The length of TU is sqrt(205) inches.
The incenter of a triangle is the point where the three angle bisectors intersect. It is also the center of the inscribed circle of the triangle.
Since T is the incenter of triangle PQR, we know that the three angle bisectors PT, QT, and RT are congruent. This means that PT = QT = RT.
We also know that the angle bisectors of a triangle divide the opposite sides into segments that are proportional to the length of the other two sides. In other words,
PT / QR = QT / RP = RT / PQ
We are given that RP = 26 and RQ = 12. Substituting these values into the above equation, we get:
PT / 26 = QT / 12 = RT / 26
Let x be the length of PT. Then, we have:
x / 26 = QT / 12 = x / 26
Solving for QT, we get:
QT = 12 * (x / 26) = 12x / 26 = 6x / 13
Since QT = RT, we also have:
RT = 6x / 13
Now, we can use the Pythagorean theorem to find the length of TU.
TU^2 = PT^2 + QT^2
TU^2 = x^2 + (6x / 13)^2
TU^2 = x^2 + 36x^2 / 169
TU^2 = 205x^2 / 169
Taking the square root of both sides, we get:
TU = sqrt(205x^2 / 169)
TU = x * sqrt(205 / 169)
Since PT = x, we can substitute x into the above equation to get the length of TU.
TU = x * sqrt(205 / 169) = sqrt(205)
Therefore, the length of TU is sqrt(205) inches.