The perimeter of triangle RXA is 39, PX=49 and AP=9.Find RX and RA.
I think it's under triangle bisector theorem. Help me answer this guys. Thanks.
where is P?
To find the lengths RX and RA, we can start by drawing a diagram to visualize the given information. Let's label the diagram as follows:
R
/ \
/ \
/ \
/ \
A----P----X
We are given the following information:
- PX = 49
- AP = 9
- The perimeter of triangle RXA is 39.
Now let's use the triangle bisector theorem to find the lengths RX and RA.
The triangle bisector theorem states that in a triangle, a line segment joining a vertex to the intersection of the opposite side divides the opposite side into segments that are proportional to the other two sides.
In this case, PX is the segment joining vertex P to the opposite side RXA. Using the theorem, we can write the following proportion:
PX / XA = BP / BA
Since we already know PX and AP, we can find XA and BP.
PX / XA = BP / BA
49 / XA = 9 / BA
Cross multiplying, we get:
49 * BA = 9 * XA
Now, we also know that the perimeter of triangle RXA is 39, which means that:
RA + RX + XA = 39
Since we've found a relationship between XA and BA, we can substitute it into the perimeter equation:
RA + RX + (49 * BA / 9) = 39
Now, we need an additional equation to solve for RA and RX.
Let's consider the triangle RAP. Using the fact that the sum of the lengths of two sides of a triangle is always greater than the length of the third side, we can write:
RA + AP > RP
RA + 9 > RX + PX
RA + 9 > RX + 49
Since we know RA + RX + XA = 39, we can substitute RA + RX with (39 - XA):
(39 - XA) + 9 > XA + 49
Now we have two equations:
(39 - XA) + 9 > XA + 49 [Equation 1]
RA + RX + (49 * BA / 9) = 39 [Equation 2]
Solving these equations simultaneously will allow us to find the values of RX and RA.