Master Circle Geometry: The Power Of Secants In Tangent And Chord Length Calculations

Secant to a circle

line, ray, or line segment that intersects the circle in exactly two points

A secant to a circle is a line that intersects the circle at two distinct points. It can be imagined as a line passed through the circle at any random point and extend it to both directions until it intersects the circle at the other point. The two points of intersection are called the endpoints of the secant.

In geometry, secants to a circle have many applications. The most important is that they can be used to find the length of a tangent from an external point. To calculate the length of the tangent, we can use the following formula:

p² = s(s+2d)

Where p is the length of the tangent, s is the length of the secant, and d is the distance between the center of the circle and the external point.

Moreover, secants also play a vital role in circle geometry while finding the lengths of chords. If we draw a line from the center of the circle perpendicular to the secant. Now, the line from the center to the midpoint of the secant is called the radius of the circle.

According to the power rule of the secant, the product of the lengths of the external part and the whole length of the secant will be equal to the product of the internal part length and the total length of the secant.

That is,

L1 * L2 = L (L+S)

Here, L1 = External part of secant

L2 = Whole-length of secant

S = Internal part of secant

L = Length of tangent.

In summary, secants to a circle intersect the circle at two distinct points and are used to find the length of tangents, as well as the lengths of chords.

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