SOLUTION: A semicircle is inscribed inside a triangle ABC and with its center O lying on the side AC and Angle B = 90° and O divides AC such that AO = 15 cm and CO = 20 cm. Find the radius o

Algebra ->  Trigonometry-basics -> SOLUTION: A semicircle is inscribed inside a triangle ABC and with its center O lying on the side AC and Angle B = 90° and O divides AC such that AO = 15 cm and CO = 20 cm. Find the radius o      Log On


   

Question 894880: A semicircle is inscribed inside a triangle ABC and with its center O lying on the side AC and Angle B = 90° and O divides AC such that AO = 15 cm and CO = 20 cm. Find the radius of the circle.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
A semicircle is inscribed inside a triangle ABC and with its center O
lying on the side AC and Angle B = 90° and O divides AC such that AO = 15 cm and CO = 20 cm. Find the radius of the circle.
  


Angles AOE and OCE are equal in measure.  Both are marked theta.

From the upper triangle,

cos%28theta%29=r%2F15
r=15cos%28theta%29

From the lower triangle,

sin%28theta%29=r%2F20
r=20sin%28theta%29

So setting the two expressions for r equal:

15cos%28theta%29=20sin%28theta%29

Divide both sides by 20cos%28theta%29

15cos%28theta%29%2F%2820cos%28theta%29%29=20sin%28theta%29%2F%2820cos%28theta%29%29



15%2F20+=+sin%28theta%29%2Fcos%28theta%29

3%2F4=tan%28theta%29

Since tangent = opposite%2F%28adjacent%29 we draw a right
triangle with opposite=3 and adjacent=4.

hypotenuse = sqrt%283%5E2%2B4%5E2%29=sqrt%289%2B16%29=sqrt%2825%29=5




And since r=15cos%28theta%29
          r=15%2Aexpr%284%2F5%29
          r=12

Edwin