New
New
Year 10
Higher

The angle in a semicircle is a right angle

I can derive and use the theorem: the angle in a semicircle is a right angle.

New
New
Year 10
Higher

The angle in a semicircle is a right angle

I can derive and use the theorem: the angle in a semicircle is a right angle.

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Lesson details

Key learning points

  1. A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments
  2. Theorems can be thought of as puzzles to solve, you are showing how to find a result
  3. In order to use this theorem, you may need to draw a diagram or add more information to an existing one

Keywords

  • Subtended - An angle can be subtended by a line segment or curve. The legs of the subtended angle meet the endpoints of the line segment or curve.

  • Arc - An arc of a circle is part of the circle’s circumference.

  • Chord - A chord is any line segment joining two points on the circumference of a circle.

  • Diameter - The diameter of a circle is any line segment that starts and ends on the circumference of the circle and passes through the centre of the circle.

Common misconception

Pupils may over-generalise the circle theorem by assume that all triangles that are inscribed in a circle are right-angled.

The triangle is only right-angled if one of its angles is subtended by the diameter. This also means that if the triangle is right-angled, then its hypotenuse is the diameter of the circle.

The lesson demonstrates how this circle theorem could be explored practically by using a circular geoboard and using dynamic geometry software. Pupils may wish to explore these further for themselves.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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6 Questions

Q1.
A chord has endpoints of A and B. Where on the circle must points A and B lie?
At the centre of the circle
On a tangent to the circle
Correct answer: On the circumference of the circle
Inside the circle
Outside the circle
Q2.
The size of the angle labelled $$x$$° on this 20-point geoboard is °.
An image in a quiz
Correct Answer: 108, 108°, 108 degrees
Q3.
The size of the angle labelled $$y$$° on this 20-point geoboard is °.
An image in a quiz
Correct Answer: 36, 36°, 36 degrees
Q4.
The size of the angle labelled $$z$$° on this 20-point geoboard is °.
An image in a quiz
Correct Answer: 54, 54°, 54 degrees
Q5.
What is the size of the angle subtended by the chord AC (in degrees)?
An image in a quiz
Correct Answer: 36, 36°, 36 degrees
Q6.
The size of the angle labelled $$m$$° is °.
An image in a quiz
Correct Answer: 68, 68°, 68 degrees

6 Questions

Q1.
Find the size of angle $$a$$° on this 8-point geoboard (in degrees).
An image in a quiz
20°
Correct answer: 45°
360°
Q2.
Find the size of angle $$b$$° on this 8-point geoboard (in degrees).
An image in a quiz
$$180° - 45°$$
Correct answer: $$\big({{180° - 45°}\over2}\big)$$
$$45°\over2$$
Correct answer: 67.5°
135°
Q3.
The size of the angle labelled $$c$$° on this 8-point geoboard is °.
An image in a quiz
Correct Answer: 135, 135°, 135 degrees
Q4.
The size of the angle $$(b+d)$$° on this 8-point geoboard is °.
An image in a quiz
Correct Answer: 90, 90°, 90 degrees
Q5.
The size of the angle labelled $$x$$° is °.
An image in a quiz
Correct Answer: 76, 76°, 76 degrees
Q6.
Chord AB has length of 4 cm. The length of chord BC is cm (2 d.p.).
An image in a quiz
Correct Answer: 23.66, 23.66 cm