Question

The perimeter of triangle rxa is 24. if rx= 4 and ra= 11, find xp and pa

Answer 1

Answer:
PA = 6.6 units
XP = 2.4 units

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:
$\frac{XP}{PA} = \frac{XR}{AR}$

We have:
XP = 9 - PA 
RX = 4
RA = 11

4- Substitute with the givens in the above ratio and solve for PA as follows:
$\frac{9-PA}{PA} = \frac{4}{11}$

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

Hope this helps :)