The perimeter of triangle RXA is 39, PX=4 and AP=9.Find RX and RA.
P is at the middle.
I think it's under triangle bisector theorem. Help me answer this guys. Thanks
To find the lengths RX and RA, you can indeed use the Triangle Bisector Theorem. Here's how you can approach the problem step by step:
1. Recall that the Triangle Bisector Theorem states that if a line divides a triangle's side into two segments, the ratio of the lengths of those segments is equal to the ratio of the lengths of the other two sides of the triangle.
2. Let's denote the length of RX as x and the length of RA as y.
3. According to the theorem, we can write the following equation:
(XP / PX) = (RA / AX)
Since PX is given as 4 and AP is given as 9, we can substitute those values into the equation:
(4 / 4) = (y / (4 + x))
Simplifying the equation:
1 = (y / (4 + x))
4. Next, we can find a similar equation using the other bisector:
(PX / XP) = (RX / AX)
Substituting the given values:
(4 / 4) = (x / (4 + y))
Simplifying the equation:
1 = (x / (4 + y))
5. Now we have a system of two equations:
1 = (y / (4 + x))
1 = (x / (4 + y))
6. We can solve this system of equations simultaneously.
One approach is to cross-multiply the equations:
y = 4 + x
x = 4 + y
Simplifying the equations:
y - x = 4
x - y = 4
Rearranging the equations:
y = x + 4
x = y + 4
7. Now you can substitute the value of y from the first equation into the second equation:
x = (x + 4) + 4
Simplifying the equation:
x = x + 8
Subtracting x from both sides:
0 = 8
This equation is not possible, which means there is no solution.
Therefore, based on the given information, it seems that there is no solution for the lengths RX and RA. Double-check the given values and conditions to ensure their accuracy.