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.

Problem

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.

circle within right triangle, problem.

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

    circle within right triangle, problem.


    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


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