Extended Mathematics

Circle Theorems

# Angles in a Semicircle (90 Degrees)

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Angles in a Semicircle (90 Degrees)

Angles in a Semicircle

Basic Principles

- The
**semicircle**is a special case among the circle theorems. It states that the angle subtended at the circumference by a semicircle is always a right angle, or**90 degrees**. - This theorem is also known as the
**angle in a semi circle is 90 degrees**theorem. - The angle in the semicircle theorem is applicable irrespective of where on the circumference the angle is placed.

Understanding the Theorem

- If you have a
**circle**with centre ‘O’ and a**diameter**from any point on its circumference, a semicircle is formed. If another point is chosen on the circumference and lines are drawn from it to the diameter's ends, a right-angled triangle is formed within the semicircle. - The
**angle**formed at the chosen point is always a right angle. - The theorem applies to all diameters and respective subtended angles on the circle, so it is not specific to any single diameter.

Using The Theorem in Problem Solving

- If a problem requires the calculation of an angle and you're given a semicircle (or you can draw a diameter to create one), it might be useful to consider the angle in a semicircle theorem.
- Remember, in a semicircle, the angle subtended at the circumference is always
**90 degrees**. So if your triangle or shape lies inside a semicircle and one of its sides is a diameter, then you have a right-angled triangle. - This could help not only to find the angles involved more conveniently but it could also bring the application of
**Pythagoras’ theorem**or the trigonometric functions into play since the triangle is right-angled.

Tangible Examples

- A real-life example of angles in a semicircle theorem could be seen in a pizza slice. If you slice a pizza across its centre to form a half circle, cut a piece from the crust to the other edge, you create a right-angled pizza slice!
- Architecturally, dome structures may implement semicircular arcs. If a brace or beam is placed from one end to another, any additional crossbeam forming an angle with the diameter would be at 90 degrees.

Understanding these principles and putting them to good use will help uncomplicate many geometrical problems involving circles and semicircles. Practice consistently to gain a good grasp.