^2 + b^2 = c^2.
a = 10
b = 15
c = x
Make your equation.
10^2 + 15^2 = x^2.
Hope this helps!
The temperature at a location was recorded for 5 days. The median temperature was 72
degrees. On four of the days, the temperatures were 67, 72, 77, and 52 degrees. Which of the
following is not a possible temperature for the fifth day?
83
79
80
70
Answers
Answer:
70
Step-by-step explanation:
First, put the values they gave you in order from least to greatest.
52 62, 72, 77
In order for the median to be 72, it has to be in the middle of the data set. Since there are 2 values on the left of 72, there must be 2 values on the right. The data set is also in order from least to greatest, so we know that the last value has to be over 72. Because it is placed to the right of 72. The only value of the options that isn't greater than 72 is 70.
The unlikely temperature of the fifith day is 70.
The The median of an observation is the number that is in the middle of a set of numbers arranged in ascending or descending order.
The median temperature is 72. So, it has to lie in the middle of the observation. Let the temperature of the fifth day be represented with x. The unknown temperature has to be greater than 72 because there are already two numbers less than 72.
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If gas costs $5.09 per gallon, how many whole gallons of gas could you buy with $54.00?
Answers
Answer:
10 gallons
Step-by-step explanation:
You spin a spinner with 4 sections numbered 1-4and then flip a coin. List all the possible outcomes with a tree diagram.
Answers
Answer:
/ heads
1
\ tails
/ heads
2
\ tails
/ heads
3
\ tails
/ heads
4
\ tails
There arent any answers for this one online
Answers
Answer:
I believe it is 240 ft
Step-by-step explanation:
A traffic study has shown that the probability that 5 cars will pass over a small bridge in a 4-minute period is 0.16.
What are the odds against exactly 5 cars passing over the bridge in that time?
Answers
The odds against exactly 5 cars passing over the bridge in that time are given as follows:
5.25:1.
What is the relation between probabilities and odds?A probability is calculated as the division of the number of desired outcomes by the number of total outcomes.
An odd, conversely, is calculated as the division of the number of desired outcomes by the number of non-desired outcomes.
The probability(in favor) that 5 cars will pass over a small bridge in a 4-minute period is 0.16, hence the odds are given as follows:
0.16 : (1 - 0.16) = 0.16 : 0.84.
To obtain the odds against, we must exchange the desired and the non-desired outcomes, meaning that the odds against is then given as follows:
0:84 : 0.16.
As the quotient of 84 by 16 is of 5.25, the simplified odd is given as follows:
5.25 : 1.
Meaning that it is 5.25 times more likely that exactly 5 cars do not pass over the bridge over that time rather that they do pass.
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how do i solve this?
Answers
Answer:
Degree: 4
Number of terms: 4
Leading Coefficient: 4
Constant: -20
Step-by-step explanation:
1) Expand (x+3)(x-4) into x^2 - x - 12
2) Combine like terms; equation becomes 4x^4 + x^2 + 4x - 20
3) Degree is looking for the highest exponent; in this case, it's 4
4) Number of terms is looking for how many terms there are in this equation; there are 4 terms
5) Leading Coefficient is looking for the coefficient of the term with the highest exponent, so the answer is 4
6) Constant is looking for the term without a variable
Hope this helps!
please assist with helping in the image below!!
Answers
Simplifying algebraic expressions
Answers
Answer:
-8
Step-by-step explanation:
7-6x+3x
You have to replace x for a 5.
7-6(5)+3(5)
After this you would multiple all of it.
7-30+15
Then add
7+15=22
-30+22=-8
Answer:
[tex] \sf \: 7 - 6x + 3x = - 8[/tex]
Step-by-step explanation:
Now we have to,
→ find the value of expression.
The expression is,
→ 7 - 6x + 3x
→ using (x = 5)
Let's solve the expression,
→ 7 - 6x + 3x
→ 7 - 6(5) + 3(5)
→ 7 - 30 + 15
→ 7 - 15
→ -8 => final answer
Hence, the answer is -8.
What is a log 10 transformation?
Answers
A log 10 transformation is a mathematical operation that is used to change the scale of a variable in a data set. The log 10 transformation is applied to the variable by taking the base-10 logarithm of each value in the data set. The result of this operation is a new variable that is on a different scale than the original variable.
The log 10 transformation is commonly used in statistics and data analysis to normalize data that is on a large scale, such as data that spans several orders of magnitude. By applying the log 10 transformation, the data is transformed into a more manageable scale that is easier to work with and analyze.
The log 10 transformation can also be used to stabilize the variance of a variable, this is particularly useful when the data is positively skewed. The log 10 transformation can also be useful for data that has a multiplicative relationship, for example, population growth or stock prices.
It is important to note that the log 10 transformation can only be applied to data that is positive and greater than zero because the logarithm of zero or a negative number is not defined.
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Please answer #6a and 6b
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The possible values of x for the given triangle ABC are 0.75<x<8.
What is the triangle inequality theorem?The triangle inequality theorem describes the relationship between the three sides of a triangle. According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side.
Given that, m∠A < m∠B < m∠C.
a) From triangle ABC,
BC<AC<AB
2x+3<6x<5x+8
Now, 2x+3<6x
Subtract 2x on both the sides of an inequality, we get
3<4x
Divide 4 on both the sides of an inequality, we get
0.75<x
Here, 6x<5x+8
Subtract 5x on both the sides of an inequality, we get
x<8
So, possible values of x are 0.75<x<8
b) From triangle ABC,
BC<AC<AB
3x+1<4x+8<7x+2
Now, 3x+1<4x+8
Subtract 3x on both the sides of an inequality, we get
1<x+8
Subtract 8 on both the sides of an inequality, we get
-7<x
Here, 4x+8<7x+2
Subtract 4x on both the sides of an inequality, we get
8<3x+2
Subtract 2 on both the sides of an inequality, we get
6<3x
Divide 3 on both the sides of an inequality, we get
2<x
So, possible values of x are -7<x<2
Therefore, the possible values of x for the given triangle ABC are 0.75<x<8.
