In tiangle ABC, centroid D is on the median line segment AM. AD=x+4 and DM=2x-4. Find AM.
To find the length of AM, we can use the properties of the centroid in a triangle. The centroid is the point of concurrency of the medians, and it divides each median into two segments, with the ratio of 2:1.
First, let's summarize the given information:
- AD = x + 4
- DM = 2x - 4
From the properties of the centroid, we know that the ratio of the lengths of AD to DM is 2:1. So, we can set up the following equation:
AD/DM = 2/1
Substituting the given values, we have:
(x + 4)/(2x - 4) = 2/1
Now, we can solve for x by cross-multiplying:
1 * (x + 4) = 2 * (2x - 4)
x + 4 = 4x - 8
Next, we will isolate the variable x by moving the terms around:
4 + 8 = 4x - x
12 = 3x
Dividing both sides by 3, we find:
x = 4
Now that we have the value of x, we can find AM, which is the sum of AD and DM:
AM = AD + DM
AM = (x + 4) + (2x - 4)
AM = (4 + 4) + (2 * 4 - 4)
AM = 8 + (8 - 4)
AM = 8 + 4
AM = 12
Therefore, the length of AM is 12.