Final answer:
The theorem used when a right angle is created by a semicircle is the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Step-by-step explanation:
The theorem that is used when a right angle is created by a semicircle is the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
This theorem can be written as: a² + b² = c², where a and b are the lengths of the legs of the triangle, and c is the length of the hypotenuse.
For example, if we have a semicircle with a radius of 5 units, the length of the diameter is 10 units. If we draw a right-angled triangle inside the semicircle, with one leg along the diameter and the other leg perpendicular to it, the length of the two legs would be 5 units each.
Using the Pythagorean theorem, we can find the length of the hypotenuse (the curved part of the semicircle) as follows:
a² + b² = c²
5² + 5² = c²
25 + 25 = c²
50 = c²
c = √50 ≈ 7.07 units