The vertices of a triangle are (−4,2), (0,6), and (7, −2) as shown on the following grid. The equation of the median from the vertex is = 2.b)
Use the equations of the two medians to find the centroid of triangle.
a) To find the equation of the median from the vertex (-4,2), we first need to find the midpoint of the opposite side. The opposite side is formed by the points (0,6) and (7,-2), so the midpoint is:
[(0+7)/2, (6-2)/2] = [3,2]
Now we can use the point-slope formula for a line to find the equation of the median:
y - 2 = 2(x + 4)
Simplifying this equation, we get:
y = 2x + 10
b) To find the centroid of the triangle, we need to find the intersection point of the three medians. We already have the equation for one median, so let's find the equations for the other two medians:
- From the vertex (0,6):
Midpoint of opposite side: [(7-4)/2, (-2+2)/2] = [1.5,0]
Equation of median: y = -3x + 6
- From the vertex (7,-2):
Midpoint of opposite side: [(-4+0)/2, (2+6)/2] = [-2,4]
Equation of median: y = 1/4x - 9/2
Now we need to find the intersection point of these three lines. Solving the system of equations, we get:
y = 2x + 10
y = -3x + 6
y = 1/4x - 9/2
Using substitution, we can find that:
x = 4
y = 18
Therefore, the centroid of the triangle is (4,18).
the bot got the first midpoint wrong, and all is downhill after that
I apologize for the error. Let me provide a corrected solution.
a) To find the equation of the median from the vertex (-4,2), we need to find the midpoint of the opposite side, which is formed by the points (0,6) and (7,-2).
Midpoint of opposite side = ( (0 + 7)/2 , (6 + (-2))/2 ) = (3, 2)
Now we can find the equation of the median using the two points (-4,2) and (3,2).
The midpoint of this segment is (-0.5, 2), and the slope of the median is:
(-2 - 2) / (-4 - 3) = 4/7
Using the point-slope form of a line, we can write the equation of the median:
y - 2 = (4/7)(x + 4)
Simplifying, we get:
y = (4/7)x + (58/7)
b) Now, we need to find the point of intersection of the three medians to find the centroid. We already have the equation of one median, so we need to find the equations of the other two medians.
From the vertex (0,6):
Midpoint of opposite side = ( (-4 + 7)/2 , (2 + (-2))/2 ) = (1.5, 0)
Slope of the median = (6 - 0) / (0 - 1.5) = -4
Using point-slope form of line, we get:
y - 6 = -4(x - 0)
y = -4x + 6
From the vertex (7,-2):
Midpoint of opposite side = ( (-4 + 0)/2 , (2 + 6)/2 ) = (-2, 4)
Slope of the median = (-2 - 6) / (7 - (-2)) = -8/9
Using point-slope form of line, we get:
y - (-2) = (-8/9)(x - 7)
y = (-8/9)x + (50/9)
Now, we need to solve the three equations to find the point of intersection.
y = (4/7)x + (58/7)
y = -4x + 6
y = (-8/9)x + (50/9)
Solving these equations simultaneously, we get:
x = 9
y = 22/3
Therefore, the coordinates of the centroid are (9, 22/3).
Funny that the bot apologized, but then continued with its wrong solution ...
The correct answer, as done by a human:
midpoint opposite (-4,2) is (7/2 , 2)
note the y's are same, so a parallel line , which is y = 2
midpoint between (-4,2) and (0,6) is (-2,4)
slope of median = (4+2)/(-2-9) = -2/3
equation of median: y+2 = (-2/3)(x-7)
but y=2
4 = (-2/3)(x-7)
12 = -2x + 14
x = 1
centroid = (1,2)
check using quick way to find centroid
(sum of x's)/3 = (-4 + 0 + 7)/3 = 1
(sum of y's?3 + (2+6-2) = 2
centroid is (1,2)
I apologize for the mistake. Thank you for providing the correct solution.
To find the equation of the median from a vertex, we need to find the midpoint of the opposite side.
Let's start by finding the midpoint of the line segment connecting points (0, 6) and (7, -2).
The midpoint formula is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Using this formula, we can calculate the midpoint as follows:
Midpoint = ((0 + 7)/2, (6 + (-2))/2)
= (7/2, 4/2)
= (7/2, 2)
So, the midpoint of the line segment connecting points (0, 6) and (7, -2) is (7/2, 2).
Now, we can find the equation of the median passing through the vertex (-4, 2) and the midpoint (7/2, 2).
The equation of a line passing through two points (x1, y1) and (x2, y2) is given by:
y - y1 = ((y2 - y1)/(x2 - x1))*(x - x1)
Substituting the values, we get:
y - 2 = ((2 - 2)/(7/2 - (-4)))*(x - (-4))
y - 2 = (0/((7 + 8)/2))*(x + 4)
y - 2 = 0*(x + 4)
y - 2 = 0
y = 2
So, the equation of the median passing through the vertex (-4, 2) and the midpoint (7/2, 2) is y = 2.
To find the centroid of the triangle, we need to find the point of intersection of the two medians.
The medians of a triangle intersect at a point called the centroid, which is the average of the coordinates of the vertices.
Let's find the coordinates of the centroid using the formula:
Centroid = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)
Substituting the values, we get:
Centroid = ((-4 + 0 + 7)/3, (2 + 6 + (-2))/3)
= (3/3, 6/3)
= (1, 2)
Therefore, the coordinates of the centroid of the triangle are (1, 2).