Angle-In-A-Semicircle Aqa Higher

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Exam code:8300

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°

Right angle in a semicicrcle, IGCSE & GCSE Maths revision notes

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

A circle with points P, Q and R on the circumference. The points are joined to form a triangle inside the circle. The angle PQR is 40º and the angle PRQ is yº.

P, Q and R are points on a circle.
RQ is a diameter.

Find the value of $y$ .

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 $y$

$y plus 90 plus 40 equals 180$

Solve for $y$

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Maths Gcse Aqa Higher

Angle-In-A-Semicircle Aqa Higher