Big Ideas

Sec 10.1: Radius-Chord relationships

Theorem 74-76 ("Two out of Three" theorem): If any two of the following facts about a line intersecting a chord are true, then the third is true also:

• Perpendicular to the chord
• Bisects the chord
• Goes through the center (contains the radius)

Sec 10.1: Basic circle formulae

The circumference of a circle is 2πr or πd; the area is πr²

Sec 10.2: Congruent chords

Th. 77 and 78: Chords are equidistant from the center iff they are congruent. ("iff," short for "if and only if" and sometimes written "<=>", means that if either is true then the other is true.)

Sec 10.3 and 10.9: Arcs and arc length

Arc measure is the measure of the arc's central angle.

Arc length is the actual length of the arc, computed by dividing the arc measure by 360° and then multiplying by the circumference. For example, an arc whose measure is 90° and whose radius is 3 has an arc length of (90/360)·3·2π = (1/4)·6π = (3/2)π.

Congruent arcs have the same measure and the same radius.

Theorem 79-84: In the same or in congruent circles, congruent chords iff congruent arcs iff congruent central angles.

Sec. 10.4: Secants and tangents

Defs: A Tangent line intersects a circle at exactly one point; a secant line intersects at exactly two points. A tangent segment is a segment that lies on a tangent line and has one endpoint at the point of tangency. A secant segment is a segment that lies on a secant line, includes both points of intersection, and has one of those points as its endpoint.

Postulates: A line is a tangent iff it is perpendicular to the radius at the point where the radius intersects the circle.

Two-tangent theorem, a.k.a. "Party Hat" theorem, a.k.a. "Dunce Cap" theorem: Two tangent segments (tangent to the same circle) sharing a common endpoint not on the circle are congruent.

A common tangent is a line tangent to two circles.

Common tangent procedure:

1. Draw the segment joining the centers.
2. Draw radii to the points of contact. (Note that, since radii are perpendicular to tangents, the radii are parallel.)
3. Draw a line through the center of the smaller circle perpendicular to the radius (parallel to the common tangent).
4. This line, the common tangent, the smaller radius, and part of the (possibly extended) larger radius, make a rectangle, so use that fact.
5. That same line, the segment joining the centers, and part of the larger radius make a right triangle, so use that fact (e.g. Pythagorean theorem). Pay attention to which angle is the right angle! (See Sample problem 3, p. 462.)

Walk-around problems: When you have a polygon circumscribed about a circle, you can use the Party Hat theorem repeatedly to "walk around" the polygon to find missing lengths. For example if you know that side AB, with point of tangency P, has length 10, you can call length AP 10−x and length BP will be x. Then, using the Party Hat Theorem, if the adjacent side is BC with point of tangency Q, you know that length BQ is also x, and so on around the polygon. (See Sample problem 4, p.463.)

Sec. 10.5: Angles related to a circle

"Sherlock Holmes page" (p. 472). All the cases break down to where the vertex of the angle is:

• Vertex inside circle: angle is 1/2 the sum of the intercepted arcs. (Chord-chord angle) (Special case: in central angle, the two intercepted arcs are congruent so 1/2 the sum of both equals the measure of one arc.
• Vertex on circle: angle is 1/2 the intercepted arc. (Inscribed angle, tangent-chord angle)
• Vertex outside circle: angle is 1/2 the difference of the intercepted arcs. (Secant-secant angle, tangent-tangent angle, secant-tangent angle)

Sec. 10.6: More angle-arc theorems

Most of these are all easily derived from Sec. 10.5 theorems.

One exception is Th. 92: The sum of the measures of a tangent-tangent angle and its minor arc is 180°.

Sec. 10.7: Inscribed & circumscribed polygons

A polygon is inscribed in a circle (or a circle is circumscribed about a polygon) if all vertices of the polygon lie on the circle.

A circle is inscribed in a polygon (or a polygon is circumscribed about a circle) if all sides of the polygon are tangent to the circle.

Consider a polygon. The center of the circumbscribed circle (if any) is the circumcenter. The center of the inscribed circle (if any) is the incenter.

Th. 93: Opposite angles of an inscribed quadrilateral are supplementary. Th. 94 is a special case: A parallelogram inscribed in a circle is a rectangle.

Sec. 10.8: Power theorems: Skip this section!

Sec. 10.9: Circumference and arc length

(See Sec. 10.3.)