Angle in a Semicircle (Cambridge International IGCSE Mathematics)

Angle in a Semicircle

What are circle theorems?

• Circle Theorems deal with angles that occur when lines are drawn within (and connected to) a circle

• You may need to use other facts and rules such as:

  • basic properties of lines and angles
  • properties of triangles and quadrilaterals
  • angles in parallel lines or polygons

Circle Theorem: The angle in a semicircle is 90°

• The lines drawn from a point on the circumference to either end of a diameter are perpendicular

  • The angle at that point on the circumference is 90°

  • This circle theorem only uses half of the circle

    • The right-angle is called the angle in a semicircle

• To spot this circle theorem on a diagram look for a triangle where

  • one side is the diameter

    • Remember that a diameter always goes through the centre

  • all three vertices are on the circumference

• The 90º angle will always be the angle opposite the diameter

• When explaining this theorem in an exam you must use the keywords:

  • The angle in a semicircle is 90°

• Questions that use this theorem may

  • appear in whole circles or in semicircles
  • require the use of Pythagoras' Theorem to find a missing length

Exam Hint

• As soon as you spot this arrangement in a question, mark the angle as 90° degrees on the diagram

  • Sometimes just doing this will earn you a mark!

Worked Example A, B, and C are points on a circle. AC is a diameter of the circle. Find the value of a.

Answer:

As the line AC is the diameter of the circle, use the circle theorem "the angle in a semicircle is 90°" to state that angle B must be 90°

We can now use the fact that the internal angles of a triangle sum to 180°, to find the unknown angle

a + 53 + 90 = 180*

a + 143 = 180*

Final answer: a = 37