Question

find the radius of circle passing through the vertices of a right angled triangle the length of whoes perpendicular sides are 8 and 15

Answer 1

If a the circle passes through the vertices of a triangle,
the angles of the triangle are inscribed angles in the circle.
Inscribed angles intercept an arc whose measure is twice the measure of the inscribed angle.
in particular, the right angle intercepts half of the circumference,
meaning that the hypotenuse of the right triangle in this problem
is the diameter of the circle.
A right triangle with legs (the perpendicular sides) of length 6 and 8,
is a 3-4-5 right triangle,
meaning a scaled up by a factor of 2 version
of a triangle with sides measuring 3 , 4 , and 5 .
In that scaled-up version,
the hypotenuse (diameter of the circle) measures 2*5 , so the radius measures 2*5/2=10/2=5