Equal Tangents Theorem
The Equal Tangents Theorem states that the lengths of two tangent segments drawn from a single external point to a circle are always equal. A tangent segment is the portion of the tangent line between the external point and the point of contact on the circle.
Key Concepts
If a point lies outside a circle, exactly two tangents can be drawn to the circle from that point.
The theorem is proven using the RHS (Right Angle-Hypotenuse-Side) congruence criterion by drawing radii to the points of contact, which form 90-degree angles with the tangents.
The line joining the external point to the center of the circle bisects the angle between the two tangents, making the figure symmetrical.
Formula & Equation
PA = PB
P is the external point, while A and B are the respective points of contact on the circle.
Common Misconceptions
Myth: The entire tangent line has a specific length that is equal to another tangent line.
Fact: A tangent is technically a line that extends infinitely. The theorem specifically refers to the length of the tangent segment, which is the measurable distance from the external point to the point of contact.
Real World Applications
Designing a V-shaped wooden stand to securely hold a spherical globe or a circular clock.
Calculating the line of sight distances from a satellite in space to the Earth's horizon.
Frequently Asked Questions
How do you prove the lengths of tangents drawn from an external point to a circle are equal?
By joining the external point to the center and drawing radii to the points of contact, two right-angled triangles are formed. These triangles are congruent by the RHS criterion, proving the tangent segments are equal.
How many tangents can be drawn from an external point to a circle?
Exactly two tangents can be drawn from any single external point to a given circle.