To find the perimeter of triangle TJP, we need to find the lengths of the three sides TJ, JP, and TP, and then add them up.
First, we can find the length of TJ. Since T is the midpoint of LM, we know that LT = TM. Therefore, we can find TM by subtracting the length of LT from the length of OL: TM = OL - LT = 32 - 9 = 23. Since J is the midpoint of MO, we know that JM = JO. Therefore, we can find JO by subtracting the length of JM from the length of OL: JO = OL - JM = 32 - 12 = 20. Finally, we can find the length of TJ by using the Pythagorean theorem on right triangle LJT: TJ^2 = LT^2 + JT^2. Since we know that LT = 9 and JT = JM/2 = 6, we can solve for TJ: TJ^2 = 9^2 + 6^2 = 81 + 36 = 117. Taking the square root of both sides, we get TJ = sqrt(117) = 3sqrt(13).
Next, we can find the length of JP. Since P is the midpoint of LO, we know that JP is half the length of LO: JP = LO/2 = 32/2 = 16.
Finally, we can find the length of TP. Since T, J, and P are all midpoints of their respective sides, we know that triangle TJP is similar to triangle LMO, with a scale factor of 1/2. Therefore, the length of TP is half the length of LM: TP = LM/2 = TM/2 = 23/2.
Now we can add up the lengths of the three sides to find the perimeter of triangle TJP: TJ + JP + TP = 3sqrt(13) + 16 + 23/2 = 3sqrt(13) + 32 + 11.5 = 3sqrt(13) + 43.5. Therefore, the perimeter of triangle TJP is 3sqrt(13) + 43.5.