Angles in a semicircle

When a triangle is formed inside a semicircle, two lines from either side of the diameter meet at a point on the circumference at a right angle.

Diagram of a right-angled scalene triangle within a circle

The angle in a semicircle is a right angle of 90^\circ.

Question

In the diagram PR is a diameter and \angle PRQ = 25^\circ.

What is the size of \angle QPR?

Diagram of a right-angled scalene triangle within a circle, the R angle is 25°

\angle PQR = 90^\circ since it is the angle in a semicircle.

The three angles in the triangle add up to 180^\circ, therefore:

\angle QPR = 180^\circ  - 90^\circ  - 25^\circ

\angle QPR = 65^\circ

Question

In the diagram KL is a diameter of the circle and is 8 cm long.

LM = 3 cm.

Calculate the size of KM.

Diagram of a right-angled scalene triangle within a circle, dimensions 8cm x 3cm x unknown

KL is a diameter so we have an angle in a semicircle therefore \angle KML = 90^\circ.

We have a right-angled triangle and so can use Pythagoras.

KM is not the hypotenuse so:

K{M^2} = {8^2} - {3^2} = 55

KM = \sqrt {55}  = 7.461...

KM = 7.4cm\,(to\,1\,d.p.)