Let P Q and R S be tangents at the extremities of the...

Let $P Q$ and $R S$ be tangents at the extremities of the diameter PR of a circle of radius $r .$ If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals (a) $\sqrt{P Q \cdot R S}$ (b) $(P Q+R S) / 2$ (c) $2 P Q \cdot R S /(P Q+R S)$ (d) $\sqrt{\left(P Q^{2}+R S^{2}\right)} / 2$

Let $P Q$ and $R S$ be tangents at the extremities of the diameter PR of a circle of radius $r .$ If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals
(a) $\sqrt{P Q \cdot R S}$
(b) $(P Q+R S) / 2$
(c) $2 P Q \cdot R S /(P Q+R S)$
(d) $\sqrt{\left(P Q^{2}+R S^{2}\right)} / 2$

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Key Concepts

Circle Geometry

This concept involves understanding the fundamental properties of circles including the relationships among diameters, chords, arcs, tangents, and the circle’s center. A firm grasp of circle geometry is essential for analyzing how various elements in a circle interact, such as how tangents and chords determine angles and segment lengths.

Tangent Properties

Tangents are lines that touch a circle at exactly one point. A key property is that a tangent at any point of a circle is perpendicular to the radius drawn to the point of tangency. This principle is crucial for solving problems where the behavior of lines touching the circle (and their intersections) leads to relationships between segment lengths.

Power of a Point

The power of a point theorem provides a method to relate distances from a point outside or on the circle to the circle itself. By equating products of segment lengths intercepted by chords, tangents, or secants, one can derive key relationships between elements in circle geometry, often simplifying complex configurations.

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