Angle in a semicircle is 90 degrees

By Martin McBride, 2023-01-01
Tags: circle semicircle angle right angle
Categories: gcse geometry


We can draw a triangle ABC where AB is a diameter of a circle, and C lies on the circumference of the circle:

Angle in a semicircle is a right angle

The angle at the circumference C is a right angle. We say the angle in a semicircle is a right angle.

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Proof

We can prove this as follows.

We start by drawing an extra line from the centre O to the point C.

Angle in a semicircle is a right angle proof

Notice that the lines OA, OB and OC are all radii of the circle, and therefore all of equal length.

Looking at the triangle AOC, this is an isosceles triangle (from the rule 2 radii form an isosceles triangle). So the two angles at the circumference are equal (we will call them a):

Angle in a semicircle is a right angle proof

By the same logic, the two angles at the circumference in triangle BOC are equal, and we will call them b:

Angle in a semicircle is a right angle proof

Looking at the original triangle ABC:

Angle in a semicircle is a right angle proof

The angle at A is a, the angle at B is b, and the angle at C is a+b. Since the three angles of a triangle add up to 180° we have:

Angle in a semicircle is a right angle formula

Gathering the terms a and b:

Angle in a semicircle is a right angle formula

Dividing both sides by 2 gives:

Angle in a semicircle is a right angle formula

Since the angle at C is a + b, this proves that the angle at C is 90°.

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