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In a classroom, 1 10 of the students are wearing blue shirts , and 2 5 are wearing white shirts. There are 20 students in the classroom. How many students are wearing shirts other than blue shirts or white shirts
Answers
Answer:
10 students are wearing shirts other than blue or white.
Step-by-step explanation:
1 10 of the students in the classroom are wearing blue shirts, which is 1/1020 = <<1/1020=2>>2 students.
2 5 of the students in the classroom are wearing white shirts, which is 2/5*20 = 8 students.
Altogether, 2+8 = <<2+8=10>>10 students are wearing either blue or white shirts.
There are 20 students in the classroom, so 20-10 = <<20-10=10>>10 students are wearing shirts other than blue or white.
100 students were surveyed about which subject they liked best. 41 students said they like math, and 25 students said they liked Science. Of those students surveyed, 19 said they liked both.
How many students is the complement of math?
Answers
In 100 students, Science was rated favorably by 25 students, while math received ratings from 41 students. 19 of the students polled indicated that they approved of both. 16 students are the complement of math.
What is complement of a number?The phrase "complement" is typically used to describe a subset of a set that does not include another subset. The entire original set is obtained by adding the complement and its taking. The complement of a set is typically represented by the notations. The complement of a set is the set that includes all the elements of the universal set that are not present in the given set.
The Total number of students were 100
Students who like maths were 41
Students who like science were 25
Students who like both maths and science were 19
Students who does not like maths are 41-25 = 16.
So the students who are the complement of math are 16.
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Need help ASAP Geometry Assignment
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The figure also has a line symmetry for the vertical line x is 5, and a line symmetry for the horizontal line y is 5 .Therefore the figure has both symmetry
What is meant by line symmetry ?A form or item is divided into two equal and symmetrical portions by a line of symmetry. Because it creates a symmetrical split in the figure and the separated pieces look like mirror reflections of one another, we also refer to this line as the axis of symmetry or the mirror line.
An imaginary line that splits a line or shape into two identical parts is called the axis or line of symmetry. The symmetry axis of a parabola will be required of you at higher level mathematics. This u-shaped line on the graph is a parabola.
B) The transformation that maps the figure unto itself are therefore;
1) Rotating the figure 180° about the point (5, 5) which is the center of the figure
2) The figure reflected across the line x = 5 and the figure reflected across the line y = 5
The complete question is : Determine if the figure below has rotational symmetry, line symmetry or both and
describe the transformation that mas the figure onto itself. BE SPECIFIC. You must
name the center of rotation and degree of rotation if it has rotational symmetry or lines of reflection if it has line symmetry
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HELP! The diameter of a circle is 12 cm. Find its area in terms of π. (pi)
Answers
The area of the given circle in terms of π is 36π cm².
What is the area of the circle?The area of the circle is equal to the product of the square of the radius of the circle and pi.
Area = πr²
where 'r' is the radius of the circle
Given that the diameter of the circle is 12 cm, we can find the radius by dividing the diameter by 2:
Radius = Diameter / 2
Radius = 12 cm / 2
Radius = 6 cm
Now, we can substitute the radius value into the formula:
Area = π x (6 cm)²
Area = π x 36 cm²
Therefore, the area of the circle in terms of π is 36π cm².
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The measures of the exterior angles of a triangle are 5x°, 7x°, and 8x°. Find the measure of the largest exterior angle.
Answers
Answer:144
Step-by-step explanation:
drag expression into order to show how to find 4 x 3/5
Answers
Answer:
4x3 12 12 3/5 0,5 12,32,98/37
What is the x-intercept of the graph of y x2 4x 4?
Answers
The x-intercept of the graph of y = x² - 4x + 4 is x = 2.
By looking at the equation, we may determine the y-intercept when the equation is expressed in the slope-intercept form (y=mx+b). The y-intercept is the value of b. This is due to the fact that the y-intercept occurs when the x value is 0. Y = B when x = 0, since mx = 0 at this point.
We set y = 0 and solve the equation for x to determine the x-intercept. This is due to the fact that the line crosses the x-axis at y=0. If an equation is not in the form y = mx + b, we can still solve for the intercepts by substituting 0 where necessary and then solving for the final variable.
The formula for the x-intercept is y = 0: The x-intercept(s) can be determined by factoring or by employing the quadratic formula because ax2 + bx + c=0.
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Given f(x) = − x² - 4x – 12, find f(-7)
Answers
Answer:
f(-7) = -33
Step-by-step explanation:
f(x) = − x² - 4x – 12
f(-7) = ?
We put -7 in for x and solve
f(-7) = -(-7)^2 -4(-7) - 12
f(-7) = -(49) + 28 - 12
f(-7) = -49 + 28 - 12
f(-7) = -33
What does FG mean mass?
Answers
FG is an abbreviation for "Femtogram," which is a unit of mass equal to one quadrillionth of a gram (10^-15 g).
An exclamation is an emphatic expression of emotion or emphasis. It is often used to express strong feelings such as surprise, joy, excitement, anger, or disbelief. Exclamations are also used to emphasize a point or give an opinion. Exclamations are most commonly used in spoken language, particularly in informal conversations. They can also be used in written language, but they should be used sparingly. Exclamations are often followed by an exclamation point, although they can also be followed by a question mark or an ellipsis. Exclamations can be used to add emphasis or emotion to a statement, making it more powerful and memorable.
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