Practice Thales' theorem
An angle whose vertex lies on a circle and whose opposite side is a diameter is a right angle.
Thales’ theorem helps you construct right-angled triangles.
Exercises with solutions
Thales’ theorem states: if point C lies on the semicircle above segment AB, then the angle at C is always a right angle. Conversely, the vertex of a right angle always lies on the semicircle above the hypotenuse.
Exercise 1
A semicircle is drawn above segment AB of length 8 cm. Point C lies on the semicircle. How large is the angle at C?
Show solution and working
90°
By Thales’ theorem the angle at every point on the semicircle is a right angle.
Exercise 2
Where is the centre of the Thales circle?
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at the midpoint of segment AB
The diameter of the circle is segment AB. The centre therefore lies exactly at the midpoint of AB.
Exercise 3
A triangle has a right angle at C and the hypotenuse AB is 10 cm long. How far is C from the midpoint of AB?
Show solution and working
5 cm
C lies on the Thales circle above AB. Its radius is half of 10 cm, so 5 cm.
Exercise 4
Why can a triangle not have two right angles?
Show solution and working
The angle sum would exceed 180°
Two right angles already make 180°. Nothing would be left for the third angle, but a triangle has three angles all greater than 0°.
Common mistake
Applying Thales’ theorem to arbitrary triangles. It holds only when C lies on the semicircle above AB. And looking for the right angle at the wrong vertex, since it is always at the point on the semicircle.
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