Angles in a semicircle
Watch this video to understand 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.
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 \(\angle QPR\)?
\(\angle PQR = 90^\circ\) 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.
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:
\(K{M^2} = {8^2} - {3^2} = 55\)
\(KM = \sqrt {55} = 7.461...\)
\(KM = 7.4cm\,(to\,1\,d.p.)\)
