Hi Grant G
x is 90°
UP is 5
QP is (5√2/2)
VW =12.51
The incenter is the one point in the triangle whose distances to the sides are equal.
According to the picture at your link, you have a series of Smaller Right Triangles inside a Larger Right Triangle TSW with incenter point P; making the triangles similar triangles. Also QPVS appears to be a square or rectangle (see other calculations below) according Triangle bisector SU which splits the 90°, angle S of Right Triangle TSW into two 45° angles with PS as a common side.
One of the smaller Right Triangles is almost completely labelled
That being Right Triangle UPW
PW = 13
UW = 12
So UP can be found using the Pythagorean Theorem
UP = √PW2 - UW2
Just plug in the values and you will find that UP = 5
Angle x is Right Angle so it is 90°
Line SU is the triangle bisector for TSW
P, the incenter of TSW, is also the midpoint of line US dividing it into two equal lengths
Line SU = UP + PS = 5 + 5 = 10
UP = PS = 5
Line SU is an angle bisector it splits angle S the 90° angle of Right Triangle TSW into two 45° degree angles. Right Triangles with 45° angles are Isosceles Right Triangle with two equal angle and two equal sides; QPS and VPS are two smaller Right Isosceles Triangles notice that these triangles share side PS as a hypotenuse or longest side
We know that PS = 5
Again we can use the Pythagorean Theorem to find the other sides because they are equal for these Isosceles Right Triangles
In Triangle QPS
QP = QS = x
QP2 + QS2 = PS2
Substituting
2x2 = 52
2x2 = 25
x2 = 25/2
x = SQRT(25/2)
x = (5√2)2 = QP
By the way since we have two Isosceles Triangles which share PS
QP = SV = (5√2)/2
QS = PV
Since we know PV and PW from above we can use it to calculate VW
PW =13
PV = (5√2)/2
PW is the Hypotenuse, longest side of Right Triangle PVW so
VW = SQRT(( 132) - ((5√2)/2))2)
VW = SQRT(169 - 25/2) = 12.51
I know the diagram looks confusing but you can view things as directional for the line containing points S, P and U
SP=PS=PU=UP; because P is the midpoint of the line
