Circle Theorems

In this lesson, we will learn

• a Circle Theorem called the Thales' Theorem or Triangle inscribed in semicircle or Angle inscribed in semicircle or “90 degrees in Semicircle” Theorem.
• how to use the Theorem to find missing angles.
• how to prove the Theorem.

Thales' theorem

The Thales' theorem states

Every angle subtended at the circumference by the diameter of a circle is a right angle (90˚).

or

The diameter of a circle always subtends a right angle to any point on the circle.

or

The angle inscribed in a semicircle is 90˚.

POQ is the diameter. ∠PAQ = ∠PBQ = ∠PCQ = 90˚.

Using the Theorem

Example:

O is the centre of the circle. Find the value of x

Solution:

∠ABC = 90˚ ( angle in a semicircle = 90˚)

63˚ + 90˚ + x = 180˚ ( sum of angles in a triangle )

x = 27˚

This video shows how we can use the Thales' Theorem to find missing angles.

Proving the Theorem

Proof of the Thales' Theorem

Proof showing that a triangle inscribed in a circle having a diameter as one side is a right triangle. This proof uses the bow theorem.

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