Tangents to a Circle (GMAT/GRE/CAT/Bank PO/SSC CGL)  Don't Memorise  Summary and Q&A
TL;DR
Tangents and circles intersect in three ways  not at all, at one point, or at two points  and the length of an arc can be calculated using the central angle and the circumference.
Questions & Answers
Q: How can tangents intersect circles and what are their properties?
Tangents can intersect circles in three ways: not at all, at one point (tangent), or at two points (secant). When a circle and a tangent meet, the radius to the point of intersection is always perpendicular to the tangent.
Q: What is the significance of drawing tangents from an external point to a circle?
When drawing tangents from an external point to a circle, the lengths of the tangents will be equal. Additionally, the radii drawn to the points of tangent intersection are perpendicular to their respective tangents.
Q: How can the length of an arc be calculated?
The length of an arc depends on the central angle it subtends and the circumference of the circle. By using the formula (central angle/360 degrees) x circumference, the length of the arc can be determined.
Q: How can the measure of a central angle be determined?
The measure of a central angle can be determined by considering the properties of a quadrilateral formed by the tangents and radii. By using the sum of the angles in the quadrilateral (360 degrees), the measure of the central angle can be calculated.
Summary & Key Takeaways

Tangents can intersect circles in three ways: not at all, at one point (tangent), or at two points (secant).

The radius to the point where the circle and the tangent meet is always perpendicular to the tangent.

When drawing tangents from an external point to a circle, the lengths of the tangents will be equal.