In Triangle ARP, centroid E is on median AX . AE = y+6 and EX = 3y-2. Find AX
13
11
5 1/2
7/3
In a triangle, the centroid (point E in this case) divides each median into two segments, where the segment joining the centroid with the vertex (AE) is two-thirds the length of the median (AX), and the segment from the centroid to the midpoint of the opposite side (EX) is one-third the length of the median.
The given lengths are AE = y + 6 and EX = 3y - 2. We also know that AX is the whole median, which is the sum of AE and EX. So,
AX = AE + EX
AX = (y + 6) + (3y - 2)
The centroid divides the median in a 2:1 ratio, which means that AE is twice as long as EX:
AE = 2 * EX
We can rewrite the above equation in terms of y, using the given values for AE and EX:
y + 6 = 2 * (3y - 2)
y + 6 = 6y - 4
Now solve for y:
6y - y = 6 + 4
5y = 10
y = 10 / 5
y = 2
With the value of y, we can now determine the lengths AE and EX:
AE = y + 6 = 2 + 6 = 8
EX = 3y - 2 = 3(2) - 2 = 6 - 2 = 4
Now we know AE and EX, so we can find the length of the median AX:
AX = AE + EX
AX = 8 + 4
AX = 12
The length of the median AX is 12. The options you provided don't include 12, so either there's a typo in the options or there's a misunderstanding in the question. However, based on the information given and the calculations above, the correct length of AX is 12.