Circle Tangent to Right Triangle - Problem With Solution
Solve a right triangle whose sides are all tangent to a circle. Both the problem and its detailed solution are presented.
ABC is a right triangle. Given one of the angles of triangle ABC and the radius of the circle tangent to all 3 sides of the right triangle, find the lengths of the two sides and the hypotenuse of triangle ABC.
Solution to Problem 1:
Triangles COM and CON are right triangles with congruent hypotenuses OC and congruent corresponding sides OM and ON and are therefore congruent. Hence angles OCM and OCN are also congruent. Hence the size of angle OCM is given by
36 degrees / 2 = 18 degrees.
We now use tan(OCM) to calculate the length of side CM as follows
tan(OCM) = r / CM
CM = r / tan(OCM) = 10 / tan(18 degrees) = 30.8 cm (rounded to one decimal place)
The size of angle PAM is given by
90 degrees - 36 degrees = 54 degrees
Triangles AOM and AOP are also right and congruent and the size of angle OAM is given by
54 degrees / 2 = 27 degrees
The side of side AM is calculated as follows
tan(OAM) = r / AM
AM = r / tan(OAM) = 10 / tan(27 degrees) = 19.6 cm (rounded to one decimal place)
The hypotenuse AC may now be calculated as follows
AC = AM + CM = 30.8 + 19.6 = 50.4 cm
Side AB may now be calculated as follows
AB = AP + r = AM + r = 19.6 + 10 = 29.6 cm
The size of side BC may be calculated using as follows
BC = BN + CN = 10 + CM = 10 + 30.8 = 40.8 cm