Inscribed and Central Angles in Circles

Definitions

Theorem

Problem 1: In the figure below chord CA has a length of 12 cm. The circle of center O has a radius of 14 cm. Find an approximate value (2 decimal places) to the size of the inscribed angle CBA.

Solution to Problem 1:

• We first calculate the central angle COA. Triangle COA is an isosceles triangle since length of CO = length of AO = radius = 14 cm. We use the cosine law to find cos (angle COA).
  
  CA ² = CO ² + AO ² - 2 CO AO cos (angle COA)
  
  
• Substitute CA, CO and AO by their numerical values and express cos(angle COA) as follows
  
  cos(angle COA) = [ 14 ² + 14 ² - 12 ² ] / [2 * 14 * 14 ]
  
  = 31 / 49
  
  
• Size of angle COA is given by.
  
  Angle COA = arcos(31 / 49)
  
  
• According to the theorem above, the size of angle CBA is equal to half the size of angle COA.
  
  Angle CBA = (1 / 2) arcos(31 / 49)
  
  = 25.38 degrees (approximated to 2 decimal places)