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

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) (Show Source): You can put this solution on YOUR website!

A triangle is formed by the vertices A(, ), B(, ), and C(,)

a.) Classify this triangle

check the distance between , , and

Solved by pluggable solver: Distance Between 2 points

The distance formula is . Plug in the numbers,

The distance is 7.81024967590665.

Solved by pluggable solver: Distance Between 2 points

The distance formula is . Plug in the numbers,

The distance is 6.

and

Solved by pluggable solver: Distance Between 2 points

The distance formula is . Plug in the numbers,

The distance is 6.08276253029822.

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

b.) Identify the centroid using the intersection of medians

find midpoints:
A(, ), B(, ) => (,)=(, ,), C(,)=> (,)= (, )=(, )

B(, ), C(,)=> (,)= (, )=(, )

find equations of the lines passing through the points (, ) and (, ), as through the points (, ) and (,)

Solved by pluggable solver: Finding the Equation of a Line

First lets find the slope through the points (,) and (,)

Start with the slope formula (note: (,) is the first point (,) and (,) is the second point (,))

Plug in ,,, (these are the coordinates of given points)

Subtract (note: if you need help with subtracting or dividing fractions, check out this solver)

Divide the fractions

So the slope is

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

------Point-Slope Formula------
where is the slope, and (,) is one of the given points

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

Plug in , , and (these values are given)

Rewrite as

Rewrite as

Distribute

Multiply and to get . Now reduce to get

Subtract from both sides to isolate y

Combine like terms and to get (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 (,) and (,) is:

The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is

Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver )

Graph of through the points (,) and (,)

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 (,) and (,)

Start with the slope formula (note: (,) is the first point (,) and (,) is the second point (,))

Plug in ,,, (these are the coordinates of given points)

Subtract the terms in the numerator to get . Subtract the terms in the denominator to get

Reduce

So the slope is

------------------------------------------------

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

------Point-Slope Formula------
where is the slope, and (,) is one of the given points

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

Plug in , , and (these values are given)

Rewrite as

Rewrite as

Distribute

Multiply and to get

Subtract from both sides to isolate y

Combine like terms and to get (note: if you need help with combining fractions, check out this solver)

------------------------------------------------------------------------------------------------------------

Answer:

So the equation of the line which goes through the points (,) and (,) is:

The equation is now in form (which is slope-intercept form) where the slope is and the y-intercept is

Notice if we graph the equation and plot the points (,) and (,), we get this: (note: if you need help with graphing, check out this solver )

Graph of through the points (,) and (,)

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

find intersection:

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

TEST

Start with the first equation

Multiply both sides by the LCD 6

Distribute and simplify

------------------------------------------

Start with the second equation

Multiply both sides by the LCD 3

Distribute and simplify

-----------------------------------------

Lets start with the given system of linear equations

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:

Multiply the top equation (both sides) by -1
Multiply the bottom equation (both sides) by -7

So after multiplying we get this:

Notice how -7 and 7 add to zero (ie )

Now add the equations together. In order to add 2 equations, group like terms and combine them

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:

Divide both sides by to solve for y

Reduce

Now plug this answer into the top equation to solve for x

Plug in

Multiply

Reduce

Subtract from both sides

Combine the terms on the right side

Multiply both sides by . This will cancel out on the left side.

Multiply the terms on the right side

So our answer is

,

which also looks like

(, )

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

we get

graph of (red) (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 (,). This verifies our answer.

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