Answer:
see attached
Step-by-step explanation:
The missing measures can be found by making use of the properties of tangents, special triangles, trig functions, and circles.
Radii
All of the incircle radii are the same length:
CJ = CK = CL = 4
Angles
All of the segments to the incenter from the triangle vertices are angle bisectors. Of course, the sum of angles in a triangle is 180°, and the acute angles in a right triangle are complementary.
∠CML = ∠CMJ = 20°
∠CPL = ∠CPK = 40° . . . . . (∠P = 180° -40° -60° = 80°)
∠CNJ = ∠CNK = 30°
The central angles are complementary:
∠MCL = ∠MCJ = 70°
∠PCK = ∠PCL = 50°
∠NCJ = ∠NCK = 60°
Tangents
The two tangents from a vertex are congruent. Each will be the radius divided by the tangent of the (half) vertex angle.
ML = MJ = 4/tan(20°) = 11.0
PL = PK = 4/tan(40°) = 4.8
NJ = NK = 4/tan(30°) = 6.9
Angle bisectors
The length from each vertex to the incenter will be the radius divided by the sine of the (half) vertex angle.
MC = 4/sin(20°) = 11.7
PC = 4/sin(40°) = 6.2
NC = 4/sin(30°) = 8.0
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Additional comment
The given dimensions are slightly off. The angle at M is 19.9831°, or the length LM is 10.9899 units, or the radius is 4.00367. In order to keep the numbers nearly consistent, we have rounded to one decimal place. The 30° triangle is the only one with an angle that has a rational trig function (sine).
