SOLUTION: A triangle is formed by the vertices A(1, 2), B(-5, -3), and C(1,-4) a.) Classify this triangle b.) Identify the centroid using the intersection of medians c.) There is a

Algebra ->  Length-and-distance -> SOLUTION: A triangle is formed by the vertices A(1, 2), B(-5, -3), and C(1,-4) a.) Classify this triangle b.) Identify the centroid using the intersection of medians c.) There is a       Log On


   

Question 1203124: A triangle is formed by the vertices A(1, 2), B(-5, -3), and C(1,-4)
a.) Classify this triangle
b.) Identify the centroid using the intersection of medians
c.) There is a formula for centroid, which is (x1+x2+x3 ÷ 3 , y1+y2+y3 ÷ 3), calculate the centroid using this formula. Do your answers match?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

A triangle is formed by the vertices A(1, 2), B(-5, -3), and C(1,-4)

a.) Classify this triangle

check the distance between AB, AC, and BC
AB
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%28-5-%281%29%29%5E2%29%2B%28%28-3-%282%29%29%5E2%29%29
sqrt%28-6%5E2%2B-5%5E2%29 The distance is 7.81024967590665.



AB=7.8


AC
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%281-%281%29%29%5E2%29%2B%28%28-4-%282%29%29%5E2%29%29
sqrt%280%5E2%2B-6%5E2%29 The distance is 6.



AC=6


and BC
Solved by pluggable solver: Distance Between 2 points
The distance formula is sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29. Plug in the numbers,
sqrt%28%28%281-%28-5%29%29%5E2%29%2B%28%28-4-%28-3%29%29%5E2%29%29
sqrt%286%5E2%2B-1%5E2%29 The distance is 6.08276253029822.



BC=6.08


so, the lengths of the sides are different and we have a scalene triangle

b.) Identify the centroid using the intersection of medians

find midpoints:
A(1, 2), B(-5, -3) => (x1,y1)=(%281-5%29%2F2, %282-3%29%2F2%29=%28%7B%7B%7B-2%2C+%7B%7B%7B-1%2F2%29%0D%0A%0D%0A%0D%0AA%28%7B%7B%7B1,+2), C(1,-4)=> (x2,y2)= (%281%2B1%29%2F2, %282-4%29%2F2)=(1, -1)

B(-5, -3), C(1,-4)=> (x3,y3)= (%28-5%2B1%29%2F2, %28-3-4%29%2F2)=(-2, -7%2F2)

find equations of the lines passing through the points (-2, -1%2F2) and (1, -4), as through the points (-5, -3) and (1,+-1)

Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (-2,-1%2F2) and (1,-4)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (-2,-1%2F2) and (x%5B2%5D,y%5B2%5D) is the second point (1,-4))


m=%28-4--1%2F2%29%2F%281--2%29 Plug in y%5B2%5D=-4,y%5B1%5D=-1%2F2,x%5B2%5D=1,x%5B1%5D=-2 (these are the coordinates of given points)


m=+%28-7%2F2%29%2F%283%2F1%29 Subtract (note: if you need help with subtracting or dividing fractions, check out this solver)




m=-7%2F6 Divide the fractions



So the slope is

m=-7%2F6





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Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


y--1%2F2=%28-7%2F6%29%28x--2%29 Plug in m=-7%2F6, x%5B1%5D=-2, and y%5B1%5D=-1%2F2 (these values are given)



y%2B1%2F2=%28-7%2F6%29%28x--2%29 Rewrite y--1%2F2 as y%2B1%2F2



y%2B1%2F2=%28-7%2F6%29%28x%2B2%29 Rewrite x--2 as x%2B2



y%2B1%2F2=%28-7%2F6%29x%2B%28-7%2F6%29%282%29 Distribute -7%2F6


y%2B1%2F2=%28-7%2F6%29x-7%2F3 Multiply -7%2F6 and 2 to get -14%2F6. Now reduce -14%2F6 to get -7%2F3

y=%28-7%2F6%29x-7%2F3-1%2F2 Subtract 1%2F2 from both sides to isolate y


y=%28-7%2F6%29x-17%2F6 Combine like terms -7%2F3 and -1%2F2 to get -17%2F6 (note: if you need help with combining fractions, check out this solver)



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Answer:



So the equation of the line which goes through the points (-2,-1%2F2) and (1,-4) is:y=%28-7%2F6%29x-17%2F6


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=-7%2F6 and the y-intercept is b=-17%2F6


Notice if we graph the equation y=%28-7%2F6%29x-17%2F6 and plot the points (-2,-1%2F2) and (1,-4), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%28-7%2F6%29x-17%2F6 through the points (-2,-1%2F2) and (1,-4)


Notice how the two points lie on the line. This graphically verifies our answer.




Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (-5,-3) and (1,-1)


m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 Start with the slope formula (note: (x%5B1%5D,y%5B1%5D) is the first point (-5,-3) and (x%5B2%5D,y%5B2%5D) is the second point (1,-1))


m=%28-1--3%29%2F%281--5%29 Plug in y%5B2%5D=-1,y%5B1%5D=-3,x%5B2%5D=1,x%5B1%5D=-5 (these are the coordinates of given points)


m=+2%2F6 Subtract the terms in the numerator -1--3 to get 2. Subtract the terms in the denominator 1--5 to get 6




m=1%2F3 Reduce



So the slope is

m=1%2F3





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Now let's use the point-slope formula to find the equation of the line:




------Point-Slope Formula------
y-y%5B1%5D=m%28x-x%5B1%5D%29 where m is the slope, and (x%5B1%5D,y%5B1%5D) is one of the given points


So lets use the Point-Slope Formula to find the equation of the line


y--3=%281%2F3%29%28x--5%29 Plug in m=1%2F3, x%5B1%5D=-5, and y%5B1%5D=-3 (these values are given)



y%2B3=%281%2F3%29%28x--5%29 Rewrite y--3 as y%2B3



y%2B3=%281%2F3%29%28x%2B5%29 Rewrite x--5 as x%2B5



y%2B3=%281%2F3%29x%2B%281%2F3%29%285%29 Distribute 1%2F3


y%2B3=%281%2F3%29x%2B5%2F3 Multiply 1%2F3 and 5 to get 5%2F3

y=%281%2F3%29x%2B5%2F3-3 Subtract 3 from both sides to isolate y


y=%281%2F3%29x-4%2F3 Combine like terms 5%2F3 and -3 to get -4%2F3 (note: if you need help with combining fractions, check out this solver)



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Answer:



So the equation of the line which goes through the points (-5,-3) and (1,-1) is:y=%281%2F3%29x-4%2F3


The equation is now in y=mx%2Bb form (which is slope-intercept form) where the slope is m=1%2F3 and the y-intercept is b=-4%2F3


Notice if we graph the equation y=%281%2F3%29x-4%2F3 and plot the points (-5,-3) and (1,-1), we get this: (note: if you need help with graphing, check out this solver)


Graph of y=%281%2F3%29x-4%2F3 through the points (-5,-3) and (1,-1)


Notice how the two points lie on the line. This graphically verifies our answer.





+y=%28-7%2F6%29x-17%2F6
y=%281%2F3%29x-4%2F3

find intersection:
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition
TEST

%287%2F6%29%2Ax%2B%281%29%2Ay=-17%2F6 Start with the first equation


6%28%287%2F6%29%2Ax%2B%281%29%2Ay%29=%286%29%2A%28-17%2F6%29 Multiply both sides by the LCD 6



7%2Ax%2B6%2Ay=-17Distribute and simplify


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%28-1%2F3%29%2Ax%2B%281%29%2Ay=-4%2F3 Start with the second equation


3%28%28-1%2F3%29%2Ax%2B%281%29%2Ay%29=%283%29%2A%28-4%2F3%29 Multiply both sides by the LCD 3



-1%2Ax%2B3%2Ay=-4 Distribute and simplify



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Lets start with the given system of linear equations

7%2Ax%2B6%2Ay=-17
-1%2Ax%2B3%2Ay=-4

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 7 and -1 to some equal number, we could try to get them to the LCM.

Since the LCM of 7 and -1 is -7, we need to multiply both sides of the top equation by -1 and multiply both sides of the bottom equation by -7 like this:

-1%2A%287%2Ax%2B6%2Ay%29=%28-17%29%2A-1 Multiply the top equation (both sides) by -1
-7%2A%28-1%2Ax%2B3%2Ay%29=%28-4%29%2A-7 Multiply the bottom equation (both sides) by -7


So after multiplying we get this:
-7%2Ax-6%2Ay=17
7%2Ax-21%2Ay=28

Notice how -7 and 7 add to zero (ie -7%2B7=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-7%2Ax%2B7%2Ax%29-6%2Ay-21%2Ay%29=17%2B28

%28-7%2B7%29%2Ax-6-21%29y=17%2B28

cross%28-7%2B7%29%2Ax%2B%28-6-21%29%2Ay=17%2B28 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

-27%2Ay=45

y=45%2F-27 Divide both sides by -27 to solve for y



y=-5%2F3 Reduce


Now plug this answer into the top equation 7%2Ax%2B6%2Ay=-17 to solve for x

7%2Ax%2B6%28-5%2F3%29=-17 Plug in y=-5%2F3


7%2Ax-30%2F3=-17 Multiply



7%2Ax-10=-17 Reduce



7%2Ax=-17%2B10 Subtract -10 from both sides

7%2Ax=-7 Combine the terms on the right side

cross%28%281%2F7%29%287%29%29%2Ax=%28-7%29%281%2F7%29 Multiply both sides by 1%2F7. This will cancel out 7 on the left side.


x=-1 Multiply the terms on the right side


So our answer is

x=-1, y=-5%2F3

which also looks like

(-1, -5%2F3)

Notice if we graph the equations (if you need help with graphing, check out this solver)

7%2Ax%2B6%2Ay=-17
-1%2Ax%2B3%2Ay=-4

we get



graph of 7%2Ax%2B6%2Ay=-17 (red) -1%2Ax%2B3%2Ay=-4 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (-1,-5%2F3). This verifies our answer.