Circle terms revision An arc is a part of the circumference.
A semicircle is a sector whose arc ends are the ends of a diameter.
A chord is a line segment joining two points on the circle.
A diameter is a special chord which passes through the centre.
B arc
If B and C are the ends of an arc of a circle, and A is another point, not on the arc, then we say that the ∠BAC is the angle at Astanding on the arc BC . We also say that it stands on the chord [BC] .
Discuss and discover
Investigate the angle in a semicircle. (i) Work with a classmate to draw the circles with the triangles inside them (inscribed), as shown. Ensure the longest side in the triangles is the diameter of the circle.
(ii) Using a protractor, fi nd the measure of each of the three interior angles of the triangles. Find the sum of these angles.
(iii) Compare the angle opposite the diameter (angle marked A) in each of the triangles.
(iv) Repeat this activity for two more triangles inscribed in a circle, where one side of the triangle is the diameter.
(v) What do you notice? What is the measure of the angle opposite the diameter in each case?
Corollary 3: Each angle in a semicircle is a right angle
Corollary 3 is sometimes called the ‘angle in the semicircle theorem’ or ‘Thales’s theorem’. An angle inscribed in a semicircle is always a right angle.
The end points are either end of a circle’s diameter; the apex point can be anywhere on the circumference.
E
In symbols, if [ AC] is a diameter of a circle, and E is any other point of the circle, then |∠AEC| = 90°.
90° A O 180° C
A corollary is a theorem that follows on from another theorem.
A A C
By the end of this section you should: ● understand how to fi nd the angle in a semicircle