Theorem 19-2

In the same or congruent circles, congruent chords are equidistant from the circle’s center.

Theorem 19-2 is illustrated in Figure 19.13 where segments AB and CD are of equal length. Segments OE and OF are radii drawn perpendicular to the chords and are therefore of equal length.

F IGURE 19.13 Congruent chords are equally distant from center of circle

E XAMPLE 19.2: Theorem 19-2

The shortest distance from the center of a hex bolt to the hex bolt’s side is inch. What is the distance across the flats?

F IGURE 19.14

Solution:

Since the six sides of a hex bolt are equal in length, a circle could be drawn circumscribing the hex bolt, making each of the sides congruent chords. By Theorem 19-2, all sides of the hex bolt are equidistant from the hex bolt’s center. Since the distance from the midpoint of one of the sides is given as inch from the center, the distance across the flats must be twice the distance given, or inch.

The converse of a theorem is formed by reversing the order of the statements in the original theorem. For example, the converse of Theorem 19-2 would state that in the same or congruent circles, chords that are equally distant from the circle’s center are congruent.

Theorem 19-3

A tangent to a circle is perpendicular to the radius drawn to the point of tangency. Conversely, a line in the plane of a circle that is perpendicular to a radius and that intersects the radius on the circle is tangent to the circle

As a motivation for Theorem 19-3, recall that a tangent to a circle is a line located outside the circle but that touches the circumference at a single point. In Figure 19.15, line AC is perpendicular to radius OB at point B , which is on the circle. Therefore, line AC is tangent to the circle.

F IGURE 19.15 Theorem 19-3

A corollary to a theorem is a statement that is related closely to and follows immediately from the theorem. Related to Theorem 19-3 is the following corollary.

Corollary to Theorem 19-3

If two or more tangents are drawn to a circle, the lines drawn perpendicular to each tangent at the point of tangency will intersect each other at the center of the circle, unless the lines are collinear.

In Figure 19.16, diameters EC and E ′ C ′ are perpendicular to tangent lines AB and A ′ B ′ , respectively, and they pass through the respective points of tangency, E and E ′ . The center of the circle is located at O , the point of intersection of diameters EC and E ′ C ′ .

F IGURE 19.16 Tangents to a circle