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 \(90^\circ\).
Example
In this diagram, PR is a diameter and \(\angle PRQ=25^\circ\).
What is the size of \(\angle QPR\) ?
Answer
\(\angle PQR=90^\circ\) since it is the angle in a semicircle.
The three angles in the triangle add up to \(180^\circ\), therefore:
\(\angle QPR=180^\circ-90^\circ-25^\circ=65^\circ\)
Question
In this diagram KL is a diameter of the circle and is 8cm long.
\(LM=3cm.\)
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:
\(K{M^2}={8^2}-{3^2}=55\)
\(KM=\sqrt{55}=7.4161...\)
\(=7.4cm\,(to\,1\,d.p.)\)
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