Objective
To verify that the angle in a semi-circle is always a right angle (90┬░) using the vector method.
Semi-circle
A semi-circle is exactly one-half of a circle formed when a circle is divided by its diameter. The curved boundary represents half of the circumference, while the straight boundary is known as the diameter.
Every point lying on the arc of the semi-circle is at an equal distance from the center of the circle. The diameter divides the circle into two equal halves called semi-circles.
What are the Angles in a Semi-circle?
The angle in a semi-circle is formed when the two ends of the diameter are joined to any point on the curved part of the semi-circle using straight line segments called chords.
Suppose AC is the diameter of a circle and B is any point on the semi-circle. By joining AB and BC, triangle ABC is formed.
According to the theorem of geometry, the angle subtended by the diameter at any point on the circumference is always a right angle.
No matter where point B is located on the semi-circle, the angle formed will always remain 90┬░. The same property holds true for any other point such as P lying on the arc.
Definition of the Dot Product
The dot product, also known as the scalar product or inner product, is a mathematical operation performed on two vectors. The result of the dot product is always a scalar quantity.
For two vectors A and B, the dot product is defined as:
Where:
- |A| = Magnitude of vector A
- |B| = Magnitude of vector B
- ╬╕ = Angle between vectors A and B
- cos(╬╕) = Cosine of the angle between the vectors
The dot product helps determine whether two vectors are perpendicular or not.
Perpendicular Vectors and the Dot Product
If two vectors are perpendicular to each other, the angle between them is exactly 90┬░.
Using the dot product formula:
Since,
cos(90┬░) = 0
Therefore,
A ┬╖ B = 0
Thus, whenever the dot product of two vectors is zero, the vectors are perpendicular to each other.
Vector Representation in a Semi-circle
To verify the theorem using vectors, consider a circle with centre O and diameter AC. Let B be any point on the semi-circle.
The vectors BA and BC are considered for verification. Using vector algebra, the dot product of these vectors is calculated.
Since the dot product is zero, vectors BA and BC are perpendicular to each other.
Therefore,
This proves that the angle subtended by the diameter at any point on the semi-circle is always a right angle.
Conclusion
The vector method verifies that the dot product of the vectors formed by the endpoints of the diameter and a point on the semi-circle is zero. Hence, the vectors are perpendicular, proving that the angle in a semi-circle is always a right angle (90┬░).