In triangle ABC, centroid D is on the median line segment AM. AD=x+5 and
DM=2x-1. Find AM.
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x+5 = 2(2x-1)
x+5 = 4 x - 2
7 = 3 x
x = 7/3
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To find AM, we need to use the relationship between centroid and median in a triangle.
The centroid divides each median into two segments, with the length of the segment from the centroid to the vertex being two-thirds of the length of the segment from the centroid to the midpoint of the opposite side.
In triangle ABC, we have AD = x + 5 and DM = 2x - 1.
Therefore, AM = AD + DM.
Substituting the given values, we get:
AM = (x + 5) + (2x - 1)
= 3x + 4.
Thus, AM = 3x + 4.
To find the length of AM, we need to find the value of x first.
In triangle ABC, D is the centroid, which means that the centroid divides the median into two segments, with the segment containing the centroid being twice as long as the segment that doesn't contain the centroid.
So, we can set up the following equation based on this property:
AD / DM = 2
Substituting the given lengths, we have:
(x + 5) / (2x - 1) = 2
Now let's solve for x:
Multiply both sides of the equation by (2x - 1):
(x + 5) = 2(2x - 1)
Expand the right side of the equation:
x + 5 = 4x - 2
Rearrange the equation by moving the terms involving x to one side:
4x - x = 5 + 2
Combine like terms:
3x = 7
Divide both sides of the equation by 3:
x = 7/3
Now that we have the value of x, we can find the length of AM by substituting the value of x back into one of the original equations. Let's use AD = x + 5:
AM = AD + DM
AM = (x + 5) + (2x - 1)
AM = 3x + 4
Substituting x = 7/3:
AM = 3(7/3) + 4
AM = 21/3 + 4
AM = 7 + 4
AM = 11
Therefore, the length of AM is 11 units.