Angle in a Semicircle (Cambridge (CIE) IGCSE International Maths)
Revision Note
Written by: Amber
Reviewed by: Dan Finlay
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Angle in a Semicircle
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
This is a special case of the circle theorem "the angle at the centre is twice the angle at the circumference"
The angle on the diameter is 180°
The angle at the circumference is halved, giving 90°
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
Worked Example
P, Q and R are points on a circle.
RQ is a diameter.
Find the value of .
Give a reason for your answer.
Use the fact that angles in a triangle add up to 180º and the circle theorem
The angle in a semicircle is 90°
Write an equation for
Solve for
The angle in a semicircle is 90°
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