An inscribed angle is an angle whose vertex lies on a circle and whose sides are chords of the circle. Thales' theorem says that any angle inscribed in a semicircle is a right angle. This result is one of the most useful links between circles, triangles, and angle measurement.
It matters because it lets you recognize right triangles from a circle diagram without measuring any angles.
The reason the theorem works comes from the central angle and inscribed angle relationship. An inscribed angle measures half the measure of the arc it intercepts, so an angle that intercepts a diameter intercepts a 180 degree arc. Half of 180 degrees is 90 degrees, so the inscribed angle is a right angle.
This idea is used in geometric proofs, construction problems, coordinate geometry, and real-world design where right angles must be created accurately.
Key Facts
- An inscribed angle has its vertex on the circle and its sides are chords of the circle.
- Inscribed angle theorem: m∠ACB = 1/2 m arc AB.
- If AB is a diameter, then m arc AB = 180°, so m∠ACB = 90°.
- Thales' theorem: If C lies on the circle with diameter AB, then ∠ACB is a right angle.
- Converse of Thales' theorem: If ∠ACB = 90°, then C lies on the circle with diameter AB.
- For a right triangle with hypotenuse AB, the circumcenter is the midpoint of AB and the circumradius is R = AB/2.
Vocabulary
- Inscribed angle
- An angle whose vertex is on a circle and whose sides intersect the circle at two other points.
- Diameter
- A chord that passes through the center of a circle and has length twice the radius.
- Semicircle
- Half of a circle, formed by cutting a circle along a diameter.
- Intercepted arc
- The arc of a circle that lies inside an angle and connects the points where the angle sides meet the circle.
- Circumcircle
- A circle that passes through all vertices of a polygon, especially the three vertices of a triangle.
Common Mistakes to Avoid
- Using the full arc measure as the inscribed angle measure. This is wrong because an inscribed angle is half the measure of its intercepted arc.
- Assuming any angle touching a circle is an inscribed angle. This is wrong because the vertex must lie on the circle and both sides must be chords or secants through the circle.
- Forgetting that Thales' theorem requires AB to be a diameter. If AB is only a chord, then an angle subtending AB is not necessarily 90 degrees.
- Placing point C at the center of the circle instead of on the semicircle. This is wrong because Thales' theorem concerns an angle with its vertex on the circle, not at the center.
Practice Questions
- 1 A circle has diameter AB. Point C lies on the circle. What is m∠ACB?
- 2 In a circle, ∠ACB is an inscribed angle that intercepts arc AB measuring 124°. Find m∠ACB.
- 3 Explain why every triangle formed by connecting the endpoints of a diameter to a third point on the circle must be a right triangle.