Explanation: 1- We are given that: perimeter of RXA = 24 units RX = 4 RA = 11 This means that: XA = 24 - 4 - 11 = 9 units
2- For XA: XA = XP + PA XA = 9 units This means that : XP + PA = 9 units XP = 9 - PA
3- Triangle angle bisector theorem: "In a triangle, the angle bisector divides the opposite side into two segments that are proportional to the other two sides"
Applying this to the given triangle, we can set the ratio as follows: [tex] \frac{XP}{PA} = \frac{XR}{AR} [/tex]
We have: XP = 9 - PA RX = 4 RA = 11
4- Substitute with the givens in the above ratio and solve for PA as follows: [tex] \frac{9-PA}{PA} = \frac{4}{11} [/tex]
11(9-PA) = 4(PA) 99 - 11PA = 4PA 99 = 15PA PA = 99/15 PA = 6.6 units
5- Substitute to get XP as follows: XP = 9 - PA XP = 9 - 6.6 XP = 2.4 units
