Circle GEOMETRY

Radius (or Radii for plural) The segment joining the center of a circle to a point on the circle. Example: OA

Diameter A chord that passes through the center of a circle. Example: AB

Chord A segment joining two points on a circle Example: AB

Chord A segment joining two points on a circle Example: AB

Secant A line that intersects the circle at exactly two points. Example: AB

Tangent A line that intersects a circle at exactly one point. Example: AB

Arc A figure consisting of two points on a circle and all the points on the circle needed to connect them by a single path. Example: arc AB

Diagram of Arcs Measure of a minor arc – the measure of its central angle Measure of a major arc – the difference between 360° and the measure of its associated minor arc.

Central Angle An angle whose vertex is at the center of a circle. Example: Angle ABC

Central angles will always equal the inscribed arc. – Example: angle ABC = arc AC

Example 2 Find the measures of the red arcs. Are the arcs congruent?

Example 1 Find the measure of each arc. 70° 360° - 70° = 290° 180°

Inscribed Angle An angle whose vertex is on a circle and whose sides are determined by two chords. Example: Angle ABC

Intercepted Arc An arc that lies in the interior of an inscribed angle. Example: arc AC

An inscribed arc will always equal twice the inscribed angle. – Ex. Arc AC= 2 times Angle ABC

Example 1 Find the measure of the blue arc or angle. a. b.

Example 1 Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius. tangent diameter chord radius

Tangent Theorem The tangent is a line or line segment that touches the perimeter of a circle at one point only and is perpendicular to the radius that contains the point.

Example 3 Use the converse of the Pythagorean Theorem to see if the triangle is right ? ?  2025

Definitions Inscribed polygon – a polygon whose vertices all lie on a circle. Circumscribed circle – A circle with an inscribed polygon. The polygon is an inscribed polygon and the circle is a circumscribed circle.

Inscribed Quadrilateral If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary.

1. Problem: Find the measure of arc GDE. Solution: By the theorem stated above, angle D and angle F are supplementary. Therefore, angle F equals 95 o. The first theorem discussed in this section tells us the measure of an arc is twice that of its inscribed angle. With that theorem, arc GDE is 190 o.

Inscribed Right Triangle Theorem If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.

Example 3 Find the value of each variable. a. b.

Chord Product Theorem If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Example 1 Find the value of x.

Try This! Find the value of x.