Circle Theorem Examples
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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 Thales' Theorem to find missing angles.
- how to prove the Thales' 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˚.
The following diagram shows the Thales' Theorem: Angles in a semi-circle are 90°. Scroll down the page for more examples and solutions.
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˚
Thales' Theorem: If the longest side of a triangle inscribed within a circle is the same length as the diameter of a circle, then that triangle is a right triangle, as well as the converse: if a right triangle is inscribed within a circle, the length of its hypotenuse is the diameter of the circle.
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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