A triangle has vertices X(0,0), Y(4,4), and Z (8,-4).
a) Write an equation for each of the three medians
b) Recall that the centroid of a triangle is the point of intersection of the medians of the triangle. Use the equations from part a) to verify that (4,0) is the centroid of triangle XYZ.
I've been trying to solve this problem for over an hour and the equations I'm getting for part a) don't match the answers in tha back of my book, y=0, x-4=0, and x+y-4=0. I really need some help, please
To find the equations of the medians, we need to find the midpoints of each side of the triangle. Let's denote the midpoints of XY, YZ, and XZ as M, N, and P, respectively.
a) Finding the equations of the medians:
1. Midpoint of XY:
The coordinates of X(0,0) and Y(4,4) can be used to find the midpoint.
M = ((0 + 4)/2, (0 + 4)/2)
M = (2, 2)
The equation of the median passing through M can be found using the coordinates of M and any other vertex. Let's use Z(8,-4):
Using the formula: (y - y₁) = m(x - x₁)
(y - (-4)) = (2 - (-4))/ (2 - 8)
y + 4 = (-6)/(-6)
y + 4 = 1
y = -3
So, the equation of the first median is y = -3.
2. Midpoint of YZ:
The coordinates of Y(4,4) and Z(8,-4) can be used to find the midpoint.
N = ((4 + 8)/2, (4 + (-4))/2)
N = (6, 0)
The equation of the median passing through N can be found using the coordinates of N and any other vertex. Let's use X(0,0):
Using the formula: (y - y₁) = m(x - x₁)
(y - 0) = (0 - 0)/ (6 - 0)
y = 0
So, the equation of the second median is y = 0.
3. Midpoint of XZ:
The coordinates of X(0,0) and Z(8,-4) can be used to find the midpoint.
P = ((0 + 8)/2, (0 + (-4))/2)
P = (4, -2)
The equation of the median passing through P can be found using the coordinates of P and any other vertex. Let's use Y(4,4):
Using the formula: (y - y₁) = m(x - x₁)
(y - 4) = (-2 - 4)/ (4 - 8)
(y - 4) = (-6)/(-4)
(y - 4) = (3/2)(x - 4)
Simplifying the equation:
2(y-4) = 3(x-4)
2y - 8 = 3x - 12
2y = 3x - 4
3x - 2y - 4 = 0
So, the equation of the third median is 3x - 2y - 4 = 0.
b) Verifying the centroid:
The centroid of a triangle is the point of intersection of the medians. So, we need to find the point of intersection of the equations obtained in part a).
From part a):
First median: y = -3
Second median: y = 0
Third median: 3x - 2y - 4 = 0
To find the centroid, we need to solve the system of equations formed by the medians.
Substituting y = -3 in the third median equation:
3x - 2(-3) - 4 = 0
3x + 6 - 4 = 0
3x + 2 = 0
3x = -2
x = -2/3
So, the x-coordinate of the centroid is -2/3.
Substituting y = 0 in the third median equation:
3x - 2(0) - 4 = 0
3x - 4 = 0
3x = 4
x = 4/3
So, the x-coordinate of the centroid is 4/3.
Therefore, the coordinates of the centroid are (4/3, 0), which means the centroid of the triangle XYZ is (4,0).
It seems like there was a mistake in the back of your book. The correct coordinates of the centroid are (4/3, 0), or (4, 0) when simplified.
Sure! I'd be happy to help you with this problem.
a) To find the equations for the medians, we need to first find the midpoints of each side of the triangle, and then find the equations of the lines passing through each of these midpoints and their opposite vertices.
Let's start by finding the midpoint of the line segment XY. The x-coordinate of the midpoint is the average of the x-coordinates of X and Y, and the y-coordinate of the midpoint is the average of the y-coordinates of X and Y. Thus, the midpoint of XY is ((0+4)/2, (0+4)/2), which simplifies to (2, 2).
Using the formulas for finding the equation of a line, we can write the equation of the median passing through (2, 2) and the opposite vertex Z(8, -4). The slope of the line passing through these two points is found by taking the difference in y-coordinates divided by the difference in x-coordinates, which is (-4 - 2) / (8 - 2) = -6/6 = -1.
Using the point-slope form of the equation of a line, the equation of the median passing through (2, 2) and Z(8, -4) is:
y - 2 = -1(x - 2)
Simplifying, we get y - 2 = -x + 2, which further simplifies to y = -x + 4. This is the equation of the first median.
Similarly, we find the midpoint of YZ, which is ((4+8)/2, (4-4)/2), simplifying to (6, 0).
The equation of the median passing through (6, 0) and X(0, 0) can be found using the slope-intercept form of the equation of a line. The slope of the line passing through these two points is found by taking the difference in y-coordinates divided by the difference in x-coordinates, which is (0 - 0) / (6 - 0) = 0.
Since the slope is 0, the equation of the median is y = 0. This is the equation of the second median.
Finally, we find the midpoint of XZ, which is ((0+8)/2, (0-4)/2), simplifying to (4, -2).
The equation of the median passing through (4, -2) and Y(4, 4) can be found using the point-slope form of the equation of a line. The slope of the line passing through these two points is found by taking the difference in y-coordinates divided by the difference in x-coordinates, which is (4 - (-2)) / (4 - 4) = 6/0.
Since the slope is undefined, the equation of the median is x - 4 = 0. This is the equation of the third median.
Now, let's move on to part b.
b) To verify that (4, 0) is the centroid of triangle XYZ, we need to find the point of intersection of the three medians.
First, let's find the equation of the line passing through the midpoints of XY and ZY, which are (2, 2) and (6, 0) respectively. The slope of this line can be found by taking the difference in y-coordinates divided by the difference in x-coordinates, which is (0 - 2) / (6 - 2) = -2/4 = -1/2.
Using the point-slope form of the equation of a line, we have:
y - 0 = (-1/2)(x - 4)
Simplifying, we get y = -1/2x + 2
Now, let's find the point of intersection of this line with the third median, which has an equation of x - 4 = 0. Since the x-coordinate of this line is fixed at 4, we substitute x = 4 into the equation of the line:
y = -1/2(4) + 2
y = -2 + 2
y = 0
Therefore, the point of intersection of the two lines is (4, 0).
Since the three medians of a triangle intersect at one point, which is the centroid, and we have shown that the point (4, 0) lies on all three medians, we have verified that (4, 0) is, indeed, the centroid of triangle XYZ.
I hope this explanation helps!
Let the midpoints of XY,YZ, and XZ be P,Q, and R respectively
Did you get P(2,2), Q(6,0) and R(4,-2) ?
XQ is obviously y = 0, the equation of the x-axis
YR has undefined slope, so its equation is x=4
Right there you already have the coordinates of the centroid, namely (4,0)
the slope of PZ is (-4-2)/(8-2) = -1
so y=-x+b, sub in (2,2) for
2=-2+b
b=4
so y = -x + 4
(notice that (4,0) satisfies this last median equation.
BTW, a quick way to find the centroid if you know the three vertices of a triangle...
centroid = (sum of x values/3,sum of y values/3)
oh, thank you so much!
it all makes sense now, my book was just so confusing on the topic and I cannot even begin to express how grateful I am; I was so desperate and overwhelmed by this.
Thank you!