SOLUTION: ABC is a right triangle and circle with the center O is tangent to all three sides of the right triangle. Given that the measurment of angle ACB is equal to 36 degrees, and that th

Algebra ->  Bodies-in-space -> SOLUTION: ABC is a right triangle and circle with the center O is tangent to all three sides of the right triangle. Given that the measurment of angle ACB is equal to 36 degrees, and that th      Log On


   

Question 189039: ABC is a right triangle and circle with the center O is tangent to all three sides of the right triangle. Given that the measurment of angle ACB is equal to 36 degrees, and that the radius of the circle is 10cm, find the lengths of all the sides of the triangle and the hypotenuse.
Answer by jojo14344(1513) About Me  (Show Source):
You can put this solution on YOUR website!


Let's see the circle tangent to all three sides of a Right Triangle:


We know, the circle tangent to the sides forms 90 deg.

Likewise, we can get angle A: A+B+C=180
A=180-B-C=180-90-36=54 deg

With these additional data, let's see the figure:
-->


Side CB = CD + DB, EQN 1

By Trigonometric function:
tan%2818%5Eo%29=opp%2Fadd=DO%2FCD
CD=DO%2Ftan18%5Eo=10cm%2Ftan18%5Eo=red%2830.8cm%29
*Take note:
red%28DB=OF=10cm=radius%29

Then, CB=CD + DB = 30.8cm + 10cm
CB = 40.8cm



Next, Side BA = BF + FA
*Take note:
red%28BF=DO=10cm=radius%29
By Trigo Function:
tan27%5Eo=opp%2Fadj=OF%2FFA
FA=OF%2Ftan27%5Eo=10cm%2Ftan27%5Eo=red%2819.6cm%29
Then,
BA= BF + FA = 10cm + 19.6cm
BA = 29.6cm



For the Hypotenuse AC = AE + EC

*Take note:
red%28DO=OE=10cm=radius%29
By Trigo Function:
tan27%5Eo=opp%2Fadj=OE%2FAE
AE=OE%2Ftan27%5Eo=10cm%2Ftan27%5Eo=red%2819.6cm%29

Also, tan18%5Eo=opp%2Fadj=OE%2FEC
EC=OE%2Ftan18%5Eo=10cm%2Ftan18%5Eo=red%2830.8cm%29

Then, AC= AE + EC = 19.6cm + 30.8cm
AC = 50.4cm, hypotenuse



Thank you,
Jojo