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.

Here is a video on this topic:

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°.

See also



Join the GraphicMaths Newletter

Sign up using this form to receive an email when new content is added:

Popular tags

adder adjacency matrix alu and gate angle area argand diagram binary maths cartesian equation chain rule chord circle cofactor combinations complex polygon complex power complex root cosh cosine cosine rule cpu cube decagon demorgans law derivative determinant diagonal directrix dodecagon ellipse equilateral triangle eulers formula exponent exponential exterior angle first principles flip-flop focus gabriels horn gradient graph hendecagon heptagon hexagon horizontal hyperbola hyperbolic function infinity integration by substitution interior angle inverse hyperbolic function inverse matrix irregular polygon isosceles trapezium isosceles triangle kite koch curve l system locus maclaurin series major axis matrix matrix algebra minor axis nand gate newton raphson method nonagon nor gate normal not gate octagon or gate parabola parallelogram parametric equation pentagon perimeter permutations polar coordinates polynomial power product rule pythagoras proof quadrilateral radians radius rectangle regular polygon rhombus root set set-reset flip-flop sine sine rule sinh sloping lines solving equations solving triangles square standard curves star polygon straight line graphs surface of revolution symmetry tangent tanh transformations trapezium triangle turtle graphics vertical volume of revolution xnor gate xor gate