<span>Based in the information given in the problem, you must apply the The Angle Bisector Theorem. Let's call the triangle: "ABC"; the internal bisector of the angle that divides its opposite side: "AP"; and "x": the longest and shortest possible lengths of the third side of the triangle.
If BP= 6 cm and CP= 5 cm, we have:
BP/CP = AB/AC
We don't know if second side of the triangle (6.9 centimeters long) is AB or AC, so:
1. If AB = 6.9 cm and AC = x:
6/5 = 6.9/x
x = (5x6.9)/6
x = 5.80 cm
2. If AC= 6.9 cm and AB= x:
6/5 = x/6.9
x = 6.9x6/5
x = 8.30 cm
Then, the answer is:
The longest possible length of the third side of the triangle is 8.30 cm and the and shortest length of it is 5.80 cm.</span>
That's the second one,
180 - x = r
Whenever two lines cross we get congruent opposite angles and supplemental adjacent angles, i.e. they add to 180 degrees.
The first one is almost right; we have r+y+z=180 and r+x=180 so x=y+z not quite what was written.
The term of humanism has been very different back during reinasance humanism and modern humanism.
Back then, the term is used to describe an act to preserve intellectual culture. In modern times, it used to describe the act to impose social justice among all human.
I would answer the problem but first upload the diagram
