To find the inradius of triangle ABC, we can use the formula:
inradius = area / semiperimeter,
where the semiperimeter is the sum of the three sides divided by 2, and the area can be found using Heron's formula:
semiperimeter = (5 + 29 + 42) / 2 = 38
area = sqrt(s * (s - a) * (s - b) * (s - c)), where s is the semiperimeter, and a, b, and c are the lengths of the sides.
area = sqrt(38 * (38 - 5) * (38 - 29) * (38 - 42)) = 90
inradius = area / semiperimeter = 90 / 38 = 45 / 19
Therefore, the inradius of triangle ABC is 45/19.
To find the circumradius of triangle ABC, we can use the formula:
circumradius = a/(2*sin(A)) = b/(2*sin(B)) = c/(2*sin(C)),
where a, b, and c are the lengths of the sides, and A, B, and C are the corresponding angles opposite those sides.
We can use the law of sines to find the angles:
sin(A) = (area * 2) / (a * b)
sin(B) = (area * 2) / (b * c)
sin(C) = (area * 2) / (c * a)
where area is the same area we found earlier using Heron's formula.
Substituting the values, we get:
sin(A) = 90 / (5 * 29) = 6/145
sin(B) = 90 / (29 * 42) = 10/87
sin(C) = 90 / (42 * 5) = 2/7
Now we can use the formula for the circumradius:
circumradius = a/(2*sin(A)) = b/(2*sin(B)) = c/(2*sin(C))
circumradius = 5/(2*6/145) = 29/(2*10/87) = 42/(2*2/7)
circumradius = 725/12
Therefore, the circumradius of triangle ABC is 725/12.