Circle Theorems - Angle at a Semi-Circle
Maths: Edexcel A Level Pure Maths, topic 3.2.6
Circle Theorems
Angle in a semicircle is 90°
Proof
- Split the triangle into two triangles which are both isosceles since they both have two sides which are radii.
- Mark one of the angles at the centre x.
Proof continued
- y = 1⁄2(180° - x) since the triangle is isosceles and all angles add up to 180°.
- Similarly z = 1⁄2(180° - (180° - x)) = 1⁄2x
- Therefore the angle at the circumference is z + y = 1⁄2 × 180° = 90° as required.
Alternative proof
- Alternatively, using the previous theorem we see that the angle at the centre is twice the angle at the circumference.
- So 180° is twice the angle at the circumference so the angle is 90°.
Test yourself on Circle Theorems - Angle at a Semi-Circle
The red line is a diameter of the circle. What is angle x?
The red line is a diameter of the circle. What is angle x?
The red line is a diameter of the circle. What is angle x?
The red line is a diameter of the circle. What is angle x?