To find the length of segment OD, we use the fact that line P bisects segment AD, making AO equal to OD. After setting up the equation 2(z+4) = 3z-11, we solve for z and discover that z equals 19, which makes the lengths of both AO and OD equal to 23 units.
If line P bisects segment AD at point O, and AO is represented by the expression z+4, and the entire segment AD is represented by the expression 3z-11, to find the length of segment OD, we need to understand that bisecting a segment means dividing it into two equal parts. Therefore, AO is equal to OD. Given AO = z+4, we can equate OD to z+4 as well.
Since AD is the whole segment, we can write the equation AO + OD = AD to represent this relationship. Substituting z+4 for both AO and OD, and 3z-11 for AD, we get:
2(z+4) = 3z-11
Expanding the left side of the equation, we obtain:
2z + 8 = 3z - 11
Now, subtracting 2z from both sides gives:
8 = z - 11
Adding 11 to both sides yields:
z = 19
Now that we have the value of z, we can find OD by substituting z back into the expression for AO, which is:
OD = z + 4 = 19 + 4 = 23
Therefore, the length of segment OD is 23 units.